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# Derivative

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The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

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The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function or dependent variable) which is determined by another quantity (the independent variable). It is a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced. The derivative measures the instantaneous rate of change of the function, as distinct from its average rate of change, and is defined as the limit of the average rate of change in the function as the length of the interval on which the average is computed tends to zero.

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. In fact, the derivative at a point of a function of a single variable is the slope of the tangent line to the graph of the function at that point.

The notion of derivative may be generalized to functions of several real variables. The generalized derivative is a linear map called the differential. Its matrix representation is the Jacobian matrix, which reduces to the gradient vector in the case of real-valued function of several variables.

The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.[1]

## Differentiation and derivative

Differentiation is the action of computing a derivative. The derivative of a function f(x) of a variable x is a measure of the rate at which the value of the function changes with respect to the change of the variable. It is called the derivative of f with respect to x. If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point.

The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y divided by x is a line. In this case, y = f(x) = m x + b, for real numbers m and b, and the slope m is given by

${\displaystyle m={\frac {{\text{change in }}y}{{\text{change in }}x}}={\frac {\Delta y}{\Delta x}},}$

where the symbol Δ (Delta) is an abbreviation for "change in." This formula is true because

y + Δy = f(x + Δx) = m (x + Δx) + b = m x + m Δx + b = y + m Δx.

It follows that Δy = m Δx.

This gives an exact value for the slope of a line. If the function f is not linear (i.e. its graph is not a line), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

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The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences Δy / Δx as Δx becomes infinitely small.

### Notation

Two distinct notations are commonly used for the derivative, one deriving from Leibniz and the other from Joseph Louis Lagrange.

In Leibniz's notation, an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written

${\displaystyle {\frac {dy}{dx}}\,\!}$

suggesting the ratio of two infinitesimal quantities. (The above expression is read as "the derivative of y with respect to x", "d y by d x", or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion.)

In Lagrange's notation, the derivative with respect to x of a function f(x) is denoted fTemplate:'(x) (read as "f prime of x") or fxTemplate:'(x) (read as "f prime x of x"), in case of ambiguity of the variable implied by the derivation. Lagrange's notation is sometimes incorrectly attributed to Newton.

### Rigorous definition

The most common approach to turn this intuitive idea into a precise definition is to define the derivative as a limit of difference quotients of real numbers.[2] This is the approach described below.

Let f be a real valued function defined in an open neighborhood of a real number a. In classical geometry, the tangent line to the graph of the function f at a was the unique line through the point (a, f(a)) that did not meet the graph of f transversally, meaning that the line did not pass straight through the graph. The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at (a, f(a)). The slope of the tangent line is very close to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)). These lines are called secant lines. A value of h close to zero gives a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. The slope m of the secant line is the difference between the y values of these points divided by the difference between the x values, that is,

${\displaystyle m={\frac {\Delta f(a)}{\Delta a}}={\frac {f(a+h)-f(a)}{(a+h)-(a)}}={\frac {f(a+h)-f(a)}{h}}.}$

This expression is Newton's difference quotient. Passing from an approximation to an exact answer is done using a limit. Geometrically, the limit of the secant lines is the tangent line. Therefore, the limit of the difference quotient as h approaches zero, if it exists, should represent the slope of the tangent line to (a, f(a)). This limit is defined to be the derivative of the function f at a:

${\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}.}$

When the limit exists, f is said to be differentiable at a. Here f′ (a) is one of several common notations for the derivative (see below).

Equivalently, the derivative satisfies the property that

${\displaystyle \lim _{h\to 0}{\frac {f(a+h)-f(a)-f'(a)\cdot h}{h}}=0,}$

which has the intuitive interpretation (see Figure 1) that the tangent line to f at a gives the best linear approximation

${\displaystyle f(a+h)\approx f(a)+f'(a)h}$

to f near a (i.e., for small h). This interpretation is the easiest to generalize to other settings (see below).

Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be found directly using this method. Instead, define Q(h) to be the difference quotient as a function of h:

${\displaystyle Q(h)={\frac {f(a+h)-f(a)}{h}}.}$

Q(h) is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from h = 0. If the limit limh→0Q(h) exists, meaning that there is a way of choosing a value for Q(0) that makes Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q(0).

In practice, the existence of a continuous extension of the difference quotient Q(h) to h = 0 is shown by modifying the numerator to cancel h in the denominator. Such manipulations can make the limit value of Q for small h clear even though Q is still not defined at h = 0. This process can be long and tedious for complicated functions, and many shortcuts are commonly used to simplify the process.

### Definition over the hyperreals

#### Moment of the fluent

Newton called o the moment of the fluent. The moment of the fluent represents the infinitely small part by which a fluent was increased in a small time interval. Once he allowed himself to divide through by o (although o can not be treated as zero because that would make the division illegitimate). Newton decided it was justifiable to drop all terms containing o.

### Euler's notation

Euler's notation uses a differential operator D, which is applied to a function f to give the first derivative Df. The second derivative is denoted D2f, and the nth derivative is denoted Dnf.

If y = f(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written

${\displaystyle D_{x}y\,}$   or   ${\displaystyle D_{x}f(x)\,}$,

although this subscript is often omitted when the variable x is understood, for instance when this is the only variable present in the expression.

Euler's notation is useful for stating and solving linear differential equations.

## Rules of computation

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones.

### Rules for basic functions

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Most derivative computations eventually require taking the derivative of some common functions. The following incomplete list gives some of the most frequently used functions of a single real variable and their derivatives.

${\displaystyle f(x)=x^{r},}$

where r is any real number, then

${\displaystyle f'(x)=rx^{r-1},}$

wherever this function is defined. For example, if ${\displaystyle f(x)=x^{1/4}}$, then

${\displaystyle f'(x)=(1/4)x^{-3/4},}$

and the derivative function is defined only for positive x, not for x = 0. When r = 0, this rule implies that f′(x) is zero for x ≠ 0, which is almost the constant rule (stated below).

${\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}$
${\displaystyle {\frac {d}{dx}}a^{x}=\ln(a)a^{x}.}$
${\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}},\qquad x>0.}$
${\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}}.}$
${\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x).}$
${\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x).}$
${\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x).}$
${\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}},-1
${\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}},-1
${\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}}$

### {{safesubst:#invoke:anchor|main}}Rules for combined functions

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In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided using differentiation rules. Some of the most basic rules are the following.

• Constant rule: if f(x) is constant, then
${\displaystyle f'=0.\,}$
${\displaystyle (\alpha f+\beta g)'=\alpha f'+\beta g'\,}$ for all functions f and g and all real numbers ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.
${\displaystyle (fg)'=f'g+fg'\,}$ for all functions f and g. By extension, this means that the derivative of a constant times a function is the constant times the derivative of the function: ${\displaystyle {\frac {d}{dr}}\pi r^{2}=2\pi r.\,}$
${\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-fg'}{g^{2}}}}$ for all functions f and g at all inputs where g ≠ 0.
${\displaystyle f'(x)=h'(g(x))\cdot g'(x).\,}$

### Computation example

The derivative of

${\displaystyle f(x)=x^{4}+\sin(x^{2})-\ln(x)e^{x}+7\,}$

is

{\displaystyle {\begin{aligned}f'(x)&=4x^{(4-1)}+{\frac {d\left(x^{2}\right)}{dx}}\cos(x^{2})-{\frac {d\left(\ln {x}\right)}{dx}}e^{x}-\ln {x}{\frac {d\left(e^{x}\right)}{dx}}+0\\&=4x^{3}+2x\cos(x^{2})-{\frac {1}{x}}e^{x}-\ln(x)e^{x}.\end{aligned}}}

Here the second term was computed using the chain rule and third using the product rule. The known derivatives of the elementary functions x2, x4, sin(x), ln(x) and exp(x) = ex, as well as the constant 7, were also used.

## Derivatives in higher dimensions

### Derivatives of vector valued functions

A vector-valued function y(t) of a real variable sends real numbers to vectors in some vector space Rn. A vector-valued function can be split up into its coordinate functions y1(t), y2(t), …, yn(t), meaning that y(t) = (y1(t), ..., yn(t)). This includes, for example, parametric curves in R2 or R3. The coordinate functions are real valued functions, so the above definition of derivative applies to them. The derivative of y(t) is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is,

${\displaystyle \mathbf {y} '(t)=(y'_{1}(t),\ldots ,y'_{n}(t)).}$

Equivalently,

${\displaystyle \mathbf {y} '(t)=\lim _{h\to 0}{\frac {\mathbf {y} (t+h)-\mathbf {y} (t)}{h}},}$

if the limit exists. The subtraction in the numerator is subtraction of vectors, not scalars. If the derivative of y exists for every value of t, then y′ is another vector valued function.

If e1, …, en is the standard basis for Rn, then y(t) can also be written as y1(t)e1 + … + yn(t)en. If we assume that the derivative of a vector-valued function retains the linearity property, then the derivative of y(t) must be

${\displaystyle y'_{1}(t)\mathbf {e} _{1}+\cdots +y'_{n}(t)\mathbf {e} _{n}}$

because each of the basis vectors is a constant.

This generalization is useful, for example, if y(t) is the position vector of a particle at time t; then the derivative y′(t) is the velocity vector of the particle at time t.

### Partial derivatives

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Suppose that f is a function that depends on more than one variable. For instance,

${\displaystyle f(x,y)=x^{2}+xy+y^{2}.\,}$

f can be reinterpreted as a family of functions of one variable indexed by the other variables:

${\displaystyle f(x,y)=f_{x}(y)=x^{2}+xy+y^{2}.\,}$

In other words, every value of x chooses a function, denoted fx, which is a function of one real number.[9] That is,

${\displaystyle x\mapsto f_{x},\,}$
${\displaystyle f_{x}(y)=x^{2}+xy+y^{2}.\,}$

Once a value of x is chosen, say a, then f(x, y) determines a function fa that sends y to a2 + ay + y2:

${\displaystyle f_{a}(y)=a^{2}+ay+y^{2}.\,}$

In this expression, a is a constant, not a variable, so fa is a function of only one real variable. Consequently the definition of the derivative for a function of one variable applies:

${\displaystyle f_{a}'(y)=a+2y.\,}$

The above procedure can be performed for any choice of a. Assembling the derivatives together into a function gives a function that describes the variation of f in the y direction:

${\displaystyle {\frac {\partial f}{\partial y}}(x,y)=x+2y.}$

This is the partial derivative of f with respect to y. Here is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".

In general, the partial derivative of a function f(x1, …, xn) in the direction xi at the point (a1 …, an) is defined to be:

${\displaystyle {\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i}+h,\ldots ,a_{n})-f(a_{1},\ldots ,a_{i},\ldots ,a_{n})}{h}}.}$

In the above difference quotient, all the variables except xi are held fixed. That choice of fixed values determines a function of one variable

${\displaystyle f_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}(x_{i})=f(a_{1},\ldots ,a_{i-1},x_{i},a_{i+1},\ldots ,a_{n}),}$

and, by definition,

${\displaystyle {\frac {df_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}}{dx_{i}}}(a_{i})={\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n}).}$

In other words, the different choices of a index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.

An important example of a function of several variables is the case of a scalar-valued function f(x1, ..., xn) on a domain in Euclidean space Rn (e.g., on R2 or R3). In this case f has a partial derivative ∂f/∂xj with respect to each variable xj. At the point a, these partial derivatives define the vector

${\displaystyle \nabla f(a)=\left({\frac {\partial f}{\partial x_{1}}}(a),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a)\right).}$

This vector is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f that takes the point a to the vector ∇f(a). Consequently the gradient determines a vector field.

### Directional derivatives

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If f is a real-valued function on Rn, then the partial derivatives of f measure its variation in the direction of the coordinate axes. For example, if f is a function of x and y, then its partial derivatives measure the variation in f in the x direction and the y direction. They do not, however, directly measure the variation of f in any other direction, such as along the diagonal line y = x. These are measured using directional derivatives. Choose a vector

${\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n}).}$

The directional derivative of f in the direction of v at the point x is the limit

${\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\rightarrow 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.}$

In some cases it may be easier to compute or estimate the directional derivative after changing the length of the vector. Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector. To see how this works, suppose that v = λu. Substitute h = k into the difference quotient. The difference quotient becomes:

${\displaystyle {\frac {f(\mathbf {x} +(k/\lambda )(\lambda \mathbf {u} ))-f(\mathbf {x} )}{k/\lambda }}=\lambda \cdot {\frac {f(\mathbf {x} +k\mathbf {u} )-f(\mathbf {x} )}{k}}.}$

This is λ times the difference quotient for the directional derivative of f with respect to u. Furthermore, taking the limit as h tends to zero is the same as taking the limit as k tends to zero because h and k are multiples of each other. Therefore Dv(f) = λDu(f). Because of this rescaling property, directional derivatives are frequently considered only for unit vectors.

If all the partial derivatives of f exist and are continuous at x, then they determine the directional derivative of f in the direction v by the formula:

${\displaystyle D_{\mathbf {v} }{f}({\boldsymbol {x}})=\sum _{j=1}^{n}v_{j}{\frac {\partial f}{\partial x_{j}}}.}$

This is a consequence of the definition of the total derivative. It follows that the directional derivative is linear in v, meaning that Dv + w(f) = Dv(f) + Dw(f).

The same definition also works when f is a function with values in Rm. The above definition is applied to each component of the vectors. In this case, the directional derivative is a vector in Rm.

### Total derivative, total differential and Jacobian matrix

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When f is a function from an open subset of Rn to Rm, then the directional derivative of f in a chosen direction is the best linear approximation to f at that point and in that direction. But when n > 1, no single directional derivative can give a complete picture of the behavior of f. The total derivative, also called the (total) differential, gives a complete picture by considering all directions at once. That is, for any vector v starting at a, the linear approximation formula holds:

${\displaystyle f(\mathbf {a} +\mathbf {v} )\approx f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {v} .}$

Just like the single-variable derivative, f ′(a) is chosen so that the error in this approximation is as small as possible.

If n and m are both one, then the derivative f ′(a) is a number and the expression f ′(a)v is the product of two numbers. But in higher dimensions, it is impossible for f ′(a) to be a number. If it were a number, then f ′(a)v would be a vector in Rn while the other terms would be vectors in Rm, and therefore the formula would not make sense. For the linear approximation formula to make sense, f ′(a) must be a function that sends vectors in Rn to vectors in Rm, and f ′(a)v must denote this function evaluated at v.

To determine what kind of function it is, notice that the linear approximation formula can be rewritten as

${\displaystyle f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} )\approx f'(\mathbf {a} )\mathbf {v} .}$

Notice that if we choose another vector w, then this approximate equation determines another approximate equation by substituting w for v. It determines a third approximate equation by substituting both w for v and a + v for a. By subtracting these two new equations, we get

${\displaystyle f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} +\mathbf {w} )+f(\mathbf {a} )\approx f'(\mathbf {a} +\mathbf {v} )\mathbf {w} -f'(\mathbf {a} )\mathbf {w} .}$

If we assume that v is small and that the derivative varies continuously in a, then f ′(a + v) is approximately equal to f ′(a), and therefore the right-hand side is approximately zero. The left-hand side can be rewritten in a different way using the linear approximation formula with v + w substituted for v. The linear approximation formula implies:

{\displaystyle {\begin{aligned}0&\approx f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} +\mathbf {w} )+f(\mathbf {a} )\\&=(f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} ))-(f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} ))-(f(\mathbf {a} +\mathbf {w} )-f(\mathbf {a} ))\\&\approx f'(\mathbf {a} )(\mathbf {v} +\mathbf {w} )-f'(\mathbf {a} )\mathbf {v} -f'(\mathbf {a} )\mathbf {w} .\end{aligned}}}

This suggests that f ′(a) is a linear transformation from the vector space Rn to the vector space Rm. In fact, it is possible to make this a precise derivation by measuring the error in the approximations. Assume that the error in these linear approximation formula is bounded by a constant times ||v||, where the constant is independent of v but depends continuously on a. Then, after adding an appropriate error term, all of the above approximate equalities can be rephrased as inequalities. In particular, f ′(a) is a linear transformation up to a small error term. In the limit as v and w tend to zero, it must therefore be a linear transformation. Since we define the total derivative by taking a limit as v goes to zero, f ′(a) must be a linear transformation.

In one variable, the fact that the derivative is the best linear approximation is expressed by the fact that it is the limit of difference quotients. However, the usual difference quotient does not make sense in higher dimensions because it is not usually possible to divide vectors. In particular, the numerator and denominator of the difference quotient are not even in the same vector space: The numerator lies in the codomain Rm while the denominator lies in the domain Rn. Furthermore, the derivative is a linear transformation, a different type of object from both the numerator and denominator. To make precise the idea that f ′(a) is the best linear approximation, it is necessary to adapt a different formula for the one-variable derivative in which these problems disappear. If f : RR, then the usual definition of the derivative may be manipulated to show that the derivative of f at a is the unique number f ′(a) such that

${\displaystyle \lim _{h\to 0}{\frac {f(a+h)-f(a)-f'(a)h}{h}}=0.}$

This is equivalent to

${\displaystyle \lim _{h\to 0}{\frac {|f(a+h)-f(a)-f'(a)h|}{|h|}}=0}$

because the limit of a function tends to zero if and only if the limit of the absolute value of the function tends to zero. This last formula can be adapted to the many-variable situation by replacing the absolute values with norms.

The definition of the total derivative of f at a, therefore, is that it is the unique linear transformation f ′(a) : RnRm such that

${\displaystyle \lim _{\mathbf {h} \to 0}{\frac {\lVert f(\mathbf {a} +\mathbf {h} )-f(\mathbf {a} )-f'(\mathbf {a} )\mathbf {h} \rVert }{\lVert \mathbf {h} \rVert }}=0.}$

Here h is a vector in Rn, so the norm in the denominator is the standard length on Rn. However, f′(a)h is a vector in Rm, and the norm in the numerator is the standard length on Rm. If v is a vector starting at a, then f ′(a)v is called the pushforward of v by f and is sometimes written fv.

If the total derivative exists at a, then all the partial derivatives and directional derivatives of f exist at a, and for all v, f ′(a)v is the directional derivative of f in the direction v. If we write f using coordinate functions, so that f = (f1, f2, ..., fm), then the total derivative can be expressed using the partial derivatives as a matrix. This matrix is called the Jacobian matrix of f at a:

${\displaystyle f'(\mathbf {a} )=\operatorname {Jac} _{\mathbf {a} }=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{ij}.}$

The existence of the total derivative f′(a) is strictly stronger than the existence of all the partial derivatives, but if the partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and depends continuously on a.

The definition of the total derivative subsumes the definition of the derivative in one variable. That is, if f is a real-valued function of a real variable, then the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a 1×1 matrix whose only entry is the derivative f′(x). This 1×1 matrix satisfies the property that f(a + h) − f(a) − f ′(a)h is approximately zero, in other words that

${\displaystyle f(a+h)\approx f(a)+f'(a)h.}$

Up to changing variables, this is the statement that the function ${\displaystyle x\mapsto f(a)+f'(a)(x-a)}$ is the best linear approximation to f at a.

The total derivative of a function does not give another function in the same way as the one-variable case. This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the tangent bundle of the source to the tangent bundle of the target.

The natural analog of second, third, and higher-order total derivatives is not a linear transformation, is not a function on the tangent bundle, and is not built by repeatedly taking the total derivative. The analog of a higher-order derivative, called a jet, cannot be a linear transformation because higher-order derivatives reflect subtle geometric information, such as concavity, which cannot be described in terms of linear data such as vectors. It cannot be a function on the tangent bundle because the tangent bundle only has room for the base space and the directional derivatives. Because jets capture higher-order information, they take as arguments additional coordinates representing higher-order changes in direction. The space determined by these additional coordinates is called the jet bundle. The relation between the total derivative and the partial derivatives of a function is paralleled in the relation between the kth order jet of a function and its partial derivatives of order less than or equal to k.

By repeatedly taking the total derivative, one obtains higher versions of the Fréchet derivative, specialized to Rp. The kth order total derivative may be interpreted as a map

${\displaystyle D^{k}f:\mathbb {R} ^{n}\to L^{k}(\mathbb {R} ^{n}\times \cdots \times \mathbb {R} ^{n},\mathbb {R} ^{m})}$

which takes a point x in Rn and assigns to it an element of the space of k-linear maps from Rn to Rm – the "best" (in a certain precise sense) k-linear approximation to f at that point. By precomposing it with the diagonal map Δ, x → (x, x), a generalized Taylor series may be begun as

{\displaystyle {\begin{aligned}f(\mathbf {x} )&\approx f(\mathbf {a} )+(Df)(\mathbf {x} )+(D^{2}f)(\Delta (\mathbf {x-a} ))+\cdots \\&=f(\mathbf {a} )+(Df)(\mathbf {x-a} )+(D^{2}f)(\mathbf {x-a} ,\mathbf {x-a} )+\cdots \\&=f(\mathbf {a} )+\sum _{i}(Df)_{i}(\mathbf {x-a} )^{i}+\sum _{j,k}(D^{2}f)_{jk}(\mathbf {x-a} )^{j}(\mathbf {x-a} )^{k}+\cdots \end{aligned}}}

where f(a) is identified with a constant function, (xa)i are the components of the vector xa, and (D f)i and (D2 f)j k are the components of D f and D2 f as linear transformations.

## Generalizations

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The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a linear approximation of the function at that point.

• An important generalization of the derivative concerns complex functions of complex variables, such as functions from (a domain in) the complex numbers C to C. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. If C is identified with R2 by writing a complex number z as x + i y, then a differentiable function from C to C is certainly differentiable as a function from R2 to R2 (in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists if the real derivative is complex linear and this imposes relations between the partial derivatives called the Cauchy Riemann equations – see holomorphic functions.
• Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold M is a space that can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R3. The derivative (or differential) of a (differentiable) map f: MN between manifolds, at a point x in M, is then a linear map from the tangent space of M at x to the tangent space of N at f(x). The derivative function becomes a map between the tangent bundles of M and N. This definition is fundamental in differential geometry and has many uses – see pushforward (differential) and pullback (differential geometry).
• Differentiation can also be defined for maps between infinite dimensional vector spaces such as Banach spaces and Fréchet spaces. There is a generalization both of the directional derivative, called the Gâteaux derivative, and of the differential, called the Fréchet derivative.
• One deficiency of the classical derivative is that not very many functions are differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all continuous functions and many other functions can be differentiated using a concept known as the weak derivative. The idea is to embed the continuous functions in a larger space called the space of distributions and only require that a function is differentiable "on average".
• The properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology — see, for example, differential algebra.
• The discrete equivalent of differentiation is finite differences. The study of differential calculus is unified with the calculus of finite differences in time scale calculus.
• Also see arithmetic derivative.

## History

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The creation of Calculus was one of the greatest achievements of the 1600s, but the inventor of calculus is widely disputed: Was it Isaac Newton or Gottfried Wilhelm Leibniz? When Isaac Newton and Gottfried Wilhelm Leibniz first formulated differential calculus they effectively made use of the concept of an infinitesimal, which they referred to as an infinitely small number. At that time before non-standard analysis, the concept of infinitesimals was very fuzzy and bothered many mathematicians. However, the concept of infinitesimals was essential to the development of differential calculus.[10]

Newton's method involved taking ratios of infinitesimals. Those terms for the ratio that which had an infinitesimal as a factor were treated as zero and thus the product of infinitesimals is equal to zero. He explained it, “terms which have [an infinitesimal] as a factor will be equivalent to nothing in respect to the others. I therefore cast them out…” Ultimately Cauchy, Weierstrass, and Riemann, reformulated Calculus in terms of limits rather than infinitesimals. Therefore the need for these infinitely small (and nonexistent) quantities was removed, and replaced by a notion of quantities being "close" to others. So the derivative and the integral were both reformulated in terms of limits.[11]

In the nineteenth century the German mathematician Karl Weierstrass introduced the epsilon-delta process, which provided a rigorous basis for Calculus and discouraged students from using the infinitesimal concept. Then in 1960 Abraham Robinson found a way to provide a foundation for infinitesimals and thus infinitesimals were acceptable. Robinson called his formulation non-standard analysis. The purpose of this material is to explain, illustrate and justify the non-standard analysis formulation of infinitesimals.[12]

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## Notes

1. Differential calculus, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Apostol 1967, Apostol 1969, and Spivak 1994.
2. Spivak 1994, chapter 10.
3. Despite this, it is still possible to take the derivative in the sense of distributions. The result is nine times the Dirac measure centered at a.
4. {{#invoke:citation/CS1|citation |CitationClass=citation }}. Cited by {{#invoke:citation/CS1|citation |CitationClass=citation }}
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6. In the formulation of calculus in terms of limits, the du symbol has been assigned various meanings by various authors. Some authors do not assign a meaning to du by itself, but only as part of the symbol du/dx. Others define dx as an independent variable, and define du by du = dx·f′(x). In non-standard analysis du is defined as an infinitesimal. It is also interpreted as the exterior derivative of a function u. See differential (infinitesimal) for further information.
7. Template:Cite web
8. Burton, D. (2011). Gottfried Leibniz: The Calculus Controversy. The History of Mathematics: An Introduction (). New York, NY: McGraw-Hill.
9. This can also be expressed as the adjointness between the product space and function space constructions.
10. Watkins, T. (n.d.). The Calculus of Infinitesimals. The Calculus of Infinitesimals. Retrieved April 17, 2014, from http://www.sjsu.edu/faculty/watkins/infincalc.htm
11. Tomforde, M. (n.d.). The History of Calculus. The History of Calculus. Retrieved April 17, 2014, from http://www.uiowa.edu/~c22m025c/history.html
12. Watkins, T. (n.d.). The Calculus of Infinitesimals. The Calculus of Infinitesimals. Retrieved April 17, 2014, from http://www.sjsu.edu/faculty/watkins/infincalc.htm

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Short and Sweet Calculus

## 2.2One-sided limits

Consider the function $$F(x)$$ whose graph is shown in Figure 2.2. If we take $$x$$ values closer and closer to 2.5, but less than 2.5, $$F(x)$$ gets closer and closer to 5. In other words, when $$x$$ approaches 2.5 through the values less than 2.5, $$F(x)$$ approaches 5. We express this by saying that “the limit of $$F(x)$$ as $$x$$ approaches 2.5 from the left is 5” or “the left-hand limit of $$F(x)$$ as $$x$$ approaches 2.5 is 5.” The notation for this is

$\lim_{x\to2.5^{-}}F(x)=5$ The minus sign that is written after 2.5 means $$x$$ approaches 2.5 from the left.

Now consider the case in which $$x$$ takes on the values close to 2.5 but larger than 2.5. As $$x$$ approaches 2.5 from the right, $$F(x)$$ approaches 2. Symbolically we write $\lim_{x\to2.5^{+}}F(x)=2,$ and say “the limit of $$F(x)$$ as $$x$$ approaches 2.5 from the right is 2” or “the right-hand limit of $$F(x)$$ as $$x$$ approaches 2.5 is 2.”

In this example, $$F(x)$$ is defined at $$x=2.5$$, but the value of $$F(2.5)$$ has no bearing on the left-hand or right-hand limit of $$F(x)$$. Even if we remove $$x=2.5$$ from the domain of $$F(x)$$ (that is, if $$F(x)$$ were not defined at $$x=2.5$$), the left-hand and right-hand limits would remain the same.

2.2. (left-hand limit) If we can make the values of $$f(x)$$ as close as we please to a number $$L$$ by taking $$x$$ sufficiently close to $$a$$ with $$\boldsymbol{x<a}$$, we say “the limit of $$f(x)$$ as $$x$$ approaches $$a$$ from the left is $$L$$” or “the left-hand limit of $$f(x)$$ as $$x$$ approaches $$a$$ is $$L$$” and write

$\lim_{x\to a^{-}}f(x)=L.$

Similarly

2.3. (right-hand limit) If we can make the values of $$f(x)$$ as close as we please to a number $$L$$ by taking $$x$$ sufficiently close to $$a$$ with $$\boldsymbol{x>a}$$, we say “the limit of $$f(x)$$ as $$x$$ approaches $$a$$ from the left is $$L$$” or “the left-hand limit of $$f(x)$$ as $$x$$ approaches $$a$$ is $$L$$” and write

$\lim_{x\to a^{-}}f(x)=L.$

Example 2.1. Consider the function $$y=\text{sgn}(x)$$ defined by ${\rm sgn}(x)=\left\{ \begin{tabular}{ll} \ensuremath{1} & if \ensuremath{x>0}\\ 0 & if \ensuremath{x=0}\\ \ensuremath{-1} & if \ensuremath{x<0} \end{tabular}\right.$ (a) Determine $${\displaystyle \lim_{x\to0^{-}}\text{sgn}(x)}$$. (b) Determine $${\displaystyle \lim_{x\to0^{+}}\text{sgn}(x)}$$.

Solution

The graph of $$y=\text{sgn}(x)$$ is shown below. (a) When $$x$$ is any negative number, the value of $$\text{sgn}(x)$$ is $$-1$$. Therefore $${\displaystyle \lim_{x\to0^{-}}\text{sgn}(x)}=-1$$.
(b) When $$x$$ is positive, the value of $$\text{sgn}(x)$$ is $$1$$. Therefore, $${\displaystyle \lim_{x\to0^{+}}\text{sgn}(x)=1}$$.

By comparing the definitions of one-sided limits and regular (or two-sided) limits, we see the following is true.

2.1. $${\displaystyle \lim_{x\to a}f(x)}$$ exists and is equal to $$L$$ if and only if $${\displaystyle \lim_{x\to a^{-}}f(x)}$$ and $${\displaystyle \lim_{x\to a^{+}}f(x)}$$ both exist and are equal to $$L$$. That is, $\lim_{x\to a}f(x)=L\quad\Longleftrightarrow\quad\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)=L.$

For instance, in the above example$${\displaystyle \lim_{x\to0^{-}}\text{sgn}(x)}\neq{\displaystyle \lim_{x\to0^{+}}\text{sgn}(x)}$$, so $${\displaystyle \lim_{x\to0}\text{sgn}(x)}$$ does not exist.

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## A Conjecture on the Collatz-Kakutani Path Length for the Mersenne Primes

Abstract

We present here a new conjecture for the nature of the Mersenne prime numbers by connecting it with the Collatz-Kakutani problem. By introducing a natural path length on the basis of the Collatz-Kakutani tree, we conjecture that this path length of a Mersenne prime from the root of the Collatz-Kakutani tree is approximately proportional to the index of the Mersenne prime. We also discuss diﬀerence of behaviors between Mersenne numbers and Mersenne primes.

### 1 Introduction

Prime numbers have attracted mathematically oriented minds. They are fountains of the interesting problems which are left unresolved to the present. As well as prime numbers, natural numbers themselves sometimes show unexpected behaviors in spite of their simple appeal. In this note, we would like to present a simple property arising from the combination of two unsolved problems in number theory; the Mersenne prime numbers and the Collatz-Kakutani conjecture. Namely, we show that there exists an approximate linear relation between an index of the Mersenne prime and its Collatz-Kakutani path length, both of which are deﬁned in the next section.

### 2 Mersenne primes and Collatz-Kakutani Conjecuture

Let us start with a brief descriptions of the Mersenne primes and the Collatz-Kakutani conjecture[1]. A Mersenne number is a positive integer given by

${M}_{n}={2}^{n}-1,$

where $n$ is a positive integer, called “index”. A Mersenne prime is a Mersenne number that is prime. It has been shown that if ${M}_{n}$ is a Mersenne prime, $n$ must be prime. However, the converse is not true. In fact, only forty-seven Mersenne primes have been found up to date with the largest one given when $n=43,112,609$. This is also the largest known prime number. There are many fundamental questions left unresolved such as whether there are inﬁnitely many Mersenne primes. The behaviors of Mersenne primes are irregular and unpredictable as seen in other types of prime numbers. Therefore, the recent discoveries of the Mersenne primes were achieved with the help of massively distributed computers, and the eﬀort for ﬁnding new Mersenne primes has been made continuously [2]. The Mersenne primes are known for their relationships with the perfect numbers, and they are also applied to create pseudorandom number generators [3].

The Collatz-Kakutani conjecture is also one of the unsolved problems in number theory. This conjecture is a halting problem that the following operations will stop for an arbitrary positive number. Consider a positive integer $X$.

If $X$ is odd, $3X+1$ is the next integer.

If $X$ is even, $X∕2$ is the next integer.

If we repeat this process, it will eventually reach the halting state $X=1$ for all positive integers. For example, if we start from $X=7$, we have a sequence as

$7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1.$

This process can be visualized as the Collatz-Kakutani tree as shown in Fig. 1A. The conjecture states that this tree covers all the positive integers. Even though this conjecture is unsolved, it is generally believed as true from arguments through probability theories and through computational veriﬁcations.

Let us introduce the “Collatz-Kakutani path length” $D\left(X\right)$ for a number of steps for $X$ to reach $X=1$ through the above operations, i.e., a number of operational steps needed to reach $1$ (“the root”) on the Collatz-Kakutani tree. For example, $D\left(7\right)=16$. The relationship between $X$ and $D\left(X\right)$ is highly irregular and no simple law for the path length is found1. As a result, the plot of $D\left(X\right)$ versus $X$ produces a irregular graph as shown in Fig. 1B. While the general relationship between $X$ and $D\left(X\right)$ is unknown, there are some trivial relations such as $D\left({2}^{n}\right)=n$. Note that, the number ${2}^{n}$ is just one larger than the Mersenne number ${M}_{n}={2}^{n}-1$.

### 3 Main Results

Our main ﬁnding to report is the fact that a path length of a Mersenne prime is approximately proportional to its index for large n, namely,

 $D\left({M}_{n}\right)\approx 13.45n.$ (1)

This is shown in Fig. 2, which we were able to compute up to the 38th Mersenne prime, ${M}_{p}\left(38\right)={M}_{6972593}$. The behavior of the path length $D$ is non monotonic for small indices, e.g., $D\left({M}_{89}\right)>D\left({M}_{107}\right)$. We expect, however, beyond $n=107$, the path length increases monotonically with $n$ for the Mersenne primes. Also, there seems no regular path to reach the root of the tree, $X=1$. For example, we have examined the path for two nearby Mersenne primes, ${M}_{p}\left(16\right)={M}_{2203}$ and ${M}_{p}\left(17\right)={M}_{2281}$. While they have the path length $D$ of approximately $30,000$, they share the last 16 steps, namely,

$22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,$ and $1$.

The relations between indices of the Mersenne primes and their path lengths are summarized in Table 1.

 Number Indices $D\left(X\right)$ $D\left(X\right)∕n$ 1 2 7 3.5 2 3 16 5.33333 3 5 106 21.2 4 7 46 6.57143 5 13 158 12.1538 6 17 224 13.1765 7 19 177 9.31579 8 31 450 14.5161 9 61 860 14.0984 10 89 1454 16.3371 11 107 1441 13.4673 12 127 1660 13.0709 13 521 6769 12.9923 14 607 8494 13.9934 15 1279 17094 13.3651 16 2203 29821 13.5365 17 2281 30734 13.4739 18 3217 43478 13.5151 19 4253 55906 13.1451 20 4423 60716 13.7273 21 9689 129608 13.3768 22 9941 134345 13.5142 23 11213 153505 13.6899 24 19937 265860 13.335 25 21701 293161 13.5091 26 23209 312164 13.4501 27 44497 598067 13.4406 28 86243 1158876 13.4373 29 110503 1482529 13.4162 30 132049 1771117 13.4126 31 216091 2906179 13.4489 32 756839 10197081 13.4732 33 859433 11568589 13.4607 34 1257787 16927967 13.4585 35 1398269 18807193 13.4503 36 2976221 40055567 13.4585 37 3021377 40663017 13.4584 38 6972593 93778449 13.4496

Table 1: Path lengths of the Mersenne primes.

This linear behavior can be understood by the following heuristic arguments. Suppose $N$ is a large random number. The standard heuristic arguments for the path length for $N$ gives

 $D\left(N\right)\approx \left(3∕\left(ln4∕3\right)\right)lnN.$ (2)

A Mersenne number ${2}^{n}-1$ becomes $3×{2}^{n-1}-1$ after two steps in Collatz-Katutani tree. Therefore, A Mersenne Number ${2}^{n}-1$ becomes ${3}^{n}-1$ after $2n$ steps. If we consider the number $N={3}^{n}-1\sim {3}^{n}$ to be a large random number for large $n$, then the path length of the number is given by

 $D\left({3}^{n}-1\right)\approx \left(3∕\left(ln4∕3\right)\right)nln3.$ (3)

Finally, we obtain the heuristic estimation of the path length of the Mersenne number to be

 $D\left({2}^{n}-1\right)\approx 2n+\left(3∕\left(ln4∕3\right)\right)nln3\approx 13.45652n,$ (4)

which is consistent with our numerical results [456]. This arguments implies that our ﬁnding is true not only for the Mersenne primes, but also for the Mersenne numbers. Our main conjecture, however, is that this linear relationship for the Mersenne primes are better than that for the general Mersenne numbers, or other general sequence of integers.

In order to show the diﬀerence between the Mersenne numbers in general and the Mersenne primes, the ratio of the path length to the index, $D\left({M}_{n}\right)∕n$, is plotted against $n$ in Fig. 2B. If the distance increases linearly to the index, then $D\left({M}_{n}\right)∕n$ becomes a constant. Figure 2B shows that the Mersenne primes show better linearity than the general Mersenne numbers.

To be more quantitative, we have performed the following statistical analyses. We have chosen thirteen Mersenne primes from the $2{6}^{\text{th}}$ (${M}_{p}\left(26\right)={M}_{23,209}$) to the $3{8}^{\text{th}}$ (${M}_{p}\left(38\right)={M}_{6,972,593}$), and computed the mean and the variance of $D\left({M}_{n}\right)∕n$. Let us call the prime indices of the Mersenne primes as the Mersenne prime indices. We compare the result with that from the following set of thirteen points approximately in the same interval.

A) Thirteen prime indices which is the next smallest to thirteen Mersenne prime indices. We note the Mersenne numbers associated with this set is not prime numbers themselves.

B) Thirteen indices which are near the mid-point of two successive Mersenne prime indices.

C) Thirteen indices which are twice the Mersenne prime indices. These are all even numbers. We have selected some which are close to the Mersenne prime indices.

D) Thirteen indices which is on the least squre best-ﬁt line based on the heuristics that, given the $k$-th Mersenne prime ${M}_{p}\left(k\right)$, the plot of ${log}_{2}\left({log}_{2}\left({M}_{p}\left(k\right)\right)\right)$ versus $k$ lies approximately on the straight line (Figure 3). We have selected some which are close to the Mersenne prime indices [7].

The results of these comparison are shown in Table 2.

 Set Indices $D\left(X\right)$ E$\left[D\left(X\right)∕n\right]$ Var[$D\left(X\right)∕n$] Mersenne Prime 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593 312164, 598067, 1158876, 1482529, 1771117, 906179, 10197081, 11568589, 16927967, 18807193, 40055567, 40663017, 93778449 13.4473 0.0002977 A 23227, 44501, 86249, 110527, 132059, 216103, 756853, 859447, 1257827, 1398281, 2976229, 3021407, 6972607 312182, 598071, 1158882, 1482553, 1771127, 2906191, 10197095, 11568603, 16928007, 18807205, 40055575, 40663047, 93778463 13.4460 0.0003194 B 22455, 33853, 65370, 98373, 121276, 174070, 486465, 808136, 1058610, 1328028, 2187245, 2998799, 4996985 299801, 457438, 875438, 1327329, 1633743, 2344640, 6524449, 10868120, 14246657, 17876449, 29428265, 40364153, 67195624 13.4485 0.0017853 C 22426, 43402, 46418, 88994, 172486, 221006, 264098, 432182, 1513678, 1718866, 2515574, 2796538, 5952442 299772, 584422, 627877, 1201650, 2320161, 2974984, 3556035, 5828307, 20384499, 23124964, 33827530, 37632788, 80085173 13.4618 0.00132591 D 20160, 43644, 64216, 94484, 139021, 204550, 442830, 651562, 958682, 1410567, 2075452, 3053739, 4493150 270868, 587280, 866299, 1265873, 1871202, 2748585, 5947053, 8769774, 12911249, 18991590, 27939124, 41095221, 60441877 13.4515 0.000502943

Table 2: Statistical analyses on path lengths of Mersenne primes and Mersenne numbers. E[$x$] denotes the arithmetic average of $x$ and Var[$x$] denotes the variance of $x$, respectively.

The data summarized in Table. 2 show that the path lengths of the Mersenne primes have the smaller variance than other sets. However, this diﬀerence is not enough to single out a Mersenne prime index.

In spite of the above, we look more detail into this property. We have examined the plot of $D\left(X={2}^{n}-1\right)$ versus $n$ more closely and noted that they are composed of a collection of “ﬂat” regions with jumps like a staircase. Correspondingly, the plot of $D\left(X\right)∕n$ versus $n$ becomes collection of stripes, which interestingly appears near parallel. They are plotted in Figure 4. These plots are made for the sampled prime indices near three Mersenne prime indices, $23209,110503,216091$ which give $2{6}^{\text{th}}$, $2{9}^{\text{th}}$, and $3{1}^{\text{th}}$ Mersenne primes respectively. It is also interesting that there are occasional “overlaps” of “stair steps”.

The nature and mechanism of these properties appeared in Figure 4 need to be studied further. We, nevertheless, conjecture that these qualitative behavior will continue to larger Mersenne numbers, and the Mersenne prime appears with its $D\left(X\right)∕n\approx 13.45$. Unfortunately, again, this is clearly not suﬃcient to select out the Mersenne primes. If this conjecture is true, however, we can heuristically rule out points that are further away from the horizontal line of $D\left(X\right)∕n\approx 13.45$ from candidates of the Mersenne primes.

### 4 Discussion

We have used the Collatz-Kakutani tree as a provider of “length” for natural numbers, and obtained a conjecture for the Mersenne primes. This may be yet another example where prime numbers can give rise to an emergence of unexpected order, such as Ulam Spiral [89].

Though the reason behind such simple behavior is left unclear, this approach of using the Collatz-Kakutani tree can be extended to other types of set of numbers known for irregularities, possibly leading to interesting insights.

Acknowledgments Authors would like to thank Dr. Masaru Suzuki, Dr. Atsushi Kamimura, and Shigenori Matsumoto for useful discussions. T.O. is also aﬃliated with Sony Computer Science Laboratories in Tokyo, and would like to thank Emeritus Prof. Yuji Ito of Keio University, for his account on his advisor, late Prof. Shizuo Kakutani of Yale University.

### References

[1]   The Math Forum Internet Mathematics Library, available at http://mathforum.org/library/.

[2]   Great Internet Mersenne Prime Search:GIMPS, available at http://www.mersenne.org/default.php/.

[3]   M. Matsumoto, T. Nishimura, Mersenne twister: A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Trans. on Modeling and Computer Simulations 8 (1998) 3.

[4]   This heuristic arguments of this paragraph is provided by one of the anonymous reviewers.

[5]   J. C. Lagarias, The $3x+1$ Problem and Its Generalizations, Amer. Math. Monthly 92 (1985) 3

[6]   A. V. Kontorovich, J. C. Lagarias, Stochastic Models for the $3x+1$ and $5x+1$ Problems, arXive:09101944 [math.NT] (2009)

[7]   M. R. Schroeder, Where Is the Next Mersenne Prime Hiding? Mathematical Intelligencer 5 (1983) 31

[8]   M. L. Stein, S. M. Ulam, M. B. Wells, A Visual Display of Some Properties of the Distribution of Primes, Amer. Math. Monthly 71 (1964) 516

[9]   M. L. Stein, S. M. Ulam, An Observation on the Distribution of Primes, Amer. Math. Monthly 74 (1967) 43.

Shirokanedai Institute, Tokyo, Japan.
ohriatoru@gmail.com

The Institute of Solid State Physics, The University of Tokyo, Kashiwa, Japan.


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# Solving Trigonometric Equation with two angles

I have the following equation, trying to solve for y:

$$A\sin(y)=(B-A\cos(y))\tan(x)$$

I got it to:

$$(A\sin(y))/B - \tan(y) = \tan(x)$$

But don't really see anything to do from there. Any help is appreciated.

simplifying your term you will get $$A\sin(y)+A\tan(x)\cos(y)=B\tan(x)$$ and now use that $$\sin(y)=2\,{\frac {\tan \left( y/2 \right) }{1+ \left( \tan \left( y/2 \right) \right) ^{2}}}$$ and $$\cos(y)={\frac {1- \left( \tan \left( y/2 \right) \right) ^{2}}{1+ \left( \tan \left( y/2 \right) \right) ^{2}}}$$ and set $$\tan(y/2)=t$$ and solve your equation for $t$

Hint

$$A\sin(y)=(B-A\cos(y))\cdot\tan(x)$$ Multiply by $\cos(x)$ $$A \sin(y)\cos(x)=B \tan(x)\cos(x)-A\cos(y)\tan(x)\cos(x)$$ $$A \sin(y)\cos(x)=B\sin(x)-A\cos(y)\sin(x)$$ $$A\sin(x+y)=B\sin(x)$$ I am sure that you can take it from here.

Hint

$$A\sin(y)=(B-A\cos(y))\cdot\tan(x)=A[\sin (y)+\tan (x)\cdot \cos (y)]=B\tan (x)$$

If you just work with $\sin (y)+\tan (x)\cdot \cos (y)$ you can use

$$\sqrt{1+\tan^2(x)}\left(\frac{1}{\sqrt{1+\tan^2(x)}}\cdot \sin (y)+\frac{\tan (x)}{\sqrt{1+\tan^2(x)}}\cdot \cos (y)\right)=\sqrt{1+\tan^2(x)}\left(\cos \theta\cdot \sin (y)+\sin \theta\cdot \cos (y)\right)=\sqrt{1+\tan^2(x)}\left(\sin(y+\theta)\right)$$


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# Tag Info

30

Suppose $x_0\in A$. Let $B := \{ d(x,K) < 1\}$. Observe that since $K$ is compact, there exists $y_0\in K$ such that $d(x_0,y_0) = 1$. By definition $B_1(y_0) = \{x: d(x,y_0) < 1\}$ is a subset of $B$. This implies that for all $\epsilon < 1/2$, we have that $$\mu(B_\epsilon(x_0) \cap B) \geq \frac1{2^n} \mu(B_\epsilon(x_0))$$ where $\mu$ is ...

11

A thin version of the usual Cantor middle third set works. The idea is just that you need to omit more than just a third of the remaining intervals as the construction proceeds, enough so as to force the Hausdorff dimension to $0$. Specifically, we construct the set in stages. At each stage, we've omitted a "middle third" from each finite interval ...

11

Negative dimension is actually much easier to talk about than complex dimension. Super vector spaces are a natural collection of objects that can have negative dimension; given a super vector space $(V_0, V_1)$ we can define its dimension to be $\dim V_0 - \dim V_1$, and this definition has many nice properties; see this blog post, for example. More ...

10

I was going to make this a comment but it occurred to me there might be sufficient interest that perhaps I should not bury it in a comment. At the beginning of the paper below Erdős gives a short proof (that he attributes to Tibor Radó) making use of the Lebesgue density theorem that $E_r$ has Lebesgue measure zero, where $E$ is a closed set in ${\mathbb ... 8 We want to mimic$f(x) = x^{-1/2}$on$[0,1]$, but then cut$[0,1]$into lots of intervals, and then on each interval remake the function so that its integral over that interval remains the same, but the support of the function is much smaller. So let$x_n \to 0$be a decreasing sequence with$x_0 = 1$. Write$x_{n} = (1+\epsilon_n) x_{n+1}$, and suppose ... 8 No. Just for good old$\mathbb R$, there is a homeomorphism$[0,1]\to[0,1]$which takes the middle-thirds Cantor set (of measure$0$) to the "fat" Smith-Volterra-Cantor set (of measure$1/2$). (The excluded intervals in the two constructions, including their endpoints, correspond naturally to each other -- simply map each of them to its counterpart ... 7 You only need to consider the case$\mathfrak{h}_{s}^\ast(A) \lt \infty$, but you need to be a bit careful in choosing the outer approximations since swapping$\inf$and$\sup$certainly isn't allowed without some thinking. If you knew that you can always take the same set$E$in the$\inf$(which I will show in \eqref{eqn:ast} below) then you'd be ... 7 A good anotated list of textbooks on geometric measure theory can be found in this blog post. Besides comments on Federer and Mattila it has several more examples. As my personal favorite I found, while lecturing geometric measure theory, "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. It is short and crisp (often you have to build ... 6 Algebraic stacks are a far-reaching generalization of algebraic varieties. If an algebraic variety is considered as a stack, then its dimension as stack is the same as its dimension as variety. However there are many stacks that do not correspond to varieties, and some of these have negative dimension. Specifically, if$V$is a variety and$G$is an ... 6 One notion of complex dimension that has been used extensively has to do with self-similar sets. A$t$-neighborhood (i.e. points within distance$t$) of such a set may have volume$v(t)$bounded above and below by constant multiples of$t^d$, where$d$is the dimension of the boundary and$t$is small, but such that$t^{-d} v(t)$is oscillatory and ... 6 A$k-$dimensional subspace$S$of$\mathbb R^n$, with$k<n$, is the range of linear transformation of rank$k$. That means that there exists a matrix$A\in\mathbb{R}^{n\times n}$, with$\mathrm{rank}(A)=k<n$, such that$\mathrm{Ran}(A)=S$. But then$\det A=0$. The theorem of change of variables $$m(S)=\int_{\mathbb R^n}\lvert\det A\rvert\,dx=0.$$ ... 6 First, notation/terminology: Talking about Lipschitz maps with respect to that nonstandard metric is a bad idea, simply because there exists a simple and much more standard way to say the same thing. Your function is Lipschitz with respect to that funny metric if $$|f(x)-f(y)|\le c|x-y|^{1/2}.$$ This is exactly the definition of $$f\in ... 5 I just checked the book and I found out that the author defines the m-dimensional density of a set E\subset \mathbb{R}^n near a point x for m\le n by$$\lim_{r\to 0} \frac{\mathcal{H}^m(E\cap B^n(x,r))}{|B^m(0,r)|}$$Note that in the numerator, the n-dimensional ball appears, whereas in the denominator it's the m-dimensional ball! For m=n the ... 4 It is, at least partially, a chicken-and-egg problem. We'd like to define the Lebesgue integral which has nicer convergence properties when we integrate sequences of functions, and take limits of the functions and integrals -- but the definition of the Lebesgue integral requires that we already have a well-behaved measure (i.e., "area") concept to build it ... 4 As Henning says, whether or not you have a Banach-Tarski paradox depends on what symmetries of the Cantor set you allow. My understanding is that you want to allow all measure preserving continuous bijections. With this understanding, the Cantor has a Banach-Tarski decomposition. Note that we can identify the Cantor set with the 2-adic integers, ... 4 In case you are still interested in an answer, I encourage you to look at Leon Simon's notes. The idea of currents, is to serve as a far reaching generalization of oriented submanifolds. They were introduced as part of the fairly long and involved program that went into solving the Plateau's problem: such a minimization problem would require three important ... 4 I will concentrate on comparing (3) and (4). The definition (1) is meant for finite signed measures, whereas all the other definitions are meant for arbitrary positive measures; (1) is equivalent to (4) in the case of finite positive measures. (2) appears to be equivalent to (4) ["locally finite" can mean "finite on compact sets", although it is sometimes ... 4 Hint: Nearest-point projection from a sphere S containing A onto the boundary of A is a contraction. 4 Be advised that Federer's book is not for newcomers to the subject (and it's not an online reference anyway). An accessible introduction to isoperimetric and related inequalities is The Brunn-Minkowski inequality by Gardner. It focuses on the volume of subsets of \mathbb R^n, thus avoiding most of the GMT machinery. A somewhat dated, but still good, ... 4 The Minkoswki content is a rather simplistic way to define an m-dimensional measure of an object immersed in the euclidean space (however mathematically speaking Minkowski content is not a measure). If \mathcal A is a regular m-dimensional surface in \mathbb R^n it is easy to understand that the Minkowski content (both lower and upper) gives the ... 4 This is false even for curves in the plane. Consider a sequence of ellipses converging to a line segment. The length of the limit curve drops by the factor of 2. If you want an example of curves without boundary, consider a similar example in 3d space. Or consider curves which are disjoint unions of pairs of concentric circles converging to a single circle. ... 4 I regret having to note that the other answer is wrong. It is well known, since work of Moran in 1946, that the Hausdorff dimension of the particular set that you describe does not depend on the location of the intervals at each step. For a proof you can see the book Dimension and Recurrence in Hyperbolic Dynamics, by Barreira (see Theorem 3.1.1). 4 The two measures are within a constant factor of each other. In K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, see the material on "net measure". In the 2nd edition, that is page 36 4 As felipeh noted, the problem reduces to p=1 by replacing f with g = |f|^p. (The case p=\infty should be treated separately.) Also, the annulus can be replaced by the sphere S^{n-1}. Indeed, define a function h on the unit sphere by h(\xi) = \int_r^1 g(t\xi)\,dt. Then h\in L^1(S^{n-1}), with a norm comparable to \|g\|_{L^1(A)} because ... 4 Assume we have a Jordan curve of positive area measure whose complement has connected components U,V as in the Jordan Curve theorem. Let U be the bounded component. Every point in \partial U = \partial V is the limit of a sequence of points in \overline U and the limit of a sequence of points in V= \overline U ^c. Thus the boundary of \overline ... 4 The answer is negative. Let's consider a Cantor-type set K\subset [0,1] built by removing the c_j proportion of each interval at jth step (c_j<1, j=1,2,\dots). It's a compact set with empty interior, so \partial K=K. The measure of K is zero provided that$$ \prod_{j=1}^\infty (1-c_j)=0 $$Hence, K is Jordan measurable in this case. On ... 4 No. As shown in Lebesgue measure of the graph of a function, there exists a function f : \mathbb{R} \to \mathbb{R} whose graph G = \{(x,f(x)) : x \in \mathbb{R}\} is a Lebesgue non-measurable subset of \mathbb{R}^2. But if we let \pi(x,y) = (x,0) be projection onto the x axis, then \pi|_G : G \to \mathbb{R} is bijective. If you can first show ... 4 Take a set C\subset\mathbb{R}^n that is not Borel. For every X\subset \mathbb{R}^n, let$$ \mu(X) = \begin{cases} 0 & \text{if } X\subset C; \\ \infty & \text{if } X\not\subset C. \\ \end{cases}$$This is a measure on the entire$P(\mathbb{R}^n)$. In particular, all Borel sets are measuable. For every Borel set$B\supset C\$ we have ...

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text:
Dimensions and World Coordinates

AstroImages are based on Dimensional Data. Each axis is assigned a dimension name and the indices are tracked.

World Coordinates

FITS files with world coordinate system (WCS) headers contain all the information necessary to map a pixel location into celestial coordinates & back.

Let's see how this works with a 2D image with RA & DEC coordinates.

using AstroImages
using Plots

"http://www.astro.uvic.ca/~wthompson/astroimages/fits/656nmos.fits",
"eagle-656nmos.fits"
);
eagle = load("eagle-656nmos.fits")

This image contains world coordinate system headers. AstroImages.jl uses WCS.jl (and wcslib under the hood) to parse these headers. We can generate a WCSTransform object to inspect:

wcs(eagle, 1) # specify which coordinate system
WCSTransform(naxis=2,cdelt=[1.0, 1.0],crval=[274.71149247724, -13.816384007184],crpix=[386.5, 396.0])

Note that we specify with an index which coordinate system we'd like to use. Most images just contain one, but some contain multiple systems.

We can lookup a coordinate from the image:

world = pix_to_world(eagle, [1, 1]) # Bottom left corner
2-element Vector{Float64}:
274.712299241082
-13.801135972688115

Or convert back from world coordinates to pixel coordinates: We can lookup a coordinate from the image:

world_to_pix(eagle, world) # Bottom left corner
2-element Vector{Float64}:
1.000000000336172
0.9999999992196535

These pixel coordinates do not necessarily have to lie within the bounds of the original image, and in general lie at a fractional pixel position.

If an image contains WCS headers, we can visualize them using implot:

implot(eagle)