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# Curvilinear coordinates

(Redirected from Curvilinear)
See orthogonal coordinates for the important special case, skew coordinates for a less common special case, and tensors in curvilinear coordinates for more generalized mathematical details.
Curvilinear, affine, and Cartesian coordinates in two-dimensional space

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.

Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R3) are Cartesian, cylindrical and spherical polar coordinates. A Cartesian coordinate surface in this space is a plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical polar coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems.

Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system.

Depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a physical problem with spherical symmetry defined in R3 (for example, motion of particles under the influence of central forces) is usually easier to solve in spherical polar coordinates than in Cartesian coordinates. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. One would for instance describe the motion of a particle in a rectangular box in Cartesian coordinates, whereas one would prefer spherical coordinates for a particle in a sphere. Spherical coordinates are one of the most used curvilinear coordinate systems in such fields as Earth sciences, cartography, and physics (in particular quantum mechanics, relativity), and engineering.

## Orthogonal curvilinear coordinates in 3d

### Coordinates, basis, and vectors

Fig. 1 - Coordinate surfaces, coordinate lines, and coordinate axes of general curvilinear coordinates.
Fig. 2 - Coordinate surfaces, coordinate lines, and coordinate axes of spherical coordinates. Surfaces: r - spheres, θ - cones, φ - half-planes; Lines: r - straight beams, θ - vertical semicircles, φ - horizontal circles; Axes: r - straight beams, θ - tangents to vertical semicircles, φ - tangents to horizontal circles

For now, consider 3d space. A point P in 3d space can be defined using Cartesian coordinates (x, y, z) [equivalently written (x1, x2, x3)], or in another system (q1, q2, q3), as shown in Fig. 1. The latter is a curvilinear coordinate system, and (q1, q2, q3) are the curvilinear coordinates of the point P.

The surfaces q1 = constant, q2 = constant, q3 = constant are called the coordinate surfaces; and the space curves formed by their intersection in pairs are called the coordinate curves. The coordinate axes are determined by the tangents to the coordinate curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates.

A basis whose vectors change their direction and/or magnitude from point to point is called local basis. All bases associated with curvilinear coordinates are necessarily local. Basis vectors that are the same at all points are global bases, and can be associated only with linear or affine coordinate systems.

Note: usually all basis vectors are denoted by e, for this article e is for the standard basis (Cartesian) and b is for the curvilinear basis.

The relation between the coordinates is given by the invertible transformations:

$x = x(q_1, q_2, q_3),\, y = y(q_1, q_2, q_3),\, z = z(q_1, q_2, q_3)$
$q_1 = q_1(x,y,z),\, q_2 = q_2(x,y,z),\, q_3 = q_3(x,y,z)$

Any point can be written as a position vector r in Cartesian coordinates:

$\mathbf{r} = x \mathbf{e}_x + y\mathbf{e}_y + z\mathbf{e}_z$

where x, y, z are the coordinates of the position vector with respect to the standard basis vectors ex, ey, ez.

However, in a general curvilinear system, there may well not be any natural global basis vectors. Instead, we note that in the Cartesian system, we have the property that

$\mathbf{e}_x = \dfrac{\partial\mathbf{r}}{\partial x}; \; \mathbf{e}_y = \dfrac{\partial\mathbf{r}}{\partial y}; \; \mathbf{e}_z = \dfrac{\partial\mathbf{r}}{\partial z}.$

We can apply the same idea to the curvilinear system to determine a system of basis vectors at P. We define

$\mathbf{h}_1 = \dfrac{\partial\mathbf{r}}{\partial q_1}; \; \mathbf{h}_2 = \dfrac{\partial\mathbf{r}}{\partial q_2}; \; \mathbf{h}_3 = \dfrac{\partial\mathbf{r}}{\partial q_3}.$

These may not have unit length, and may also not be orthogonal. In the case that they are orthogonal at all points where the derivatives are well-defined, we define the Lamé coefficients (after Gabriel Lamé) by

$h_1 = |\mathbf{h}_1|; \; h_2 = |\mathbf{h}_2|; \; h_3 = |\mathbf{h}_3|$

and the curvilinear orthonormal basis vectors by

$\mathbf{b}_1 = \dfrac{\mathbf{h}_1}{h_1}; \; \mathbf{b}_2 = \dfrac{\mathbf{h}_2}{h_2}; \; \mathbf{b}_3 = \dfrac{\mathbf{h}_3}{h_3}.$

It is important to note that these basis vectors may well depend upon the position of P; it is therefore necessary that they are not assumed to be constant over a region. (They technically form a basis for the tangent bundle of $\mathbb{R}^3$ at P, and so are local to P.)

In general, curvilinear coordinates allow the generality of basis vectors not all mutually perpendicular to each other, and not required to be of unit length: they can be of arbitrary magnitude and direction. The use of an orthogonal basis makes vector manipulations simpler than for non-orthogonal. However, some areas of physics and engineering, particularly fluid mechanics and continuum mechanics, require non-orthogonal bases to describe deformations and fluid transport to account for complicated directional dependences of physical quantities. A discussion of the general case appears later on this page.

### Vector calculus

#### Differential elements

Since the total differential change in r is

$d\mathbf{r}=\dfrac{\partial\mathbf{r}}{\partial q_1}dq_1 + \dfrac{\partial\mathbf{r}}{\partial q_2}dq_2 + \dfrac{\partial\mathbf{r}}{\partial q_3}dq_3 = h_1 dq_1 \mathbf{b}_1 + h_2 dq_2 \mathbf{b}_2 + h_3 dq_3 \mathbf{b}_3$

so scale factors are

$h_i = \left|\frac{\partial\mathbf{r}}{\partial q_i}\right|$

They can also be written for each component of r:

$h^i{}_k = \cfrac{\partial x^i}{\partial q^k}$.

However, this designation is very rarely used, largely replaced with the components of the metric tensor gik (see below).

### Covariant and contravariant bases

A vector v (red) represented by • a vector basis (yellow, left: e1, e2, e3), tangent vectors to coordinate curves (black) and • a covector basis or cobasis (blue, right: e1, e2, e3), normal vectors to coordinate surfaces (grey) in general (not necessarily orthogonal) curvilinear coordinates (q1, q2, q3). Note the basis and cobasis do not coincide unless the coordinate system is orthogonal.[1]

The basis vectors, gradients, and scale factors are all interrelated within a coordinate system by two methods:

1. the basis vectors are unit tangent vectors along the coordinate curves:
$\mathbf{b}_i=\dfrac{1}{\left|\dfrac{\partial\mathbf{r}}{\partial q_i}\right|}\dfrac{\partial\mathbf{r}}{\partial q_i}=\frac{1}{h_i}\dfrac{\partial\mathbf{r}}{\partial q_i}$
which transform like covariant vectors (denoted by lowered indices), or
2. the basis vectors are unit normal vectors to the coordinate surfaces:
$\mathbf{b}^i=\dfrac{\nabla q_i}{\left|\nabla q_i\right|} = h_i\nabla q_i$
which transform like contravariant vectors (denoted by raised indices), ∇ is the del operator.

So depending on the method by which they are built, for a general curvilinear coordinate system there are two sets of basis vectors for every point: {b1, b2, b3} is the covariant basis, and {b1, b2, b3} is the contravariant basis.

A vector v can be given in terms either basis, i.e.,

$\mathbf{v} = v^1\mathbf{b}_1 + v^2\mathbf{b}_2 + v^3\mathbf{b}_3 = v_1\mathbf{b}^1 + v_2\mathbf{b}^2 + v_3\mathbf{b}^3$

The basis vectors relate to the components by[2](pp30–32)

$\mathbf{v}\cdot\mathbf{b}^i = v^k\mathbf{b}_k\cdot\mathbf{b}^i = v^k\delta^i_k = v^i$
$\mathbf{v}\cdot\mathbf{b}_i = v_k\mathbf{b}^k\cdot\mathbf{b}_i = v_k\delta_i^k = v_i$

and

$\mathbf{v}\cdot\mathbf{b}_i = v^k\mathbf{b}_k\cdot\mathbf{b}_i = g_{ki}v^k$
$\mathbf{v}\cdot\mathbf{b}^i = v_k\mathbf{b}^k\cdot\mathbf{b}^i = g^{ki}v_k$

where g is the metric tensor (see below).

A vector is covariant or contravariant if, respectively, its components are covariant (lowered indices, written vk) or contravariant (raised indices, written vk). From the above vector sums, it can be seen that contravariant vectors are represented with covariant basis vectors, and covariant vectors are represented with contravariant basis vectors.

A key convention in the representation of vectors and tensors in terms of indexed components and basis vectors is invariance in the sense that vector components which transform in a covariant manner (or contravariant manner) are paired with basis vectors that transform in a contravariant manner (or covariant manner).

### Covariant basis

#### Constructing a covariant basis in one dimension

Fig. 3 – Transformation of local covariant basis in the case of general curvilinear coordinates

Consider the one-dimensional curve shown in Fig. 3. At point P, taken as an origin, x is one of the Cartesian coordinates, and q1 is one of the curvilinear coordinates (Fig. 3). The local (non-unit) basis vector is b1 (notated h1 above, with b reserved for unit vectors) and it is built on the q1 axis which is a tangent to that coordinate line at the point P. The axis q1 and thus the vector b1 form an angle α with the Cartesian x axis and the Cartesian basis vector e1.

It can be seen from triangle PAB that

$\cos \alpha = \cfrac{|\mathbf{e}_1|}{|\mathbf{b}_1|} \quad \Rightarrow \quad |\mathbf{e}_1| = |\mathbf{b}_1|\cos \alpha$

where |e1|, |b1| are the magnitudes of the two basis vectors, i.e., the scalar intercepts PB and PA. Note that PA is also the projection of b1 on the x axis.

However, this method for basis vector transformations using directional cosines is inapplicable to curvilinear coordinates for the following reasons:

1. By increasing the distance from P, the angle between the curved line q1 and Cartesian axis x increasingly deviates from α.
2. At the distance PB the true angle is that which the tangent at point C forms with the x axis and the latter angle is clearly different from α.

The angles that the q1 line and that axis form with the x axis become closer in value the closer one moves towards point P and become exactly equal at P.

Let point E be located very close to P, so close that the distance PE is infinitesimally small. Then PE measured on the q1 axis almost coincides with PE measured on the q1 line. At the same time, the ratio PD/PE (PD being the projection of PE on the x axis) becomes almost exactly equal to cos α.

Let the infinitesimally small intercepts PD and PE be labelled, respectively, as dx and dq1. Then

$\cos \alpha = \cfrac{dx}{dq^1} = \frac{|\mathbf{e}_1|}{|\mathbf{b}_1|}$.

Thus, the directional cosines can be substituted in transformations with the more exact ratios between infinitesimally small coordinate intercepts. It follows that the component (projection) of b1 on the x axis is

$p^1 = \mathbf{b}_1\cdot\cfrac{\mathbf{e}_1}{|\mathbf{e}_1|} = |\mathbf{b}_1|\cfrac{|\mathbf{e}_1|}{|\mathbf{e}_1|}\cos\alpha = |\mathbf{b}_1|\cfrac{dx}{dq^1} \quad \Rightarrow \quad \cfrac{p^1}{|\mathbf{b}_1|} = \cfrac{dx}{dq^1}$.

If qi = qi(x1, x2, x3) and xi = xi(q1, q2, q3) are smooth (continuously differentiable) functions the transformation ratios can be written as $\cfrac{\partial q^i}{\partial x_j}$ and $\cfrac{\partial x_i}{\partial q^j}$. That is, those ratios are partial derivatives of coordinates belonging to one system with respect to coordinates belonging to the other system.

#### Constructing a covariant basis in three dimensions

Doing the same for the coordinates in the other 2 dimensions, b1 can be expressed as:

$\mathbf{b}_1 = p^1\mathbf{e}_1 + p^2\mathbf{e}_2 + p^3\mathbf{e}_3 = \cfrac{\partial x_1}{\partial q^1} \mathbf{e}_1 + \cfrac{\partial x_2}{\partial q^1} \mathbf{e}_2 + \cfrac{\partial x_3}{\partial q^1} \mathbf{e}_3$

Similar equations hold for b2 and b3 so that the standard basis {e1, e2, e3} is transformed to a local (ordered and normalised) basis {b1, b2, b3} by the following system of equations:

\begin{align} \mathbf{b}_1 & = \cfrac{\partial x_1}{\partial q^1} \mathbf{e}_1 + \cfrac{\partial x_2}{\partial q^1} \mathbf{e}_2 + \cfrac{\partial x_3}{\partial q^1} \mathbf{e}_3 \\ \mathbf{b}_2 & = \cfrac{\partial x_1}{\partial q^2} \mathbf{e}_1 + \cfrac{\partial x_2}{\partial q^2} \mathbf{e}_2 + \cfrac{\partial x_3}{\partial q^2} \mathbf{e}_3 \\ \mathbf{b}_3 & = \cfrac{\partial x_1}{\partial q^3} \mathbf{e}_1 + \cfrac{\partial x_2}{\partial q^3} \mathbf{e}_2 + \cfrac{\partial x_3}{\partial q^3} \mathbf{e}_3 \end{align}

By analogous reasoning, one can obtain the inverse transformation from local basis to standard basis:

\begin{align} \mathbf{e}_1 & = \cfrac{\partial q^1}{\partial x_1} \mathbf{b}_1 + \cfrac{\partial q^2}{\partial x_1} \mathbf{b}_2 + \cfrac{\partial q^3}{\partial x_1} \mathbf{b}_3 \\ \mathbf{e}_2 & = \cfrac{\partial q^1}{\partial x_2} \mathbf{b}_1 + \cfrac{\partial q^2}{\partial x_2} \mathbf{b}_2 + \cfrac{\partial q^3}{\partial x_2} \mathbf{b}_3 \\ \mathbf{e}_3 & = \cfrac{\partial q^1}{\partial x_3} \mathbf{b}_1 + \cfrac{\partial q^2}{\partial x_3} \mathbf{b}_2 + \cfrac{\partial q^3}{\partial x_3} \mathbf{b}_3 \end{align}

#### Jacobian of the transformation

The above systems of linear equations can be written in matrix form as

$\cfrac{\partial x_i}{\partial q^k} \mathbf{e}_i = \mathbf{b}_k, \quad \cfrac{\partial q^i}{\partial x_k} \mathbf{b}_i = \mathbf{e}_k$.

This coefficient matrix of the linear system is the Jacobian matrix (and its inverse) of the transformation. These are the equations that can be used to transform a Cartesian basis into a curvilinear basis, and vice versa.

In three dimensions, the expanded forms of these matrices are

$\mathbf{J} = \begin{bmatrix} \cfrac{\partial x_1}{\partial q^1} & \cfrac{\partial x_1}{\partial q^2} & \cfrac{\partial x_1}{\partial q^3} \\ \cfrac{\partial x_2}{\partial q^1} & \cfrac{\partial x_2}{\partial q^2} & \cfrac{\partial x_2}{\partial q^3} \\ \cfrac{\partial x_3}{\partial q^1} & \cfrac{\partial x_3}{\partial q^2} & \cfrac{\partial x_3}{\partial q^3} \\ \end{bmatrix},\quad \mathbf{J}^{-1} = \begin{bmatrix} \cfrac{\partial q^1}{\partial x_1} & \cfrac{\partial q^1}{\partial x_2} & \cfrac{\partial q^1}{\partial x_3} \\ \cfrac{\partial q^2}{\partial x_1} & \cfrac{\partial q^2}{\partial x_2} & \cfrac{\partial q^2}{\partial x_3} \\ \cfrac{\partial q^3}{\partial x_1} & \cfrac{\partial q^3}{\partial x_2} & \cfrac{\partial q^3}{\partial x_3} \\ \end{bmatrix}$

In the inverse transformation (second equation system), the unknowns are the curvilinear basis vectors. For all points there can only exist one and only one set of basis vectors (else vectors are not well defined at those points). This condition is satisfied if and only if the equation system has a single solution, from linear algebra, a linear equation system has a single solution (non-trivial) only if the determinant of its system matrix is non-zero:

$\det(\mathbf{J}^{-1}) \neq 0$

which shows the rationale behind the above requirement concerning the inverse Jacobian determinant.

## Generalization to n dimensions

The formalism extends to any finite dimension as follows.

Consider the real Euclidean n-dimensional space, that is Rn = R × R × ... × R (n times) where R is the set of real numbers and × denotes the Cartesian product, which is a vector space.

The coordinates of this space can be denoted by: x = (x1, x2,...,xn). Since this is a vector (an element of the vector space), it can be written as:

$\mathbf{x} = \sum_{i=1}^n x_i\mathbf{e}^i$

where e1 = (1,0,0...,0), e2 = (0,1,0...,0), e3 = (0,0,1...,0),...,en = (0,0,0...,1) is the standard basis set of vectors for the space Rn, and i = 1, 2,...n is an index labelling components. Each vector has exactly one component in each dimension (or "axis") and they are mutually orthogonal (perpendicular) and normalized (has unit magnitude).

More generally, we can define basis vectors bi so that they depend on q = (q1, q2,...,qn), i.e. they change from point to point: bi = bi(q). In which case to define the same point x in terms of this alternative basis: the coordinates with respect to this basis vi also necessarily depend on x also, that is vi = vi(x). Then a vector v in this space, with respect to these alternative coordinates and basis vectors, can be expanded as a linear combination in this basis (which simply means to multiply each basis vector ei by a number viscalar multiplication):

$\mathbf{v} = \sum_{j=1}^n \bar{v}^j\mathbf{b}_j = \sum_{j=1}^n \bar{v}^j(\mathbf{q})\mathbf{b}_j(\mathbf{q})$

The vector sum that describes v in the new basis is composed of different vectors, although the sum itself remains the same.

### Transformation of coordinates

From a more general and abstract perspective, a curvilinear coordinate system is simply a coordinate patch on the differentiable manifold En (n-dimensional Euclidean space) that is diffeomorphic to the Cartesian coordinate patch on the manifold.[3] Note that two diffeomorphic coordinate patches on a differential manifold need not overlap differentiably. With this simple definition of a curvilinear coordinate system, all the results that follow below are simply applications of standard theorems in differential topology.

The transformation functions are such that there's a one-to-one relationship between points in the "old" and "new" coordinates, that is, those functions are bijections, and fulfil the following requirements within their domains:

1. They are smooth functions: qi = qi(x)
2. The inverse Jacobian determinant
$J^{-1}=\begin{vmatrix} \dfrac{\partial q^1}{\partial x_1} & \dfrac{\partial q^1}{\partial x_2} & \cdots & \dfrac{\partial q^1}{\partial x_n} \\ \dfrac{\partial q^2}{\partial x_1} & \dfrac{\partial q^2}{\partial x_2} & \cdots & \dfrac{\partial q^2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac{\partial q^n}{\partial x_1} & \dfrac{\partial q^n}{\partial x_2} & \cdots & \dfrac{\partial q^n}{\partial x_n} \end{vmatrix} \neq 0$

is not zero; meaning the transformation is invertible: xi(q).

according to the inverse function theorem. The condition that the Jacobian determinant is not zero reflects the fact that three surfaces from different families intersect in one and only one point and thus determine the position of this point in a unique way.[4]

## Vector and tensor algebra in three-dimensional curvilinear coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid-1900s, for example the text by Green and Zerna.[5] Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden,[6] Naghdi,[7] Simmonds,[2] Green and Zerna,[5] Basar and Weichert,[8] and Ciarlet.[9]

### Tensors in curvilinear coordinates

A second-order tensor can be expressed as

$\boldsymbol{S} = S^{ij}\mathbf{b}_i\otimes\mathbf{b}_j = S^i{}_j\mathbf{b}_i\otimes\mathbf{b}^j = S_i{}^j\mathbf{b}^i\otimes\mathbf{b}_j = S_{ij}\mathbf{b}^i\otimes\mathbf{b}^j$

where $\scriptstyle\otimes$ denotes the tensor product. The components Sij are called the contravariant components, Si j the mixed right-covariant components, Si j the mixed left-covariant components, and Sij the covariant components of the second-order tensor. The components of the second-order tensor are related by

$S^{ij} = g^{ik}S_k{}^j = g^{jk}S^i{}_k = g^{ik}g^{j\ell}S_{k\ell}$

### The metric tensor in orthogonal curvilinear coordinates

Main article: Metric tensor

At each point, one can construct a small line element dx, so the square of the length of the line element is the scalar product dx • dx and is called the metric of the space, given by:

$d\mathbf{x}\cdot d\mathbf{x} = \cfrac{\partial x_i}{\partial q^j}\cfrac{\partial x_i}{\partial q^k}dq^jdq^k$

and the symmetric quantity

$g_{ij}(q^i,q^j) = \cfrac{\partial x_k}{\partial q^i}\cfrac{\partial x_k}{\partial q^j} = \mathbf{b}_i\cdot\mathbf{b}_j$

is called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.

Indices can be raised and lowered by the metric:

$v^i = g^{ik}v_k$

#### Relation to Lamé coefficients

Defining the scale factors hi by

$h_ih_j = g_{ij} = \mathbf{b}_i\cdot\mathbf{b}_j \quad \Rightarrow \quad h_i =\sqrt{g_{ii}}= \left|\mathbf{b}_i\right|=\left|\cfrac{\partial\mathbf{x}}{\partial q^i}\right|$

gives a relation between the metric tensor and the Lamé coefficients. Note also that

$g_{ij} = \cfrac{\partial\mathbf{x}}{\partial q^i}\cdot\cfrac{\partial\mathbf{x}}{\partial q^j} = \left( h_{ki}\mathbf{e}_k\right)\cdot\left( h_{mj}\mathbf{e}_m\right) = h_{ki}h_{kj}$

where hij are the Lamé coefficients. For an orthogonal basis we also have:

$g = g_{11}g_{22}g_{33} = h_1^2h_2^2h_3^2 \quad \Rightarrow \quad \sqrt{g} = h_1h_2h_3 = J$

#### Example: Polar coordinates

If we consider polar coordinates for R2, note that

$(x, y)=(r \cos \theta, r \sin \theta) \,\!$

(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (r,θ) → (r cos θ, r sin θ) is r.

The orthogonal basis vectors are br = (cos θ, sin θ), bθ = (−r sin θ, r cos θ). The normalized basis vectors are er = (cos θ, sin θ), eθ = (−sin θ, cos θ) and the scale factors are hr = 1 and hθ= r. The fundamental tensor is g11 =1, g22 =r2, g12 = g21 =0.

### The alternating tensor

In an orthonormal right-handed basis, the third-order alternating tensor is defined as

$\boldsymbol{\mathcal{E}} = \epsilon_{ijk}\mathbf{e}^i\otimes\mathbf{e}^j\otimes\mathbf{e}^k$

In a general curvilinear basis the same tensor may be expressed as

$\boldsymbol{\mathcal{E}} = \mathcal{E}_{ijk}\mathbf{b}^i\otimes\mathbf{b}^j\otimes\mathbf{b}^k = \mathcal{E}^{ijk}\mathbf{b}_i\otimes\mathbf{b}_j\otimes\mathbf{b}_k$

It can also be shown that

$\mathcal{E}^{ijk} = \cfrac{1}{J}\epsilon_{ijk} = \cfrac{1}{+\sqrt{g}}\epsilon_{ijk}$

### Christoffel symbols

Christoffel symbols of the first kind
$\mathbf{b}_{i,j} = \frac{\partial \mathbf{b}_i}{\partial q^j} = \Gamma_{ijk}\mathbf{b}^k \quad \Rightarrow \quad \mathbf{b}_{i,j} \cdot \mathbf{b}_k = \Gamma_{ijk}$

where the comma denotes a partial derivative (see Ricci calculus). To express Γijk in terms of gij we note that

\begin{align} g_{ij,k} & = (\mathbf{b}_i\cdot\mathbf{b}_j)_{,k} = \mathbf{b}_{i,k}\cdot\mathbf{b}_j + \mathbf{b}_i\cdot\mathbf{b}_{j,k} = \Gamma_{ikj} + \Gamma_{jki}\\ g_{ik,j} & = (\mathbf{b}_i\cdot\mathbf{b}_k)_{,j} = \mathbf{b}_{i,j}\cdot\mathbf{b}_k + \mathbf{b}_i\cdot\mathbf{b}_{k,j} = \Gamma_{ijk} + \Gamma_{kji}\\ g_{jk,i} & = (\mathbf{b}_j\cdot\mathbf{b}_k)_{,i} = \mathbf{b}_{j,i}\cdot\mathbf{b}_k + \mathbf{b}_j\cdot\mathbf{b}_{k,i} = \Gamma_{jik} + \Gamma_{kij} \end{align}

Since

$\mathbf{b}_{i,j} = \mathbf{b}_{j,i}\quad\Rightarrow\quad\Gamma_{ijk} = \Gamma_{jik}$

using these to rearrange the above relations gives

$\Gamma_{ijk} = \frac{1}{2}(g_{ik,j} + g_{jk,i} - g_{ij,k}) = \frac{1}{2}[(\mathbf{b}_i\cdot\mathbf{b}_k)_{,j} + (\mathbf{b}_j\cdot\mathbf{b}_k)_{,i} - (\mathbf{b}_i\cdot\mathbf{b}_j)_{,k}]$
Christoffel symbols of the second kind
$\Gamma_{ij}{}^k = \Gamma_{ji}{}^k,\quad \cfrac{\partial \mathbf{b}_i}{\partial q^j} = \Gamma_{ij}{}^k\mathbf{b}_k$

This implies that

$\Gamma_{ij}{}^k = \cfrac{\partial \mathbf{b}_i}{\partial q^j}\cdot\mathbf{b}^k = -\mathbf{b}_i\cdot\cfrac{\partial \mathbf{b}^k}{\partial q^j}$

$\cfrac{\partial \mathbf{b}^i}{\partial q^j} = -\Gamma^i{}_{jk}\mathbf{b}^k,\quad \boldsymbol{\nabla}\mathbf{b}_i = \Gamma_{ij}{}^k\mathbf{b}_k\otimes\mathbf{b}^j,\quad \boldsymbol{\nabla}\mathbf{b}^i = -\Gamma_{jk}{}^i\mathbf{b}^k\otimes\mathbf{b}^j$

### Vector operations

1. Dot product:

The scalar product of two vectors in curvilinear coordinates is[2](p32)

$\mathbf{u}\cdot\mathbf{v} = u^iv_i = u_iv^i = g_{ij}u^iv^j = g^{ij}u_iv_j$
2. Cross product:

The cross product of two vectors is given by[2](pp32–34)

$\mathbf{u}\times\mathbf{v} = \epsilon_{ijk}{u}_j{v}_k\mathbf{e}_i$

where $\epsilon_{ijk}$ is the permutation symbol and $\mathbf{e}_i$ is a Cartesian basis vector. In curvilinear coordinates, the equivalent expression is

$\mathbf{u}\times\mathbf{v} = [(\mathbf{b}_m\times\mathbf{b}_n)\cdot\mathbf{b}_s]u^mv^n\mathbf{b}^s = \mathcal{E}_{smn}u^mv^n\mathbf{b}^s$
where $\mathcal{E}_{ijk}$ is the third-order alternating tensor.

## Vector and tensor calculus in three-dimensional curvilinear coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

Adjustments need to be made in the calculation of line, surface and volume integrals. For simplicity, the following restricts to three dimensions and orthogonal curvilinear coordinates. However, the same arguments apply for n-dimensional spaces. When the coordinate system is not orthogonal, there are some additional terms in the expressions.

Simmonds,[2] in his book on tensor analysis, quotes Albert Einstein saying[10]

The magic of this theory will hardly fail to impose itself on anybody who has truly understood it; it represents a genuine triumph of the method of absolute differential calculus, founded by Gauss, Riemann, Ricci, and Levi-Civita.

Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity,[11] in the mechanics of curved shells,[9] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials[12][13] and in many other fields.

Some useful relations in the calculus of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden,[14] Simmonds,[2] Green and Zerna,[5] Basar and Weichert,[8] and Ciarlet.[9]

Let φ = φ(x) be a well defined scalar field and v = v(x) a well-defined vector field, and λ1, λ2... be parameters of the coordinates

### Geometric elements

1. Tangent vector: If x(λ) parametrizes a curve C in Cartesian coordinates, then
${\partial \mathbf{x} \over \partial \lambda} = {\partial \mathbf{x} \over \partial q^i}{\partial q^i \over \partial \lambda} = \left( h_{ki}\cfrac{\partial q^i}{\partial \lambda}\right)\mathbf{b}_k$

is a tangent vector to C in curvilinear coordinates (using the chain rule). Using the definition of the Lamé coefficients, and that for the metric gij = 0 when ij, the magnitude is:

$\left|{\partial \mathbf{x} \over \partial \lambda} \right| = \sqrt{h_{ki}h_{kj}\cfrac{\partial q^i}{\partial \lambda}\cfrac{\partial q^j}{\partial \lambda}} = \sqrt{ g_{ij}\cfrac{\partial q^i}{\partial \lambda}\cfrac{\partial q^j}{\partial \lambda}} = \sqrt{h_{i}^2\left(\cfrac{\partial q^i}{\partial \lambda}\right)^2}$
2. Tangent plane element: If x(λ1, λ2) parametrizes a surface S in Cartesian coordinates, then the following cross product of tangent vectors is a normal vector to S with the magnitude of infinitesimal plane element, in curvilinear coordinates. Using the above result,
${\partial \mathbf{x} \over \partial \lambda_1}\times {\partial \mathbf{x} \over \partial \lambda_2} =\left({\partial \mathbf{x} \over \partial q^i}{\partial q^i \over \partial \lambda_1}\right) \times \left({\partial \mathbf{x} \over \partial q^j}{\partial q^j \over \partial \lambda_2}\right) = \mathcal{E}_{kmp}\left( h_{ki}{\partial q^i \over \partial \lambda_1}\right)\left(h_{mj}{\partial q^j \over \partial \lambda_2}\right) \mathbf{b}_p$

where $\mathcal{E}$ is the permutation symbol. In determinant form:

${\partial \mathbf{x} \over \partial \lambda_1}\times {\partial \mathbf{x} \over \partial \lambda_2} =\begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 \\ h_{1i} \dfrac{\partial q^i}{\partial \lambda_1} & h_{2i} \dfrac{\partial q^i}{\partial \lambda_1} & h_{3i} \dfrac{\partial q^i }{\partial \lambda_1} \\ h_{1j} \dfrac{\partial q^j}{\partial \lambda_2} & h_{2j} \dfrac{\partial q^j}{\partial \lambda_2} & h_{3j} \dfrac{\partial q^j }{\partial \lambda_2} \end{vmatrix}$

### Integration

Operator Scalar field Vector field
Line integral $\int_C \varphi(\mathbf{x}) ds = \int_a^b \varphi(\mathbf{x}(\lambda))\left|{\partial \mathbf{x} \over \partial \lambda}\right| d\lambda$ $\int_C \mathbf{v}(\mathbf{x}) \cdot d\mathbf{s} = \int_a^b \mathbf{v}(\mathbf{x}(\lambda))\cdot\left({\partial \mathbf{x} \over \partial \lambda}\right) d\lambda$
Surface integral $\int_S \varphi(\mathbf{x}) dS = \iint_T \varphi(\mathbf{x}(\lambda_1, \lambda_2)) \left|{\partial \mathbf{x} \over \partial \lambda_1}\times {\partial \mathbf{x} \over \partial \lambda_2}\right| d\lambda_1 d\lambda_2$ $\int_S \mathbf{v}(\mathbf{x}) \cdot dS = \iint_T \mathbf{v}(\mathbf{x}(\lambda_1, \lambda_2)) \cdot\left({\partial \mathbf{x} \over \partial \lambda_1}\times {\partial \mathbf{x} \over \partial \lambda_2}\right) d\lambda_1 d\lambda_2$
Volume integral $\iiint_V \varphi(x,y,z) dV = \iiint_V \chi(q_1,q_2,q_3) Jdq_1dq_2dq_3$ $\iiint_V \mathbf{u}(x,y,z) dV = \iiint_V \mathbf{v}(q_1,q_2,q_3) Jdq_1dq_2dq_3$

### Differentiation

The expressions for the gradient, divergence, and Laplacian can be directly extended to n-dimensions, however the curl is only defined in 3d.

The vector field bi is tangent to the qi coordinate curve and forms a natural basis at each point on the curve. This basis, as discussed at the beginning of this article, is also called the covariant curvilinear basis. We can also define a reciprocal basis, or contravariant curvilinear basis, bi. All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point x.

Operator Scalar field Vector field 2nd order tensor field
Gradient $\nabla\varphi = \cfrac{1}{h_i}{\partial\varphi \over \partial q^i} \mathbf{b}^i$ $\nabla\mathbf{v} = \cfrac{1}{h_i^2}{\partial \mathbf{v} \over \partial q^i}\otimes\mathbf{b}_i$ $\boldsymbol{\nabla}\boldsymbol{S} = \cfrac{\partial \boldsymbol{S}}{\partial q^i}\otimes\mathbf{b}^i$
Divergence N/A $\nabla \cdot \mathbf{v} = \cfrac{1}{\prod_j h_j} \frac{\partial }{\partial q^i}(v^i\prod_{j\ne i} h_j)$ $(\boldsymbol{\nabla}\cdot\boldsymbol{S})\cdot\mathbf{a} = \boldsymbol{\nabla}\cdot(\boldsymbol{S}\cdot\mathbf{a})$

where a is an arbitrary constant vector. In curvilinear coordinates,

$\boldsymbol{\nabla}\cdot\boldsymbol{S} = \left[\cfrac{\partial S_{ij}}{\partial q^k} - \Gamma^l_{ki}S_{lj} - \Gamma^l_{kj}S_{il}\right]g^{ik}\mathbf{b}^j$

Laplacian $\nabla^2 \varphi = \cfrac{1}{\prod _j h_j}\frac{\partial }{\partial q^i}\left(\cfrac{\prod _j h_j}{h_i^2}\frac{\partial \varphi}{\partial q^i}\right)$
Curl N/A For vector fields in 3d only,

$\nabla\times\mathbf{v} = \frac{1}{h_1h_2h_3} \mathbf{e}_i \epsilon_{ijk} h_i \frac{\partial (h_k v_k)}{\partial q^j}$

where $\epsilon_{ijk}$ is the Levi-Civita symbol.

N/A

## Fictitious forces in general curvilinear coordinates

An inertial coordinate system is defined as a system of space and time coordinates x1x2x3t in terms of which the equations of motion of a particle free of external forces are simply d2xj/dt2 = 0.[15] In this context, a coordinate system can fail to be “inertial” either due to non-straight time axis or non-straight space axes (or both). In other words, the basis vectors of the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both. When equations of motion are expressed in terms of any non-inertial coordinate system (in this sense), extra terms appear, called Christoffel symbols. Strictly speaking, these terms represent components of the absolute acceleration (in classical mechanics), but we may also choose to continue to regard d2xj/dt2 as the acceleration (as if the coordinates were inertial) and treat the extra terms as if they were forces, in which case they are called fictitious forces.[16] The component of any such fictitious force normal to the path of the particle and in the plane of the path’s curvature is then called centrifugal force.[17]

This more general context makes clear the correspondence between the concepts of centrifugal force in rotating coordinate systems and in stationary curvilinear coordinate systems. (Both of these concepts appear frequently in the literature.[18][19][20]) For a simple example, consider a particle of mass m moving in a circle of radius r with angular speed w relative to a system of polar coordinates rotating with angular speed W. The radial equation of motion is mr” = Fr + mr(w + W)2. Thus the centrifugal force is mr times the square of the absolute rotational speed A = w + W of the particle. If we choose a coordinate system rotating at the speed of the particle, then W = A and w = 0, in which case the centrifugal force is mrA2, whereas if we choose a stationary coordinate system we have W = 0 and w = A, in which case the centrifugal force is again mrA2. The reason for this equality of results is that in both cases the basis vectors at the particle’s location are changing in time in exactly the same way. Hence these are really just two different ways of describing exactly the same thing, one description being in terms of rotating coordinates and the other being in terms of stationary curvilinear coordinates, both of which are non-inertial according to the more abstract meaning of that term.

When describing general motion, the actual forces acting on a particle are often referred to the instantaneous osculating circle tangent to the path of motion, and this circle in the general case is not centered at a fixed location, and so the decomposition into centrifugal and Coriolis components is constantly changing. This is true regardless of whether the motion is described in terms of stationary or rotating coordinates.

## References

1. ^ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. ISBN 0-7167-0344-0.
2. Simmonds, J. G. (1994). A brief on tensor analysis. Springer. ISBN 0-387-90639-8.
3. ^ Boothby, W. M. (2002). An Introduction to Differential Manifolds and Riemannian Geometry (revised ed.). New York, NY: Academic Press.
4. ^ McConnell, A. J. (1957). Application of Tensor Analysis. New York, NY: Dover Publications, Inc. Ch. 9, sec. 1. ISBN 0-486-60373-3.
5. ^ a b c Green, A. E.; Zerna, W. (1968). Theoretical Elasticity. Oxford University Press. ISBN 0-19-853486-8.
6. ^ Ogden, R. W. (2000). Nonlinear elastic deformations. Dover.
7. ^ Naghdi, P. M. (1972). "Theory of shells and plates". In S. Flügge. Handbook of Physics. VIa/2. pp. 425–640.
8. ^ a b Basar, Y.; Weichert, D. (2000). Numerical continuum mechanics of solids: fundamental concepts and perspectives. Springer.
9. ^ a b c Ciarlet, P. G. (2000). Theory of Shells 1. Elsevier Science.
10. ^ Einstein, A. (1915). "Contribution to the Theory of General Relativity". In Laczos, C. The Einstein Decade. p. 213. ISBN 0-521-38105-3.
11. ^ Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1973). Gravitation. W. H. Freeman and Co. ISBN 0-7167-0344-0.
12. ^ Greenleaf, A.; Lassas, M.; Uhlmann, G. (2003). "Anisotropic conductivities that cannot be detected by EIT". Physiological measurement 24 (2): 413–419. doi:10.1088/0967-3334/24/2/353. PMID 12812426.
13. ^ Leonhardt, U.; Philbin, T.G. (2006). "General relativity in electrical engineering". New Journal of Physics 8 (10): 247. doi:10.1088/1367-2630/8/10/247.
14. ^ Ogden
15. ^ Friedman, Michael (1989). The Foundations of Space–Time Theories. Princeton University Press. ISBN 0-691-07239-6.
16. ^ Stommel, Henry M.; Moore, Dennis W. (1989). An Introduction to the Coriolis Force. Columbia University Press. ISBN 0-231-06636-8.
17. ^ Beer; Johnston (1972). Statics and Dynamics (2nd ed.). McGraw–Hill. p. 485. ISBN 0-07-736650-6.
18. ^ Hildebrand, Francis B. (1992). Methods of Applied Mathematics. Dover. p. 156. ISBN 0-13-579201-0.
19. ^ McQuarrie, Donald Allan (2000). Statistical Mechanics. University Science Books. ISBN 0-06-044366-9.
20. ^ Weber, Hans-Jurgen; Arfken, George Brown (2004). Essential Mathematical Methods for Physicists. Academic Press. p. 843. ISBN 0-12-059877-9.


text:
Properties Of Relations

The question asks if there can be a relation on a set that is neither reflexive or irreflexive. The example the book give makes perfect sense: "Yes, for instance{(1,1)} on{1,2}."

I was wondering, if I had the relation on that same set {(1, 2)}, would that be irreflexive?

-
Yes. In fact, the relation $\{\langle 1,2\rangle\}$ is irreflexive on every set that contains both $1$ and $2$: it contains no ordered pairs of the form $\langle x,x\rangle$. –  Brian M. Scott Nov 3 '12 at 17:14
Thank you very much. That was an insightful response. –  Mack Nov 3 '12 at 17:22
You’re very welcome. –  Brian M. Scott Nov 3 '12 at 17:23

format: plain


text:
# Literal part of a term Calculator

## Get detailed solutions to your math problems with our Literal part of a term step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here!

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### Difficult Problems

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Solved example of literal part of a term

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text:
# Physics Commons™

Selected Works

Strongly correlated electrons

Articles 1 - 16 of 16

## Full-Text Articles in Physics

A Spin-Polarized Scheme For Obtaining Quasi-Particle Energies Within The Density Functional Theory, B. Barbiellini, A. Bansil Apr 2012

#### A Spin-Polarized Scheme For Obtaining Quasi-Particle Energies Within The Density Functional Theory, B. Barbiellini, A. Bansil

##### Bernardo Barbiellini

We discuss an efficient scheme for obtaining spin-polarized quasi-particle excitation energies within the general framework of the density functional theory (DFT). Our approach is to correct the DFT eigenvalues via the electrostatic energy of a majority or minority spin electron resulting from its interaction with the associated exchange and correlation holes by using appropriate spin-resolved pair correlation functions. A version of the method for treating systems with localized orbitals, including the case of partially filled metallic bands, is considered. Illustrative results on Cu are presented.

Direct Observation Of Localization Of The Minority-Spin-Band Electrons In Magnetite Below The Verwey Temperature, Hisao Kobayashi, Toshihiro Nagao, Masayoshi Itou, Sakae Todo, Bernardo Barbiellini, Peter E. Mijnarends, Arun Bansil, Nobuhiko Sakai Apr 2012

#### Direct Observation Of Localization Of The Minority-Spin-Band Electrons In Magnetite Below The Verwey Temperature, Hisao Kobayashi, Toshihiro Nagao, Masayoshi Itou, Sakae Todo, Bernardo Barbiellini, Peter E. Mijnarends, Arun Bansil, Nobuhiko Sakai

##### Bernardo Barbiellini

Two-dimensional spin-uncompensated momentum density distributions, $\rho_{\rm s}^{2D}({\bf p})$s, were reconstructed in magnetite at 12K and 300K from several measured directional magnetic Compton profiles. Mechanical de-twinning was used to overcome severe twinning in the single crystal sample below the Verwey transition. The reconstructed $\rho_{\rm s}^{2D}({\bf p})$ in the first Brillouin zone changes from being negative at 300 K to positive at 12 K. This result provides the first clear evidence that electrons with low momenta in the minority spin bands in magnetite are localized below the Verwey transition temperature.

Compton Scattering Beyond The Impulse Approximation, I. G. Kaplan, B. Barbiellini, A. Bansil Apr 2012

#### Compton Scattering Beyond The Impulse Approximation, I. G. Kaplan, B. Barbiellini, A. Bansil

##### Bernardo Barbiellini

We treat the non-relativistic Compton scattering process in which an incoming photon scatters from an N-electron many-body state to yield an outgoing photon and a recoil electron, without invoking the commonly used frameworks of either the impulse approximation (IA) or the independent particle model (IPM). An expression for the associated triple differential scattering cross section is obtained in terms of Dyson orbitals, which give the overlap amplitudes between the N-electron initial state and the (N-1) electron singly ionized quantum states of the target. We show how in the high energy transfer regime, one can recover from our general formalism the ...

Effect Of Hole Doping On The Electronic Structure Of Tl2201, S. Sahrakorpi, Hsin Lin, R. S. Markiewicz, A. Bansil Apr 2012

#### Effect Of Hole Doping On The Electronic Structure Of Tl2201, S. Sahrakorpi, Hsin Lin, R. S. Markiewicz, A. Bansil

##### Hsin Lin

We discuss doping dependencies of the electronic structure and Fermi surface of the monolayer Tl$_{2-x}$Cu$_x$Ba$_2$CuO$_{6+\delta}$ (Tl2201). The TlO bands are found to be particularly sensitive to doping in that these bands rapidly move to higher energies as holes are added into the system. Such doping effects beyond the rigid band picture should be taken into account in analyzing and modeling the electronic spectra of the cuprates.

Direct Observation Of Localization Of The Minority-Spin-Band Electrons In Magnetite Below The Verwey Temperature, Hisao Kobayashi, Toshihiro Nagao, Masayoshi Itou, Sakae Todo, Bernardo Barbiellini, Peter E. Mijnarends, Arun Bansil, Nobuhiko Sakai Feb 2011

#### Direct Observation Of Localization Of The Minority-Spin-Band Electrons In Magnetite Below The Verwey Temperature, Hisao Kobayashi, Toshihiro Nagao, Masayoshi Itou, Sakae Todo, Bernardo Barbiellini, Peter E. Mijnarends, Arun Bansil, Nobuhiko Sakai

##### Arun Bansil

Two-dimensional spin-uncompensated momentum density distributions, $\rho_{\rm s}^{2D}({\bf p})$s, were reconstructed in magnetite at 12K and 300K from several measured directional magnetic Compton profiles. Mechanical de-twinning was used to overcome severe twinning in the single crystal sample below the Verwey transition. The reconstructed $\rho_{\rm s}^{2D}({\bf p})$ in the first Brillouin zone changes from being negative at 300 K to positive at 12 K. This result provides the first clear evidence that electrons with low momenta in the minority spin bands in magnetite are localized below the Verwey transition temperature.

Effect Of Hole Doping On The Electronic Structure Of Tl2201, S. Sahrakorpi, Hsin Lin, R. Markiewicz, A. Bansil Feb 2011

#### Effect Of Hole Doping On The Electronic Structure Of Tl2201, S. Sahrakorpi, Hsin Lin, R. Markiewicz, A. Bansil

##### Arun Bansil

We discuss doping dependencies of the electronic structure and Fermi surface of the monolayer Tl$_{2-x}$Cu$_x$Ba$_2$CuO$_{6+\delta}$ (Tl2201). The TlO bands are found to be particularly sensitive to doping in that these bands rapidly move to higher energies as holes are added into the system. Such doping effects beyond the rigid band picture should be taken into account in analyzing and modeling the electronic spectra of the cuprates.

A Spin-Polarized Scheme For Obtaining Quasi-Particle Energies Within The Density Functional Theory, B. Barbiellini, A. Bansil Feb 2011

#### A Spin-Polarized Scheme For Obtaining Quasi-Particle Energies Within The Density Functional Theory, B. Barbiellini, A. Bansil

##### Arun Bansil

We discuss an efficient scheme for obtaining spin-polarized quasi-particle excitation energies within the general framework of the density functional theory (DFT). Our approach is to correct the DFT eigenvalues via the electrostatic energy of a majority or minority spin electron resulting from its interaction with the associated exchange and correlation holes by using appropriate spin-resolved pair correlation functions. A version of the method for treating systems with localized orbitals, including the case of partially filled metallic bands, is considered. Illustrative results on Cu are presented.

Compton Scattering Beyond The Impulse Approximation, I. G. Kaplan, B. Barbiellini, A. Bansil Feb 2011

#### Compton Scattering Beyond The Impulse Approximation, I. G. Kaplan, B. Barbiellini, A. Bansil

##### Arun Bansil

We treat the non-relativistic Compton scattering process in which an incoming photon scatters from an N-electron many-body state to yield an outgoing photon and a recoil electron, without invoking the commonly used frameworks of either the impulse approximation (IA) or the independent particle model (IPM). An expression for the associated triple differential scattering cross section is obtained in terms of Dyson orbitals, which give the overlap amplitudes between the N-electron initial state and the (N-1) electron singly ionized quantum states of the target. We show how in the high energy transfer regime, one can recover from our general formalism the ...

Dielectric Signatures Of Lattice Instabilities At 32k And 245k In $La_{2-Y}Sr_{Y}Mo_{4+X}$ (M=Cu,Ni) Cuprates And Nickelates, P. Parimi, N. Hakim, F. Chou, S. Cheong, S. Sridhar Feb 2011

#### Dielectric Signatures Of Lattice Instabilities At 32k And 245k In $La_{2-Y}Sr_{Y}Mo_{4+X}$ (M=Cu,Ni) Cuprates And Nickelates, P. Parimi, N. Hakim, F. Chou, S. Cheong, S. Sridhar

##### Srinivas Sridhar

New dielectric transitions are observed at common temperatures 32K and 245K, in isostructural $La_{2}CuO_{4+x}$ and $La_{5/3}Sr_{1/3}NiO_{4},$ that are signatures of local lattice (octahedral) instabilities. The present dielectric transitions reveal new aspects of the phase diagram of the perovskite cuprates and nickelates. They suggest that competition and coexistence of superconductivity with dielectricity occurs that is analogous to that between superconductivity and anti-ferromagnetism. These results also indicate that inhomogeneous electronic states, such as charge stripes and oxygen ordering, are strongly connected to underlying lattice instabilities.

Effect Of Hole Doping On The Electronic Structure Of Tl2201, S. Sahrakorpi, Hsin Lin, R. S. Markiewicz, A. Bansil Feb 2011

#### Effect Of Hole Doping On The Electronic Structure Of Tl2201, S. Sahrakorpi, Hsin Lin, R. S. Markiewicz, A. Bansil

##### Robert Markiewicz

We discuss doping dependencies of the electronic structure and Fermi surface of the monolayer Tl$_{2-x}$Cu$_x$Ba$_2$CuO$_{6+\delta}$ (Tl2201). The TlO bands are found to be particularly sensitive to doping in that these bands rapidly move to higher energies as holes are added into the system. Such doping effects beyond the rigid band picture should be taken into account in analyzing and modeling the electronic spectra of the cuprates.

Compressibility Of A Two-Dimensional Hole Gas In A Tilted Magnetic Field, Maryam Rahimi, M. Sakr, S. Kravchenko, S. Dultz, H. Jiang Jan 2011

#### Compressibility Of A Two-Dimensional Hole Gas In A Tilted Magnetic Field, Maryam Rahimi, M. Sakr, S. Kravchenko, S. Dultz, H. Jiang

##### Sergey Kravchenko

We have measured compressibility of a two-dimensional hole gas in p-GaAs/AlGaAs heterostructure, grown on a (100) surface, in the presence of a tilted magnetic field. It turns out that the parallel component of magnetic field affects neither the spin splitting nor the density of states. We conclude that: (a) g-factor in the parallel magnetic field is nearly zero in this system; and (b) the level of the disorder potential is not sensitive to the parallel component of the magnetic field.

Conductivity Of A Spin-Polarized Two-Dimensional Electron Liquid In The Ballistic Regime, A. A. Shashkin, E. V. Deviatov, V. T. Dolgopolov, A. A. Kapustin, S. Anissimova, A. Venkatesan, S. V. Kravchenko, T. M. Klapwijk Jan 2011

#### Conductivity Of A Spin-Polarized Two-Dimensional Electron Liquid In The Ballistic Regime, A. A. Shashkin, E. V. Deviatov, V. T. Dolgopolov, A. A. Kapustin, S. Anissimova, A. Venkatesan, S. V. Kravchenko, T. M. Klapwijk

##### Sergey Kravchenko

In the ballistic regime, the metallic temperature dependence of the conductivity in a twodimensional electron system in silicon is found to change non-monotonically with the degree of spin polarization. In particular, it fades away just before the onset of complete spin polarization but reappears again in the fully spin-polarized state, being, however, suppressed relative to the zero-field case. Analysis of the degree of the suppression allows one to distinguish between the screening and the interaction-based theories.

A Metal-Insulator Transition In 2d: Established Facts And Open Questions, S. V. Kravchenko, M. P. Sarachik Jan 2011

#### A Metal-Insulator Transition In 2d: Established Facts And Open Questions, S. V. Kravchenko, M. P. Sarachik

##### Sergey Kravchenko

The discovery of a metallic state and a metal-insulator transition (MIT) in two-dimensional (2D) electron systems challenges one of the most influential paradigms of modern mesoscopic physics, namely, that "there is no true metallic behavior in two dimensions". However, this conclusion was drawn for systems of noninteracting or weakly interacting carriers, while in all 2D systems exhibiting the metal-insulator transition, the interaction energy greatly exceeds all other energy scales. We review the main experimental findings and show that, although significant progress has been achieved in our understanding of the MIT in 2D, many open questions remain.

"Forbidden" Transitions Between Quantum Hall And Insulating Phases In P-Sige Heterostructures, M. R. Sakr, Maryam Rahimi, S. V. Kravchenko, P. T. Coleridge, R. L. Williams, J. Lapointe Jan 2011

#### "Forbidden" Transitions Between Quantum Hall And Insulating Phases In P-Sige Heterostructures, M. R. Sakr, Maryam Rahimi, S. V. Kravchenko, P. T. Coleridge, R. L. Williams, J. Lapointe

##### Sergey Kravchenko

We show that in dilute metallic p-SiGe heterostructures, magnetic field can cause multiple quantum Hall-insulator-quantum Hall transitions. The insulating states are observed between quantum Hall states with filling factors = 1 and 2 and, for the first time, between = 2 and 3 and between = 4 and 6. The latter are in contradiction with the original global phase diagram for the quantum Hall effect. We suggest that the application of a (perpendicular) magnetic field induces insulating behaviour in metallic p-SiGe heterostructures in the same way as in Si MOSFETs. This insulator is then in competition with, and interrupted by, integer quantum Hall ...

Comment On "Theory Of Metal-Insulator Transitions In Gated Semiconductors", S. V. Kravchenko, M. P. Sarachik, D. Simonian Jan 2011

#### Comment On "Theory Of Metal-Insulator Transitions In Gated Semiconductors", S. V. Kravchenko, M. P. Sarachik, D. Simonian

##### Sergey Kravchenko

No abstract provided.

Jan 2011

#### Absorption Of Nuclear Gamma Radiation By Heavy Electrons On Metallic Hydride Surfaces, A. Widom, L. Larsen

##### Allan Widom

Low energy nuclear reactions in the neighborhood of metallic hydride surfaces may be induced by ultra-low momentum neutrons. Heavy electrons are absorbed by protons or deuterons producing ultra low momentum neutrons and neutrinos. The required electron mass renormalization is provided by the interaction between surface electron plasma oscillations and surface proton oscillations. The resulting neutron catalyzed low energy nuclear reactions emit copious prompt gamma radiation. The heavy electrons which induce the initially produced neutrons also strongly absorb the prompt nuclear gamma radiation, re-emitting soft photons. Nuclear hard photon radiation away from the metallic hydride surfaces is thereby strongly suppressed.


text:
# Understanding of the probability

I have a problem with understanding some of my statistics homework. I hope that some of you could help me understand.

In summary the question is as follows:

There are 30 people in a group, which are split up in 2 groups of 15. The first group we call group1 and the other group2. Group1 has been placed in an order from 1 till 15 and group 2 is ordered 16 till 30. In the question we need to find two persons. We do this by removing each time the 3rd person till we only have 2 people left.

So for example:

1 2 3 4 5 6 7 8 9 10 11.. 30; We start with number 4, first number 7 will be removed. Then number 10 will be removed. Next will be number 13. Etc, And after number 28 number 1 will be removed. till there are only 2 people left.

The start position is random, so it could be 1 till 30.

I need to answer the following questions:

1. What is the probability that there are 0 persons left from group1?
2. What is the probability that there is 1 person left from group1?
3. What is the probability that the two persons, who are still there are both from group1?

Someone told me the answers, but I do not understand them. So I hope that someone could explain it to me. I got the following answers:

1) 4/30

2) 22/30

3) 4/30

• "We start with number 4, first number 7 will be removed. Then number 10 will be removed. Next will be number 13..." Who is removed after number 28? – Mike Pierce May 4 '15 at 20:03
• Thank for the reply. Number 1 will be removed. – Danique May 4 '15 at 20:06

First, we look at an example. If we start at number $1$, we'll remove elements from the sequence $$(1, 2, 3, \dots, 19, 20, 21, \dots, 28, 29, 30)$$ until we end up with persons $20$ and $30$ (please correct me if I'm wrong).

When we start with person $2$, we could use the same sequence and redo the calculation. Instead, we modify the order as follows: $$(2,3,4, \dots, 20, \underset{\underset{20\text{th}}{\uparrow}}{21}, 22, \dots, 29, 30, \underset{\underset{30\text{th}}{\uparrow}}{1})$$ After removing elements from this sequence, we observe that we have discarded the same positions as in the last example. In particular, the persons on the $20$th and $30$th place would remain (numbers $21$ and $1$). Especially, we conclude that the two positions have a constant distance of $10$.

If we start with number $n \in \{1, \dots, 30 \}$, we obtain the persons $p_1, p_2 \in \{1, \dots, 30\}$ from the $20$th and $30$th position, respectively.

$$(n, \dots, \underset{\underset{20\text{th}}{\uparrow}}{p_1}, \dots, \underset{\underset{30\text{th}}{\uparrow}}{p_2})$$

There are $15$ different choices for $n$, s.t. $p_1 \leq 15$. Then, we have $p_2 = p_1 + 10$ (because of the distance of the indices). Consequently, there are only $5$ possibilities for our starting number $n$, s.t. both persons are in group #$1$. It follows, that the answer to question 3 is $5/30$.

Because of symmetry, the answer to question 1 is $5/30$, too. The remaining probability (question 2) has to be $1- 5/30 - 5/30= 20 / 30$.

• I think your right. Thanks for the clear explanation! – Danique May 5 '15 at 15:41


text:
# Solve trigonometric equation

I am interested in finding all $t$ in $[0,2\pi]$ with $\cos t = - \sin t$. Is there a way to do this without using a calculator? I know $- \sin t = \sin (-t)$ but it doesn't seem to help. Thank you.

-
You could make tangent appear, if it makes it easier for you. –  julien Feb 26 '13 at 16:11
I have found answer on my own. By drawing the curves. It is not as difficult to draw as it might seem. Please delete this question. –  colollary Feb 26 '13 at 16:17
How can I delete this question? I have answered it. –  colollary Feb 26 '13 at 16:19
Like many questions you will be asked, this one is easier without a calculator. –  André Nicolas Feb 26 '13 at 18:38

You can use $\cos t + \sin t=\sqrt 2 \sin (t + \frac \pi 4)$ and set it to zero to find $t=\frac {3 \pi}4, \frac {7 \pi}4$

-
Thanks, it is good to see different way of getting to same solution. After posting question I found solution by drawing the curves. It works also. –  colollary Feb 26 '13 at 16:29

Hint: $\cos t = \sin t \;\implies \;\;t_1 = \pi /4, \;\;t_2 = 5\pi/4\;\;$ for $0 \leq t \leq 2\pi$

Added: Now what does this mean if $\cos t = -\sin t$?

It means that the solutions for $\;\;0 \leq t \leq 2\pi\;\;$ are the solutions $\;\;t_1 + \pi/2\;\;\;\text{and}\;\;\;t_2 + \pi/2$

Alternatively, $$\cos t = - \sin t \;\; \iff \;\; 1 = -\frac{\sin t}{\cos t}$$ $$\iff\;\; 1 = -\tan t \;\;\iff \tan t = -1 \;; \iff t = \tan^{-1}(-1)$$

-
Yes I know how to use a calculator. I don't see how your hint is answering my question. Meanwhile I have found answer using picture. One correct solution is $t= 3 \pi / 4$ –  colollary Feb 26 '13 at 16:16
My hint was about the value of t when $\cos t = \sin t$ (angle in first and third quadrant)...so, I was trying to hint that if $\cos t = \bf{-}\sin t$, we have a rotation of $\pi/2$ from the values I listed: angles in the second and fourth quadrant: which yes in the second quadrant $t = \pi/4 + \pi/2 = 3\pi/4$ and the other being $t = 5\pi/4 + \pi/2 = 7\pi/4$ which is in the fourth quadrant. I did not mean to be demeaning, if that's the way you took my post. –  amWhy Feb 26 '13 at 16:20
@Ross, please see my comment above. I KNOW that there is a minus sign on $\sin$ in the given equation. I was trying to HINT that if you know the values of $t$ for which $\cos t = \sin t$, one can easily deduce the values for $t$ when $\cos t = - \sin t$ –  amWhy Feb 26 '13 at 16:24
@colollary I did not intend to be demeaning. I hope you read by comment above. I wouldn't have answered unless I was trying to be helpful. –  amWhy Feb 26 '13 at 16:33
No, not demeaning. But my question is how to compute this without calculator. Your answer is beside the point and therefore useless: you don't explain how to compute it you post a solution. As for "Alternatively...": how do I calculate $\tan^{-1}$ without calculator? I am not a calculator. –  colollary Feb 26 '13 at 16:35

$\cos t=-\sin t\implies \tan t=-1=-\tan\frac\pi4=\tan \left(-\frac\pi4\right)$ as $\tan(-x)=-\tan x$

$\implies t=n\pi-\frac\pi4$ where $n$ is any integer.

Now, we need $0\le n\pi-\frac\pi4\le 2\pi\implies 0\le4n-1\le 8$

So, $4n-1\ge 0\implies n\ge \frac14\implies n\ge1$ as $n$ is any integer

and $4n-1\le 8\implies n\le\frac94\implies n\le2$ as $n$ is any integer.

So, $n=1,2\implies t=\pi-\frac\pi4=\frac{3\pi}4$ or $t=2\pi-\frac\pi4=\frac{7\pi}4$

-


text:
# how to find the order of an element in a quotient group

Consider the quotient group $\mathbb{Q}/\mathbb{Z}$ of the additive group of rational numbers. Then how to find the order of the element $2/3 + \mathbb{Z}$ in $\mathbb{Q}/\mathbb{Z}$.

-

Supplementing Makoto's answer, we can generalize: the order of an element $p/q \in \mathbb{Q}/\mathbb{Z}$ is the least integer $n$ such that $(p/q)n$ is an integer. If the fraction is written in reduced form, this is the same as the denominator.
Let $a = 2/3 + \mathbb{Z}$. Since $3a = 0$ and $a \neq 0$, the order of $a$ is $3$.
@ram: Note that in Makoto's nice answer(+1); $3a=0=0_{\mathbb Q/\mathbb Z}=\mathbb Z$. – Babak S. Sep 16 '12 at 9:26


text:
# Continuity of the radius of convergence of p-adic differential equations on Berkovich analytic spaces - Mathematics > Number Theory

Abstract: We consider a vector bundle with integrable connection \cE, a on ananalytic domain U in the generic fiber \cX {\eta} of a smooth formal p-adicscheme \cX, in the sense of Berkovich. We define the \emph{diameter}\delta {\cX}\xi,U of U at \xi\in U, the \emph{radius} ho {\cX}\xi of thepoint \xi\in\cX {\eta}, the \emph{radius of convergence} of solutions of\cE, a at \xi, R\xi = R {\cX}\xi, U,\cE, a. We discuss semi-continuity of these functions with respect to the Berkovich topology. Inparticular, under we prove under certain assumptions that \delta {\cX}\xi,U, ho {\cX}\xi and R {\xi}U,\cE, a are upper semicontinuous functions of\xi; for Laurent domains in the affine space, \delta {\cX}-,U is continuous.In the classical case of an affinoid domain U of the analytic affine line, R isa continuous function.

Author: Francesco Baldassarri, Lucia Di Vizio

Source: https://arxiv.org/


text:
• The Electromagnetic Counterpart of the Binary Neutron Star Merger LIGO/Virgo GW170817. I. Dark Energy Camera Discovery of the Optical Counterpart(1710.05459)

Oct. 16, 2017 astro-ph.HE
We present the Dark Energy Camera (DECam) discovery of the optical counterpart of the first binary neutron star merger detected through gravitational wave emission, GW170817. Our observations commenced 10.5 hours post-merger, as soon as the localization region became accessible from Chile. We imaged 70 deg$^2$ in the $i$ and $z$ bands, covering 93\% of the initial integrated localization probability, to a depth necessary to identify likely optical counterparts (e.g., a kilonova). At 11.4 hours post-merger we detected a bright optical transient located $10.6''$ from the nucleus of NGC\,4993 at redshift $z=0.0098$, consistent (for $H_0 = 70$\, km s$^{-1}$ Mpc$^{-1}$) with the distance of $40 \pm 8$\, Mpc reported by the LIGO Scientific Collaboration and the Virgo Collaboration (LVC). At detection the transient had magnitudes $i\approx 17.30$ and $z\approx 17.45$, and thus an absolute magnitude of $M_i = -15.7$, in the luminosity range expected for a kilonova. We identified 1,500 potential transient candidates. Applying simple selection criteria aimed at rejecting background events such as supernovae, we find the transient associated with NGC\,4993 as the only remaining plausible counterpart, and reject chance coincidence at the 99.5\% confidence level. We therefore conclude that the optical counterpart we have identified near NGC\,4993 is associated with GW170817. This discovery ushers in the era of multi-messenger astronomy with gravitational waves, and demonstrates the power of DECam to identify the optical counterparts of gravitational-wave sources.
• The Electromagnetic Counterpart of the Binary Neutron Star Merger LIGO/VIRGO GW170817. II. UV, Optical, and Near-IR Light Curves and Comparison to Kilonova Models(1710.05840)

Oct. 16, 2017 astro-ph.HE
We present UV, optical, and NIR photometry of the first electromagnetic counterpart to a gravitational wave source from Advanced LIGO/Virgo, the binary neutron star merger GW170817. Our data set extends from the discovery of the optical counterpart at $0.47$ days to $18.5$ days post-merger, and includes observations with the Dark Energy Camera (DECam), Gemini-South/FLAMINGOS-2 (GS/F2), and the {\it Hubble Space Telescope} ({\it HST}). The spectral energy distribution (SED) inferred from this photometry at $0.6$ days is well described by a blackbody model with $T\approx 8300$ K, a radius of $R\approx 4.5\times 10^{14}$ cm (corresponding to an expansion velocity of $v\approx 0.3c$), and a bolometric luminosity of $L_{\rm bol}\approx 5\times10^{41}$ erg s$^{-1}$. At $1.5$ days we find a multi-component SED across the optical and NIR, and subsequently we observe rapid fading in the UV and blue optical bands and significant reddening of the optical/NIR colors. Modeling the entire data set we find that models with heating from radioactive decay of $^{56}$Ni, or those with only a single component of opacity from $r$-process elements, fail to capture the rapid optical decline and red optical/NIR colors. Instead, models with two components consistent with lanthanide-poor and lanthanide-rich ejecta provide a good fit to the data, the resulting "blue" component has $M_\mathrm{ej}^\mathrm{blue}\approx 0.01$ M$_\odot$ and $v_\mathrm{ej}^\mathrm{blue}\approx 0.3$c, and the "red" component has $M_\mathrm{ej}^\mathrm{red}\approx 0.04$ M$_\odot$ and $v_\mathrm{ej}^\mathrm{red}\approx 0.1$c. These ejecta masses are broadly consistent with the estimated $r$-process production rate required to explain the Milky Way $r$-process abundances, providing the first evidence that BNS mergers can be a dominant site of $r$-process enrichment.
• The detection of GW170817 in both gravitational waves and electromagnetic waves heralds the age of gravitational-wave multi-messenger astronomy. On 17 August 2017 the Advanced LIGO and Virgo detectors observed GW170817, a strong signal from the merger of a binary neutron-star system. Less than 2 seconds after the merger, a gamma-ray burst (GRB 170817A) was detected within a region of the sky consistent with the LIGO-Virgo-derived location of the gravitational-wave source. This sky region was subsequently observed by optical astronomy facilities, resulting in the identification of an optical transient signal within $\sim 10$ arcsec of the galaxy NGC 4993. These multi-messenger observations allow us to use GW170817 as a standard siren, the gravitational-wave analog of an astronomical standard candle, to measure the Hubble constant. This quantity, which represents the local expansion rate of the Universe, sets the overall scale of the Universe and is of fundamental importance to cosmology. Our measurement combines the distance to the source inferred purely from the gravitational-wave signal with the recession velocity inferred from measurements of the redshift using electromagnetic data. This approach does not require any form of cosmic "distance ladder;" the gravitational wave analysis can be used to estimate the luminosity distance out to cosmological scales directly, without the use of intermediate astronomical distance measurements. We determine the Hubble constant to be $70.0^{+12.0}_{-8.0} \, \mathrm{km} \, \mathrm{s}^{-1} \, \mathrm{Mpc}^{-1}$ (maximum a posteriori and 68% credible interval). This is consistent with existing measurements, while being completely independent of them. Additional standard-siren measurements from future gravitational-wave sources will provide precision constraints of this important cosmological parameter.
• Cosmography with Galaxy Clusters(0810.3689)

Oct. 21, 2008 astro-ph
In the present work we focus on future experiments using cluster abundance observations to constraint the Dark Energy equation of state parameter, w. To obtain tight constraints from this kind of experiment, a reliable sample of galaxy clusters must be obtained from deep and wide-field images. We therefore present the computational environment (2DPHOT) that allow us to build the galaxy catalog from the images and the Voronoi Tessellation cluster finding algorithm that we use to identify the galaxy clusters on those catalogs. To test our pipeline with data similar in quality to what will be gathered by future wide field surveys, we process images from the Deep fields obtained as part of the LEGACY Survey (four fields of one square degree each, in five bands, with depth up to r'=25). We test our cluster finder by determining the completeness and purity of the finder when applied to mock galaxy catalogs made for the Dark Energy Survey cluster finder comparison project by Risa Wechsler and Michael Busha. This procedure aims to understand the selection function of the underlying dark matter halos.
• The Northern Sky Optical Cluster Survey II: An Objective Cluster Catalog for 5800 Square Degrees(astro-ph/0301274)

Jan. 14, 2003 astro-ph
We present a new, objectively defined catalog of candidate galaxy clusters based on the galaxy catalogs from the Digitized Second Palomar Observatory Sky Survey (DPOSS). This cluster catalog, derived from the best calibrated plates in the high latitude (|b|>30) Northern Galactic Cap region, covers 5,800 square degrees, and contains 8,155 candidate clusters. A simple adaptive kernel density mapping technique, combined with the SExtractor object detection algorithm, is used to detect galaxy overdensities, which we identify as clusters. Simulations of the background galaxy distribution and clusters of varying richnesses and redshifts allow us to optimize detection parameters, and measure the completeness and contamination rates for our catalog. Cluster richnesses and photometric redshifts are measured, using integrated colors and magnitudes for each cluster. An extensive spectroscopic survey is used to confirm the photometric results. This catalog, with well-characterized sample properties, provides a sound basis for future studies of cluster physics and large scale structure.


text:
# Are these conditions sufficient for a self-distributive algebra to occur in the algebras of elementary embeddings?

Suppose that $$\mathcal{E}_{\lambda}$$ is the set of all elementary embeddings $$j:V_{\lambda}\rightarrow V_{\lambda}$$ and $$*$$ is the operation on $$\mathcal{E}_{\lambda}$$ defined by $$j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$$. If $$\gamma$$ is a limit ordinal with $$\gamma<\lambda$$, define an equivalence relation $$\equiv^{\gamma}$$ on $$\mathcal{E}_{\lambda}$$ by letting $$j\equiv^{\gamma}k$$ if and only if $$j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$$. Then $$\equiv^{\gamma}$$ is a congruence on $$\mathcal{E}_{\lambda}$$. Now, a natural question to ask is if there is a natural characterization of the subalgebras and quotient algebras of the algebras that may arise as $$\mathcal{E}_{\lambda}/\equiv^{\gamma}.$$

Let us now attempt an axiomatization of the subalgebras of $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ that does not require set theory.

An algebra $$(X,*)$$ is said to be self-distributive if $$x*(y*z)=(x*y)*(x*z)$$ for all $$x,y,z\in X$$. A self-distributive algebra $$(X,*)$$ is said to be a reduced Laver-like algebra if there is some $$1\in X$$ where $$1*x=x,x*1=1$$ for $$x\in X$$ and where if $$x_{n}\in X$$ for each $$n\in\omega$$, then there is some $$N\in\omega$$ with $$x_{0}*\dots*x_{N}=1$$ (here parentheses are grouped on the left, so $$a*b*c=(a*b)*c$$).

$$\mathbf{Theorem:}$$ If $$\lambda$$ is a cardinal and $$\gamma$$ is a limit ordinal with $$\gamma<\lambda$$, then $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ is Laver-like.

reduced Laver-like algebras behave a lot like the algebras $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ in the sense that Laver-like algebras have a notion of a critical point, a composition operation, and an analogue of $$\equiv^{\gamma}$$. Furthermore, the notion of a critical point for Laver-like algebras behaves in many ways the same as it does in $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ and the notion of a critical point is the magic that makes Laver-like algebras run.

If $$(X,*)$$ is a reduced Laver-like algebra, then there is a unique ordinal $$\alpha$$ and function $$\mathrm{crit}:X\rightarrow\alpha+1$$ such that

1. $$\mathrm{crit}(x)=\alpha$$ if and only if $$x=1$$,

2. $$\mathrm{crit}(y)=\mathrm{crit}(x*y)$$ whenever $$\mathrm{crit}(y)<\mathrm{crit}(x)$$, and

3. $$\mathrm{crit}(y)<\mathrm{crit}(x*y)$$ whenever $$\mathrm{crit}(x)\leq\mathrm{crit}(y)<\alpha$$.

$$\mathbf{Theorem:}$$ Suppose that $$j\in\mathcal{E}_{\lambda}$$. Then $$(j*j)(\alpha)\leq j(\alpha)$$ whenever $$\alpha<\lambda$$.

$$\mathbf{Corollary:}$$ If $$X$$ is isomorphic to a quotient of a subalgebra of $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$, then $$\mathrm{crit}(x*x*y)\leq\mathrm{crit}(x*y)$$ for all $$x,y\in X$$.

$$\mathbf{Example:}$$ There exists an $$11$$ element reduced Laver-like algebra $$X$$ along with $$x,y\in X$$ where $$\mathrm{crit}(x*x*y)>\mathrm{crit}(x*y)$$. Therefore, $$X$$ is not isomorphic to any quotient of a subalgebra of $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$.

$$\mathbf{Question:}$$ If $$(X,*)$$ is a finite reduced Laver-like algebra such that $$\mathrm{crit}(x*x*y)>\mathrm{crit}(x*y)$$, then is $$X$$ isomorphic to some subalgebra of some $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ in some model of ZFC? If not, then what is the correct purely algebraic characterization of the finite subalgebras of $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ or at least the finite quotients of subalgebras of $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$?


text:
# Relay current flow

I'm new to electronics.

I found the tutorial about the control of a relay using GPIO in raspberry pi from http://arnholm.org/raspberry-pi-controlling-a-relay/ . It has the following electric circuit :

I wonder the existence of the diode laid down parallel with relay. I searched about this and got the answer from Why is there a diode connected in parallel to a relay coil?

Since an inductor (the relay coil) cannot change it's current instantly, the flyback diode provides a path for the current when the coil is switched off. Otherwise, a voltage spike will occur causing arcing on switch contacts or possibly destroying switching transistors.

It says eventually when the switch is off, the flow of relay current should go through the flyback diode to prevent destroying the switching transistor because it causes the undesirable spark on it.

But I am worried about if the relay current from REL1 go through D1 diode, the sudden excess of current might make the pi die after the current going up above the diode.

So I thought one of the solution is if we put 2 diodes in the position D1 and draw a line between them, we can release the current safely.

So my question is

What if we put a resistor above the D1 diode? Does the current can be attenuated with dissipation in the resistor? Can this be also a solution?

• They say "don't do this bit" because that relay would need to be listed as equipment (not components) in mains power switching. The UL listing on a relay from Mouser is a "component" listing, not sufficient. Contrast with a RiB relay, which starts with a listed component, but packages it to work in mains wiring methods, and has achieved UL listing as equipment for direct mains use. I wish someone would make more stuff like this for the Raspberry Pi'ers of the world. – Harper - Reinstate Monica Mar 26 '18 at 18:56
• voltage spike will occur causing arcing on switch contacts .... it does not refer to the relay contacts in your diagram ... if you used a smaller relay, instead of the transistor, to drive the main relay, then the note about arcing would refer to the contacts of the smaller relay ... arcing would happen because of the inductive load of the bigger relay coil – jsotola Mar 26 '18 at 19:24

This so-called arc from breaking the coil current Only occurs with the dry contact . The time could be in the nanosecond range and since V equals V=LdI/dt with dt going to zero V can arc on dry contacts. Because the transistor has some capacitance it cannot shut off that quickly so the diode is only to protect the transistor and there’s very little capacitance that would feedback any significant charge back to the microcontroller so no worries there .

The 100mA transistor current is immediately switched through the diode along the same wires can be handled easily by 100 mA small signal diode. The diode should be placed close to the transistor rather than the coil so that the current path is the same . For EMI pulse attenuation it’s best to have the switched current ground and power signals to the relay coil interface close together to minimize current loop area and antenna-like interference to high impedance nearby signals.

Furthermore 3 mA into the base is more than enough to saturate the transistor. The BC 337 will rise from 50 mV to less than 200 mV which is within 5% of your 5 V supply with only one milliamp of base current @Ic=100mA

That sudden excess of current flows only through the diode and the relay coil, hence it has no influence on the rest of the circuit, provided that the diode is close to the relay coil pins.

What might cause some harm is that the transistor is switched of, and it suddenly draws less current from the 5V supply. If you are worried aboyut this you could put a beefy capacitor (for instance 470uF) between gnd and 5V. (Or, when you have multiple relays, use a separate 5V supply for the relays.)

With a Pi this effect will be less pronounced, because most (all?) Pi circuitry is powered by a switcher that converts the 5V to 3.3V.


text:
Uh Oh! It seems you’re using an Ad blocker!

We always struggled to serve you with the best online calculations, thus, there's a humble request to either disable the AD blocker or go with premium plans to use the AD-Free version for calculators.

Or

# Cross Multiply Calculator

$$\dfrac{A}{B}=\dfrac{C}{D}$$

Enter any 3 known numbers to find the fourth value

A

C

B

D

Table of Content

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Get the Widget!

Add this calculator to your site and lets users to perform easy calculations.

Feedback

How easy was it to use our calculator? Did you face any problem, tell us!

Make free use of this cross multiply calculator to determine the unknown variable value by using the technique of cross multiply fractions. Now the fraction multiplication is now fast and accurate enough with the assistance of this best cross multiplication calculator.

Want to delve into the technique in order to understand? Stay focused and it’s time to move by!

## How Do You Cross Multiply?

In mathematical theory:

“Cross multiplication means:

• The product of numerator in first fraction with the denominator of the second fraction
• The product of the denominator of the first fraction with the numerator of the second fraction”

This free cross multiply calculator also works on the same criteria aforementioned to generate accurate answers against any fraction problem. Except these calculations, you can also use our simple fraction calculator to add, subtract, multiply, and divide fractions.

### Cross Multiplication Formula:

Now you could easily find the missing value of the variable by either using this cross multiplication calculator or the following formula:

$$\frac{A}{B} = \frac{C}{D}$$

$$A*D = B*C$$

### How To Cross Multiply Fractions?

Let us resolve a few examples to understand the calculations done in this particular method.

Let’s move on!

Example  01:

How to cross multiply to find x in the proportion pair given below:

$$\frac{4}{B} = \frac{7}{9}$$

Solution:

If you want instant calculations, use this best cross multiply calculator. And if you want to understand manually, give a read further!

Cross multiplying fractions:

$$\frac{4}{B} = \frac{7}{9}$$

$$A*D = B*C$$

$$4*9 = B*7$$

$$B = \frac{4*9}{7}$$

$$B = 5.142$$

Example  02:

How to do cross multiplication for the following set of fractions?

$$\frac{1}{6} = \frac{C}{2}$$

Solution:

Here we have:

$$\frac{1}{6} = \frac{C}{2}$$

$$A*D = B*C$$

$$1*2 = 6*C$$

$$C = \frac{1*2}{6}$$

$$C = \frac{1}{3}$$

### How Cross Multiply Calculator Works?

This cross multiplication calculator will take a couple of seconds to solve for an unknown entity in a set of proportions. Let’s find how!

Input:

• You are given four fields to enter numbers
• Now what you need to do is enter any three of these numbers in their respective designated fields and leave the fourth as empty
• Now it’s time to hit the calculate button, so go for it

Output:

The free solve for x calculator fractions does the following calculations:

• Determines the value of the unknown variable in a fraction combination

## FAQ’s:

### Why does cross-multiplication compare fractions?

Whenever we make use of the cross product method, the equivalent fractions are not covered in it. This is the reason this particular method does relate all the ratios in order to generate a specific report against the data being provided.

### Who invented cross multiplication?

Joseph Louis Langrage, a famous mathematician, introduced the concept of both dot and cross multiplication of fractions by comparing the tetrahedron properties.

### Is cross multiplication is used in unitary method?

Yes of course! Cross multiplication is a part of this specific mathematical operation.

### Can you cross multiply and divide fractions?

Yes of course! In this particular method, what we need to consider in mind is:

The numerator of the first fraction is multiplied by the denominator of the second fraction. After you get the number, you need to write it as the numerator of the resulting fractions. And in case you find it tricky enough, let this cross multiplication calculator do that for you in a blink of moments.

## Conclusion:

Usually complex and larger fractions are not easy to explore and reduce. That is why the method of cross multiplication is being introduced to resolve and find solutions to such complicated mathematical problems. And rest when it comes to most accurate and fast outputs of the fraction combinations, this cross multiply calculator will be there to assist you in this regard.

## References:

From the source of Wikipedia: Cross-multiplication, Procedure, Use, Rule of three, Double rule of three,

From the source of Khan Academy: system of equations, elimination, substitution

From the source of Lumen Learning: Solving Proportions, Applications Using Proportions


text:
# Portable Python CSV class

I have been working on a project where I needed to analyze multiple, large datasets contained inside many CSV files at the same time. I am not a programmer but an engineer, so I did a lot of searching and reading. Python's stock CSV module provides the basic functionality, but I had a lot of trouble getting the methods to run quickly on 50k-500k rows since many strategies were simply appending. I had lots of problems getting what I wanted and I saw the same questions asked over and over again. I decided to spend some time and write a class that performed these functions and would be portable. If nothing else, myself and other people I work with could use it.

I would like some input on the class and any suggestions you may have. I am not a programmer and don't have any formal background so this has been a good OOP intro for me. The end result is in two lines you can read all CSV files in a folder into memory as either pure Python lists or, as lists of NumPy arrays. I have tested it in many scenarios and hopefully found most of the bugs. I'd like to think this is good enough that other people can just copy and paste into their code and move on to the more important stuff. I am open to all critiques and suggestions. Is this something you could use? If not, why?

You can try it with generic CSV data. The standard Python lists are flexible in size and data type. NumPy will only work with numeric (float specifically) data that is rectangular in format:

x, y, z,
1, 2, 3,
4, 5, 6,
...

import numpy as np
import csv
import os
import sys

class EasyCSV(object):
"""Easily open from and save CSV files using lists or numpy arrays.

Initiating and using the class is as easy as CSV = EasyCSV('location').
The class takes the following arguements:

EasyCSV(location, width=None, np_array='false', skip_rows=0)

location is the only mandatory field and is string of the folder location
containing .CSV file(s).

width is optional and specifies a constant width. The default value None
will return a list of lists with variable width. When used with numpy the
array will have the dimensions of the first valid numeric row of data.

np_array will create a fixed-width numpy array of only float values.

skip_rows will skip the specified rows at the top of the file.

"""

def __init__(self, location, width=None, np_array='false', skip_rows=0):
# Initialize default vairables
self.np_array = np_array
self.skip_rows = skip_rows
self.loc = str(location)
os.chdir(self.loc)
self.dataFiles = []
self.width = width
self.i = 0
#Find all CSV files in chosen directory.
for files in os.listdir(loc):
if files.endswith('CSV') or files.endswith('csv'):
self.dataFiles.append(files)
#Preallocate array to hold csv data later
self.allData = [0] * len(self.dataFiles)

'''Reads all files contained in the folder into memory.
'''
self.Dict = {}    #Stores names of files for later lookup
#Main processig loop
for files in self.dataFiles:
self.trim = 0
self.j = 0
with open(files,'rb') as self.rawFile:
print files
self.dataList = []
#Extend iterates through CSV object and passes to datalist
self.dataList.extend(self.newData)
#Trims off pre specified lines at the top
if self.skip_rows != 0:
self.dataList = self.dataList[self.skip_rows:]

#Numpy route, requires all numeric input
if self.np_array == 'true':
#Finds width if not specified
if self.width is None:
self.width = len(self.dataList[self.skip_rows])
self.CSVdata = np.zeros((len(self.dataList),self.width))

#Iterate through data and adds it to numpy array
self.k = 0
for data in self.dataList:
try:
self.CSVdata[self.j,:] = data
self.j+=1
except ValueError: #non numeric data
if self.width < len(data):
sys.exit('Numpy array too narrow. Choose another width')
self.trim+=1
pass
self.k+=1
#trims off excess
if not self.trim == 0:
self.CSVdata = self.CSVdata[:-self.trim]
#Python nested lists route; tolerates multiple data types
else:
#Declare required empty str arrays
self.CSVdata = [0]*len(self.dataList)
for rows in self.dataList:
self.k = 0
self.rows = rows
#Handle no width imput, flexible width
if self.width is None:
self.numrow = [0]*len(self.rows)
else:
self.numrow = [0]*self.width
#Try to convert to float, fall back on string.
for data in self.rows:
try:
self.numrow[self.k] = float(data)
except ValueError:
try:
self.numrow[self.k] = data
except IndexError:
pass
except IndexError:
pass
self.k+=1

self.CSVdata[self.j] = self.numrow
self.j+=1
#append file to allData which contains all files
self.allData[self.i] = self.CSVdata
#trim CSV off filename and store in Dict for indexing of allData
self.dataFiles[self.i] = self.dataFiles[self.i][:-4]
self.Dict[self.dataFiles[self.i]] = self.i
self.i+=1

def write(self, array, name, destination=None):
'''Writes array in memory to file.

EasyCSV.write(array, name, destination=None)

array is a pointer to the array you want written to CSV

name will be the name of said file

destination is optional and will change the directory to the location
specified. Leaving it at the default value None will overwrite any CSVs
that may have been read in by the class earlier.
'''
self.array = array
self.name = name
self.dest = destination
#Optional change directory
if self.dest is not None:
os.chdir(self.dest)
#Dict does not hold CSV, check to see if present and trim
if not self.name[-4:] == '.CSV' or self.name[-4:] == '.csv':
self.name = name + '.CSV'
#Create files and write data, 'wb' binary req'd for Win compatibility
with open(self.name,'wb') as self.newCSV:
self.CSVwrite = csv.writer(self.newCSV,dialect='excel')
for data in self.array:
self.CSVwrite.writerow(data)
os.chdir(self.loc) #Change back to original __init__.loc

def lookup(self, key=None):
'''Prints a preview of data to the console window with just a key input
'''
self.key = key
#Dict does not hold CSV, check to see if present and trim
if self.key[-4:] == '.CSV' or self.key[-4:] == '.csv':
self.key = key[:-4]
#Print None case
elif self.key is None:
print self.allData[0]
print self.allData[0]
print '... ' * len(self.allData[0][-2])
print self.allData[0][-2]
print self.allData[0]
#Print everything else
else:
self.index = self.Dict[self.key]
print self.allData[self.index][0]
print self.allData[self.index][1]
print '... ' * len(self.allData[self.index][-2])
print self.allData[self.index][-2]
print self.allData[self.index][-1]

def output(self, key=None):
'''Returns the array for assignment to a var with just a key input
'''
self.key = key
#Dict does not hold CSV, check to see if present and trim
if self.key is None:
return self.allData[0]
elif self.key[-4:] == '.CSV' or self.key[-4:] == '.csv':
self.key = key[:-4]
#Return file requested
self.index = self.Dict[self.key]
return self.allData[self.Dict[self.key]]

################################################

loc = 'C:\Users\Me\Desktop'
CSV = EasyCSV(loc, np_array='false', width=None, skip_rows=0)

target = 'somecsv' #with or without .csv/.CSV
CSV.lookup(target)

A = CSV.output(target)

loc2 = 'C:\Users\Me\Desktop\New folder'
for keys in CSV.Dict:
print keys
CSV.write(CSV.output(keys),keys,destination=loc2)

• Also you should investigate csvkit and pandas, or maybe import CSVs into a relational or key-value database instead of using them directly. – Francis Avila Apr 7 '13 at 20:28
• Pandas is interesting but not very light weight. I wish I saw a suggestion for it before. Can't use CSV kit because I'm primarily on Win. Thanks for the useful info. – devfarce Apr 7 '13 at 21:26
• csvkit is supported on all platforms. – Francis Avila Apr 8 '13 at 14:55
• Please use comments to respond to people rather than editing your question. – Winston Ewert Apr 8 '13 at 19:11

Some observations:

• You expect read to be called exactly once (otherwise it reads the same files again, right?). You might as well call it from __init__ directly. Alternatively, read could take location as parameter, so one could read multiple directories into the object.
• You use strings 'true', 'false' where you should use actual bool values True, False
• You set instance variables such as self.key = key that you use only locally inside the function, where you could simply use the local variable key.
• The read method is very long. Divide the work into smaller functions and call them from read.
• You have docstrings and a fair amount of comments, good. But then you have really cryptic statements such as self.i = 0.
• Some variable names are misleading, such as files which is actually a single filename.
• Don't change the working directory (os.chdir). Use os.path.join(loc, filename) to construct paths. (If you think it's OK to change it, think what happens if you combine this module with some other module that also thinks it's OK)
• This was exactly the sort of stuff I was looking for. Thanks for taking the time to look through it. You hit on a lot of the issues that came up during. Allowing for separate read paths would be really helpful. Also I need to research more on the local vars for a function. I was having problems getting them to work and found it easier to declare a self.xyz. Tan – devfarce Apr 8 '13 at 15:05

Janne's points are good. In addition:

1. When I try running this code, it fails:

>>> e = EasyCSV('.')
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "cr24836.py", line 37, in __init__
for files in os.listdir(loc):
NameError: global name 'loc' is not defined

I presume that loc is a typo for self.loc. This makes me suspicious. Have you actually used or tested this code?

2. The width and skip_rows arguments to the constructor apply to all CSV files in the directory. But isn't it likely that different CSV files will have different widths and need different numbers of rows to be skipped?

3. Your class requires NumPy to be installed (otherwise the line import numpy as np will fail). But since it has a mode of operation that doesn't require NumPy (return lists instead), it would be nice if it worked even if NumPy is not installed. Wait until you're just about to call np.zeros before importing NumPy.

4. location is supposed to be the name of a directory, so name it directory.

5. You write self.key[-4:] == '.CSV' but why not use .endswith like you did earlier in the program? Or better still, since you are testing this twice, write a function:

def filename_is_csv(filename):
"""Return True if filename has the .csv extension."""
_, ext = os.path.splitext(filename)
return ext.lower() == '.csv'

But having said that, do you really want to insist that this can only read CSV files whose names end with .csv? What if someone has CSV stored in a file named foo.data? They'd never be able to read it with your class.

6. There's nothing in the documentation for the class that explains that I am supposed to call the read() method. (If I don't, nothing happens.)

7. There's nothing in the documentation for the class that explains how I am supposed to access the data that has been loaded into memory.

8. If I want to access the data for a filename, I have look up the filename in the Dict attribute to get the index, and then I could look up the index in the allData attribute to get the data. Why this double lookup? Why not have a dictionary that maps filename to data instead of going via an index?

9. There is no need to preallocate arrays in Python. Wait to create the array until you have some data to put in it, and then append each entry to it. Python is not Fortran!

10. In your read() method, you read all the CSV files into memory. This seems wasteful. What if I had hundreds of files but only wanted to read one of them? Why not wait to read a file until the caller needs it?

11. You convert numeric elements to floating-point numbers. This might not be what I want. For example, if I have a file containing:

Apollo,Launch
7,19681011
8,19681221
9,19690303
10,19690518
11,19690716
12,19691114
13,19700411
14,19710131
15,19710726
16,19720416
17,19721207

and then I try to read it, all the data has been wrongly converted to floating-point:

>>> e = EasyCSV('.')
apollo.csv
>>> from pprint import pprint
>>> pprint(e.allData[e.Dict['apollo']])
[['Apollo', 'Launch'],
[7.0, 19681011.0],
[8.0, 19681221.0],
[9.0, 19690303.0],
[10.0, 19690518.0],
[11.0, 19690716.0],
[12.0, 19691114.0],
[13.0, 19700411.0],
[14.0, 19710131.0],
[15.0, 19710726.0],
[16.0, 19720416.0],
[17.0, 19721207.0]]

This can go wrong in other ways. For example, suppose I have a CSV file like this:

product code,inventory
1a0,81
7b4,61
9c2,32
8d3,90
1e9,95
2f4,71

When I read it with your class, look at what happens to the sixth row:

>>> e = EasyCSV('.')


text:
# Contradiction between energy conservation and radiation damping force

We know that if charged particle is accelerated, it will radiate. According to Larmor formula, the power is $$P = {2 \over 3} \frac{q^2 a^2}{ c^3}$$ in the Gauss units, where $a$ is acceleration.

And accelerated particle will experience radiation damping force(Abraham–Lorentz force) $$\mathbf{F}_\mathrm{rad} = { 2 \over 3} \frac{ q^2}{ c^3} \mathbf{\dot{a}}$$

The equation of motion of particle should be $$m \frac{d \mathbf{v}}{dt}= \mathbf{F}_{ext}+ { 2 \over 3} \frac{ q^2}{ c^3} \mathbf{\dot{a}} \tag{1}$$

But a contradiction arises when the external force is a constant force. In this case, the particle has constant acceleration ($\mathbf{\dot a}=0$). But it will still radiate energy ($P\neq0$). The work done by external force $\mathbf{F}_{ext}$ will be totally translated to particle's kinetic energy(due to uniform acceleration), but the particle will still radiate energy to infinity. Does this violate the energy conservation?

Note1: For constant force, the uniform acceleration solution $\mathbf{\dot a}=0$ is still the new equation(1) 's solution. That is, $\dot a=0$ is a solution for $a=b$. Certainly this solution($\dot a=0$ ) can also be the solution of $a= b+d \dot a =b$.

Note2: Even though you take the relativistic effect into consideration, you still cannot explain this contradiction. The relativistic generalization of radiation damping force is in Landau's The classical theory of fields (76.2) $$F_{rad}^\mu= \frac{2 e^2}{3 c}(\frac{d^2 u^{\mu}}{ds^2}-(u^\mu u^\alpha)\frac{d^2 u_\alpha}{ds^2})$$ So we see with constant external force $F_{ext}^\mu=\text{cosntant}$, equation of motion, $$m\frac{du^{\mu}}{ds}=F^{\mu}_{ext}+F^{\mu}_{rad}\tag{2}$$

Constant $4$-acceleration $\frac{du^{\mu}}{ds}=\frac{F^{\mu}_{ext}}{m}$, $\frac{d^2 u^{\mu}}{ds^2}=0$ is still the equation(2)'s solution. So there is still no radiation damping force.

• Your differential equation becomes of the form $a=b+d\dot a$ where $b$ and $d$ are constants. It is a first order linear differential equation. – Farcher Apr 30 '17 at 5:16
• @Farcher But original solution is still new equation's solution. – user153663 Apr 30 '17 at 5:17
• Where did you get the idea that the acceleration is constant.? – Farcher Apr 30 '17 at 5:19
• @Farcher $\dot a=0$ is a solution for $a=b$. Certainly this solution can also be the solution of $a= b+d \dot a =b$ – user153663 Apr 30 '17 at 5:20
• Apparently, this is a hard question: physics.stackexchange.com/q/70915 – Mark H Apr 30 '17 at 11:29

## 1 Answer

The equation (1) is only an approximate equation of motion of a charged body of non-zero dimensions, because the expression

$$\frac{2}{3}\frac{q^2}{c^3}\dot{\mathbf a}$$

only approximates total internal EM force that is acting on such body due to non-rectilinear motion of its parts. Because of this approximation, this equation does not reflect the law of conservation of energy, it only gives approximate description of motion of the body as a whole.

The exact expression for net internal force would be much more complicated, it would involve internal degrees of freedom of the charged body and interaction of its parts. For highly simple models of such charged body (like uniformly charged sphere), the net force can be expressed as infinite series and the above term is just one term of this series.

In an exact model of the charged body where conservation of energy is present and can be proven, the energy that the system radiates is balanced by decrease of internal energy inside and near the charged body. That includes the EM energy in the vicinity of the body.

But if the internal degrees and dimensions of the body are neglected so it becomes a point, this internal energy becomes invisible and the remaining visible degrees of freedom manifest violation of energy conservation. It is just because of simplifying the model of the extended charged body so much, it is not an indication there is something wrong with the general theory.

See also my answer here: Does a constantly accelerating charged particle emit EM radiation or not?