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# Define resistivity of a material. Does it depend on temperature?

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Hint: Here, we will proceed by defining the term resistivity of a material. Then, we will derive the equation for resistivity in terms of resistance. Finally, we will discuss the variation of resistivity of conductors, semiconductors and insulators with temperature.

Complete Step-by-Step solution:
Electrical resistivity is the reciprocal of electrical conductivity. It is the measure of the ability of a material to oppose the flow of current. Materials resist the flow of electric current. Some of the materials are better at conducting electric current than the others.
Materials that conduct electrical current easily are called conductors and have a low resistivity. Those that do not conduct electricity easily are called insulators and these materials have a high resistivity.
Resistivity is defined as the electrical resistance of a conductor of unit cross-sectional area and unit length.
The resistance of any conductor is directly proportional to the length of the conductor and inversely proportional to the area of cross-section of the conductor.
If L is the length of the conductor and A is the area of cross-section of the conductor. Then, the resistance R of the conductor is given by
${\text{R}} \propto \dfrac{{\text{L}}}{{\text{A}}} \\ \Rightarrow {\text{R}} = \rho \dfrac{{\text{L}}}{{\text{A}}} \\$
Where $\rho$ is known as the resistivity of the conductor
The general formula to determine the resistivity of any conductor will be given by
$\Rightarrow \rho = \dfrac{{{\text{RA}}}}{{\text{L}}}$
The SI unit of resistance (R) is ohm ($\Omega$) , SI unit of area of cross-section (A) is square meters (${{\text{m}}^2}$), SI unit of length is meters (m). Then, the SI unit of resistivity will be given by
SI unit of resistivity = $\dfrac{{\left( \Omega \right)\left( {{{\text{m}}^2}} \right)}}{{\left( {\text{m}} \right)}} = \Omega {\text{m}}$
SI unit of resistivity is ohm-meters.
The resistivity of materials depends on the temperature according to the below mentioned equation.
$\rho = {\rho _0}\left[ {1 + \alpha \left( {{\text{T}} - {{\text{T}}_0}} \right)} \right]$
where $\rho$ denotes the resistivity of the material at the temperature T${}^0{\text{C}}$, ${\rho _0}$ denotes the resistivity of the material at the reference temperature ${{\text{T}}_0}{}^0{\text{C}}$ and $\alpha$ is the temperature coefficient of resistivity.
Variation of Resistivity in Conductors
When the temperature increases the vibrations of the metal ions in the lattice structure increases. The atoms start to vibrate with higher amplitude. These vibrations in turn cause frequent collisions between the free electrons and the other electrons. Each collision drains out some energy of the free electrons and causes them unable to move. Thus it restricts the movement of the delocalized electrons. When the collision happens the drift velocity of the electrons decreases. This means that the resistivity of the metal increases and thus current flow in the metal is decreased. The resistivity increases mean that the conductivity of the material decreases.

Variation of Resistivity in Semiconductors
When the temperature is increased the forbidden gap between the two bands becomes very less and the electrons move from the valence band to the conduction band. Thus some electrons from the covalent bonds between the atoms are free to move within the structure. This increases the conductivity of the material. The conductivity increases mean the resistivity decreases. Thus when the temperature is increased in a semiconductor, the density of the charge carriers also increases and the resistivity decreases.

Variation of Resistivity in Insulators
When the temperature is increased, the atoms of the material vibrate and it makes the valence electrons present in the valence band to shift to the conduction band. This in turn increases the conductivity of the material. When the conductivity of the material increases, it means that the resistivity decreases and so the current flow increases. Thus some insulators at room temperatures change to conductors at high temperature.

Note- Resistivity is a measure of the resistance of a given size of a specific material to electrical conduction. Resistivity may also be referred to as the specific electrical resistance, or volume resistivity, although these terms are less widely used. The resistivity is a figure that enables comparisons of the way in which different materials allow or resist current flow.

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# Help me

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Calculate the distance between the points =M, −44 and =P, 2−1 in the coordinate plane.

Guest Feb 11, 2017
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What are the coordinates of M ....????.....you only listed one......

CPhill  Feb 11, 2017
#2
+5907
+1

If I understand your question correctly, the two points are:

(M,-44) and (P,[2-1])

That is:

(M,-44) and (P,1)

All you need to do is put this info into the distance formula.

d=$$\sqrt{(M-P)^{2}+(-44-1)^{2}}$$

d=$$\sqrt{(M - P)^{2}+(-45)^{2}}$$

d= $$\sqrt{(M - P)^{2}+2025}$$

hectictar  Feb 11, 2017
edited by Guest  Feb 11, 2017
edited by Guest  Feb 11, 2017

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# Fraktur symbols for Lie algebras: $\mathfrak{g}$, etc.

Does anyone know when and why the Fraktur script was introduced for Lie and other algebras—$\mathfrak{g}$, $\mathfrak{gl}_n$, $X/\mathfrak{g}$, $\mathfrak{g}\oplus\mathfrak{g}$, $\mathfrak{su}$, $\mathfrak{M}_g$, etc.? And introduced by whom? Is its use pretty much restricted to algebra, or was it in the past employed in, say, geometry as well, but has only survived to the current time within algebra? (Or maybe it is currently used outside of algebra and I am just ignorant of those areas.)

The typeface itself goes back to the 15th century. The generally illuminating website "Earliest Uses of Various Mathematical Symbols" seems not to shed light on this issue.

I find the Fraktur font adds a certain elegance and mystery to the subjects that utilize it! I'm a bit envious, not working in those fields... —$\mathfrak{Joseph}$ :-)

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Various German algebraists (Noether for example) used Fraktur script in their work at least back to the 1910s. So one reasonable guess would be Hermann Weyl, who advanced the theory of Lie algebras in the 1930s. –  Eric Rowell Feb 6 '12 at 2:20
Reminds me of johnlangdon.net/angelsanddemons.php if you click on any of the "ambigrams" it rotates it 180 degrees. In response to your question, I think unreadable letters are introduced out of malice, to make life difficult for other people. In the other direction, the Korean alphabet was commissioned (work done by a group of scholars) by King Sejong, to help a largely illiterate populace who did not speak (or write) Chinese, when learned discourse was written in Chinese. Sort of like Latin. –  Will Jagy Feb 6 '12 at 4:26
Will, I remember my algebraic number theory class, where the Fraktur letters were written with such elegance that I couldn't tell a "$\mathfrak p$" from a "$\mathfrak q$". I guess it did add to the mystery :) –  B R Feb 6 '12 at 5:47
@BR, I took German in school, at some point my mother got an inexpensive box of books at an auction, so I had a dozen books in Fraktur. I couldn't make heads or tails of them. –  Will Jagy Feb 6 '12 at 6:34

Some of what's been said so far about the history makes good sense, but by no means all. Let me add my own perspective, for what it's worth. The font called Fraktur by LaTeX (also known as "gothic") was widely used historically in German printing (though I don't own a Gutenberg Bible). It naturally crept into mathematical usage and notation. For instance, the upper case Fraktur letter G was commonly used to denote a group, while the ordinary roman G denoted an element of the group.

This convention persisted among emigres like Walter Feit who grew up in Vienna (and escaped on the last children's train though his parents didn't). In his course at Yale which I took as a graduate student he filled the blackboard elegantly with ornate symbols, which I sort of learned to copy down (see his Benjamin lecture notes on character theory from that era). But I had actually encountered Fraktur when I first learned some German grammar in high school. It was a mediocre working class public school but located among various ethnic enclaves (including Italian and German), so those languages got taught for a while in two year sequences. The principal wouldn't let me and a classmate of German descent start with the second course, so we sat in the back of the classroom in the first year course and worked ahead on our own. The old German textbooks available in that postwar era were all in Fraktur, which had been promoted during the Third Reich as the "correct" way to print the language of the master race. So I did learn to distinguish upper case B and V, etc.

The point is that group theory and Lie groups in particular were actively developed by German mathematicians in the nineteenth century; they were not inventing exotic notation when they used these particular letters as symbols. In number theory there is still a widespread tendency to use even the lower case letters like m in Fraktur, which most people find impossible to imitate by hand. But for Hilbert and others this was quite natural notation, as was the lower case Fraktur f (for Fuhrer, now usually called "conductor").

By the way, in Lie algebra theory the lower case letter was naturally used because the Lie algebra was first regarded as an infinitesimal group.

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I want to upvote this more than once. –  Ketil Tveiten Mar 2 '12 at 12:41
Incidentally, the wartime German mathematical journal Deutsche Mathematik used the Roman font for running text in the first few issues, but soon switched over to black letter ("Fraktur"), presumably because it was thought to be more Teutonic. I don't know whether they had the time to switch back to Roman before closing down definitively; the library in my current place of work doesn't have copies of the journal. –  Chandan Singh Dalawat Mar 2 '12 at 14:41
May I point out a mistake: During the third Reich the Nazi tried to abolish Fraktur. In fact they succeeded. I do not know why today Fraktur is associated with Nazi - historically this is false. I have read many books in fraktur and like it. My grandmother' handwriting was "Kurrent" which was more beautiful and easier to read that todays handwriting. The aim was to minimize hand movements in directions other than the main direction (NE-SW). In very old manuscripts, Fraktur was always the type for German texts, and Latin type was the type for Latin texts. You find texts containing both. –  Peter Michor Jan 1 '13 at 20:11
@Peter: My sources are mainly anecdotal, so I can't disagree with your version of the history which seems to be acquired more directly. Growing up as I did among so many people with Italian or German heritage, I never learned any really impartial version of early 20th century history. It's certainly possible that the textbooks in Fraktur which I used in high school were made obsolete by the Nazis, but for some reason I grew up thinking the opposite. Our public schools never taught any history after 1920, since FDR too was a toxic subject. –  Jim Humphreys Jan 3 '13 at 23:50
@Jim: The german Wikipedia <A HREF="de.wikipedia.org/wiki/…; has a good historical description. If you switch to English, the historical description becomes much shorter. I was not completely correct: In the beginning the Nazi favored Fraktur a little (out of romantic feeling), but there was a sudden change against Fraktur in 1941. –  Peter Michor Jan 9 '13 at 8:13

I don't know the mathematical history of Fraktur, but the following story (I'm not sure whether it's true but it's at least imaginable, and I didn't make it up) might make you feel better about working in a Fraktur-less field. The Detroit Free Press (one of Detroit's two major daily newspapers) had its name in very large Fraktur type in its masthead. It took a long time (years, not days) before someone pointed out that it said "Vetroit", not "Detroit".

I can vouch for a similar confusion on the basis of my own experience. In mathematical logic, we often use Fraktur capital letters for models and the corresponding ordinary (italic) letters for the underlying sets of the models. Far too many students assume that the Fraktur A ($\mathfrak A$) is intended to be a U.

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:-) $\mbox{} \mbox{}$ –  Joseph O'Rourke Feb 6 '12 at 1:45
Best MO anecdote EVAR. –  Mariano Suárez-Alvarez Feb 6 '12 at 2:08
@Mariano, shouldn't it be EDER? Which, as I had to look up, is a river. In Germany. So it all fits. –  Will Jagy Feb 6 '12 at 4:49
–  Frank Thorne Mar 2 '12 at 2:22

I am pretty sure that the first use of Fraktur by Sophus Lie occurs in 1869, which is before he invented Lie groups or Lie algebras. It appears in his paper Repraesentation der Imaginaeren der Plangeometrie, in the first volume of his collected works, to represent the plane. I assume that it was standard practice in German mathematics to use German script letters, because they were used in number theory by Dirichlet and others before 1869. Lie groups slowly evolved in the mid to late 1870's, entering their final form in the 1880s. But in discussing Lie algebras, he rarely uses Fraktur fonts. He usually talks about a group G and then writes out its Lie algebra. I didn't run into Fraktur fonts before 1891, Die linearen homogenen gewohnlichen Differentialgleichungen, used to describe a sort of generating function for a Lie algebra. Maybe an expert (Thomas Hawkins or Peter Olver) would have better luck.

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Fraktur was the standard font for cardinal numbers, for example, in writing the continuum hypothesis as $\aleph_1=\frak c$.

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text: Suppose $$R$$ is the region between the two simple closed curves $$C_1$$ and $$C_2$$. The contour integral becomes I C 1 z − z0 dz = Z2π 0 1 z(t) − z0 dz(t) dt dt = Z2π 0 ireit reit Let $f(z) = f(x + yi) = x - yi = \overline{z}$. 1. Then $u(x, y) = e^{x^2 - y^2} \cos (2xy)$ and $v(x, y) = e^{x^2 - y^2} \sin (2xy)$. )©@¤Ä@T\A!sbM°1q¼GY*|z¹ô\mT¨sd. FÀX¥Q.Pu -PAFhÔ(¥  Residues and evaluation of integrals 9. all of its elements have order p for some natural number k) if and only if G has order p for some natural number n. One may use the abelian case of Cauchy's Theorem in an inductive proof of the first of Sylow's theorems, similar to the first proof above, although there are also proofs that avoid doing this special case separately. Change the name (also URL address, possibly the category) of the page. The interior of a square or a circle are examples of simply connected regions. Theorem (Some Consequences of MVT): Example (Approximating square roots): Mean value theorem finds use in proving inequalities. (4) is analytic inside C, J= 0: (5) On the other hand, J= JI +JII; (6) where JI is the integral along the segment of the positive real axis, 0 x 1; JII is the z +i(z −2)2. . Identity principle 6. Let Cbe the unit circle. Recall from The Cauchy-Riemann Theorem page that if $A \subseteq \mathbb{C}$ is open, $f : A \to \mathbb{C}$ with $f = u + iv$, and $z_0 \in A$ then $f$ is analytic at $z_0$ if and only if there exists a neighbourhood $\mathcal N$ of $z_0$ with the following properties: We also stated an important result that can be proved using the Cauchy-Riemann theorem called the complex Inverse Function theorem which says that if $f'(z_0) \neq 0$ then there exists open neighbourhoods $U$ of $z_0$ and $V$ of $f(z_0)$ such that $f : U \to V$ is a bijection and such that $\displaystyle{\frac{d}{dw} f^{-1}(w) = \frac{1}{f'(z)}}$ where $w = f(z)$. If we assume that f0 is continuous (and therefore the partial derivatives of u … Cauchy's Integral Theorem Examples 1. Let f ( z) = e 2 z. C have continuous partial derivatives and they satisfy the Cauchy Riemann equations then Z @U f(z)dz= 0: Proof. However note that $\displaystyle{1 = \frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y} = -1}$ ANYWHERE. View wiki source for this page without editing. Then $u(x, y) = x$ and $v(x, y) = -y$. They are: So the first condition to the Cauchy-Riemann theorem is satisfied. We will now look at some example problems in applying the Cauchy-Riemann theorem. Then $u(x, y) = x$ and $v(x, y) = -y$. Q.E.D. The following theorem says that, provided the ﬁrst order partial derivatives of u and v are continuous, the converse is also true — if u(x,y) and v(x,y) obey the Cauchy–Riemann equations then Compute Z C 1 (z2 + 4)2 Im(z) Im(z) 2i 2i C Solution: Let f(z) = cos(z)=(z2 + 8). Recall from the Cauchy's Integral Theorem page the following two results: The Cauchy-Goursat Integral Theorem for Open Disks: If $f$ is analytic on an open disk $D(z_0, r)$ then for any closed, piecewise smooth curve $\gamma$ in $D(z_0, r)$ we have that: (1) An illustration is Hadamard's example: The Cauchy problem for the Laplace equation $$\Delta u = \ \frac{\partial ^ {2} u }{\partial x ^ {2} } + \frac{\partial ^ {2} u }{\partial y ^ {2} } + \frac{\partial ^ {2} u }{\partial z ^ {2} } = 0$$ It is a very simple proof and only assumes Rolle’s Theorem. Example 4.4. Cauchy's vs Lagrange's theorem in Group Theory. Re(z) Im(z) C. 2. 4.3.2 More examples Example 4.8. examples, which examples showing how residue calculus can help to calculate some deﬁnite integrals. However note that $\displaystyle{1 = \frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y} = -1}$ ANYWHERE. The stronger (better) version of Cauchy's Extension of the MVT eliminates this condition. Corollary of Cauchy's theorem … Watch headings for an "edit" link when available. Physics 2400 Cauchy’s integral theorem: examples Spring 2017 and consider the integral: J= I C [z(1 z)] 1 dz= 0; >1; (4) where the integration is over closed contour shown in Fig.1. Click here to toggle editing of individual sections of the page (if possible). A practically immediate consequence of Cauchy's theorem is a useful characterization of finite p-groups, where p is a prime. For example, for consider the function . New content will be added above the current area of focus upon selection dz, where. So, we rewrite the integral as Z C cos(z)=(z2 + 8) z dz= Z C f(z) z dz= 2ˇif(0) = 2ˇi 1 8 = ˇi 4: Example 4.9. Theorem 2.9 Let Mbe an oriented smooth manifold with corners and Bbe an n-dimensional body in M. Suppose that and are bounded n-forms on B and ˝is a continuous function on the bundle of oriented hyperplanes! Cauchy saw that it was enough to show that if the terms of the sequence got suﬃciently close to each other. The first order partial derivatives of $u$ and $v$clearly exist and are continuous. Theorem 23.3 we know that all of the derivatives of f are also analytic in D.Inparticular, this implies that all the partials of u and v of all orders are continuous. The mean value theorem says that there exists a time point in between and when the speed of the body is actually . Determine whether the function $f(z) = \overline{z}$is analytic or not. Determine whether the function $f(z) = e^{z^2}$ is analytic or not using the Cauchy-Riemann theorem. One can use the Cauchy integral formula to compute contour integrals which take the form given in the integrand of the formula. Laurent expansions around isolated singularities 8. Cauchy’s Theorem Cauchy’s theorem actually analogue of the second statement of the fundamental theorem of calculus and integration of familiar functions is facilitated by this result In this example, it is observed that is nowhere analytic and so need not be independent of choice of the curve connecting the points 0 and . Theorem 7.4.If Dis a simply connected domain, f 2A(D) and is any loop in D;then Z f(z)dz= 0: Proof: The proof follows immediately from the fact that each closed curve in Dcan be shrunk to a point. Cauchy’s theorem Simply-connected regions A region is said to be simply-connected if any closed curve in that region can be shrunk to a point without any part of it leaving a region. Click here to edit contents of this page. f(z) is analytic on and inside the curve C. That is, the roots of z2 + 8 are outside the curve. Let $f(z) = f(x + yi) = x - yi = \overline{z}$. Example 5.2. The partial derivatives of these functions exist and are continuous. Suppose that $f$ is analytic. Then from the proof of the Cauchy-Riemann theorem we have that: The other formula can be derived by using the Cauchy-Riemann equations or by the fact that in the proof of the Cauchy-Riemann theorem we also have that: \begin{align} \quad \frac{\partial u}{\partial x} = 1 \quad , \quad \frac{\partial u}{\partial y} = 0 \quad , \quad \frac{\partial v}{\partial x} = 0 \quad , \quad \frac{\partial v}{\partial y} = -1 \end{align}, \begin{align} \quad f(z) = f(x + yi) = e^{(x + yi)^2} = e^{(x^2 - y^2) + 2xyi} = e^{x^2 - y^2} e^{2xyi} = e^{x^2 - y^2} \cos (2xy) + e^{x^2 - y^2} \sin (2xy) i \end{align}, \begin{align} \quad \frac{\partial u}{\partial x} = 2x e^{x^2 - y^2} \cos (2xy) - 2y e^{x^2 - y^2} \sin (2xy) = e^{x^2 - y^2} [2x \cos (2xy) - 2y \sin (2xy)] \end{align}, \begin{align} \quad \frac{\partial v}{\partial y} = -2ye^{x^2 - y^2} \sin(2xy) + 2x e^{x^2 - y^2} \cos (2xy) = e^{x^2 - y^2}[2x \cos (2xy) - 2y \sin (2xy)] \end{align}, \begin{align} \quad \frac{\partial u}{\partial y} =-2ye^{x^2 - y^2} \cos (2xy) - 2x e^{x^2 - y^2} \sin (2xy) = -e^{x^2 - y^2}[2x \sin (2xy) + 2y \cos (2xy)] \end{align}, \begin{align} \quad \frac{\partial v}{\partial x} = 2xe^{x^2 - y^2}\sin(2xy) + 2ye^{x^2 - y^2}\cos(2xy) = e^{x^2 - y^2}[2x \sin (2xy) + 2y \cos(2xy)] \end{align}, \begin{align} \quad f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} \end{align}, \begin{align} \quad \mid f'(z) \mid = \sqrt{ \left( \frac{\partial u}{\partial x} \right )^2 + \left ( \frac{\partial v}{\partial x} \right )^2} \end{align}, \begin{align} \quad \mid f'(z) \mid^2 = \left( \frac{\partial u}{\partial x} \right )^2 + \left ( \frac{\partial v}{\partial x} \right )^2 \end{align}, \begin{align} \quad f'(z) = \frac{\partial v}{\partial y} -i\frac{\partial u}{\partial y} \end{align}, Unless otherwise stated, the content of this page is licensed under. In particular, a finite group G is a p-group (i.e. Solution: Since ( ) = e 2 ∕( − 2) is analytic on and inside , Cauchy’s theorem says that the integral is 0. View/set parent page (used for creating breadcrumbs and structured layout). 2. I have deleted my non-Latex post on this theorem. In cases where it is not, we can extend it in a useful way. Solution The circle can be parameterized by z(t) = z0 + reit, 0 ≤ t ≤ 2π, where r is any positive real number. The path is traced out once in the anticlockwise direction. Do the same integral as the previous example with the curve shown. Example Evaluate the integral I C 1 z − z0 dz, where C is a circle centered at z0 and of any radius. Thus by the Cauchy-Riemann theorem, $f(z) = e^{z^2}$ is analytic everywhere. ∫ C ( z − 2) 2 z + i d z, \displaystyle \int_ {C} \frac { (z-2)^2} {z+i} \, dz, ∫ C. . See pages that link to and include this page. Since the integrand in Eq. Check out how this page has evolved in the past. Determine whether the function $f(z) = \overline{z}$ is analytic or not. Existence of a strange Group. Argument principle 11. HBsuch What is an intuitive way to think of Cauchy's theorem? So one of the Cauchy-Riemann equations is not satisfied anywhere and so $f(z) = \overline{z}$ is analytic nowhere. Compute. 0. example link > This is a quote: This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. Cauchy Mean Value Theorem Let f(x) and g(x) be continuous on [a;b] and di eren-tiable on (a;b). A remarkable fact, which will become a theorem in Chapter 4, is that complex analytic functions automatically possess all Re(z) Im(z) C. 2. Logarithms and complex powers 10. They are: So the first condition to the Cauchy-Riemann theorem is satisfied. If the real and imaginary parts of the function f: V ! Except for the proof of the normal form theorem, the material is contained in standard text books on complex analysis. 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So one of the Cauchy-Riemann equations is not satisfied anywhere and so $f(z) = \o… ANALYSIS I 9 The Cauchy Criterion 9.1 Cauchy’s insight Our diﬃculty in proving “a n → ‘” is this: What is ‘?$\displaystyle{\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}}$,$\displaystyle{\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}}$,$\displaystyle{\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}}$,$\displaystyle{\frac{d}{dw} f^{-1}(w) = \frac{1}{f'(z)}}$,$f(z) = f(x + yi) = x - yi = \overline{z}$,$\displaystyle{1 = \frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y} = -1}$,$\displaystyle{\mid f'(z) \mid^2 = \left (\frac{\partial u}{\partial x} \right )^2 + \left ( \frac{\partial v}{\partial x} \right )^2}$,$\displaystyle{\mid f'(z) \mid^2 = \left (\frac{\partial u}{\partial y} \right )^2 + \left ( \frac{\partial v}{\partial y} \right )^2}$, Creative Commons Attribution-ShareAlike 3.0 License. 3. The Riemann Mapping Theorem; Complex Integration; Complex Integration: Examples and First Facts; The Fundamental Theorem of Calculus for Analytic Functions; Cauchy's Theorem and Integral Formula; Consequences of Cauchy's Theorem and Integral Formula; Infinite Series of Complex Numbers; Power Series; The Radius of Convergence of a Power Series Solution: This one is trickier. Cauchy’s theorem requires that the function $$f(z)$$ be analytic on a simply connected region. Group of order$105$has a subgroup of order$21$5. Now let C be the contour shown below and evaluate the same integral as in the previous example. 3. THE CAUCHY MEAN VALUE THEOREM JAMES KEESLING In this post we give a proof of the Cauchy Mean Value Theorem. Related. The notes assume familiarity with partial derivatives and line integrals. Cauchy’s theorem 3. 3)¸%ÀÄ¡*Å2:à)Ã2 Do the same integral as the previous examples with the curve shown. Compute Z C cos(z) z(z2 + 8) dz over the contour shown. In particular, has an element of order exactly . Cauchy’s integral theorem An easy consequence of Theorem 7.3. is the following, familiarly known as Cauchy’s integral theorem. The Cauchy-Goursat Theorem Cauchy-Goursat Theorem. If you want to discuss contents of this page - this is the easiest way to do it. Example 4.3. This should intuitively be clear since$f$is a composition of two analytic functions. Then there is … If a function f is analytic at all points interior to and on a simple closed contour C (i.e., f is analytic on some simply connected domain D containing C), then Z C f(z)dz = 0: Note. View and manage file attachments for this page. Prove that if$f$is analytic at then$\displaystyle{\mid f'(z) \mid^2 = \left (\frac{\partial u}{\partial x} \right )^2 + \left ( \frac{\partial v}{\partial x} \right )^2}$and$\displaystyle{\mid f'(z) \mid^2 = \left (\frac{\partial u}{\partial y} \right )^2 + \left ( \frac{\partial v}{\partial y} \right )^2}$. To do so, we have to adjust the equation in the theorem just a bit, but the meaning of the theorem is still the same. For example, a function of one or more real variables is real-analytic if it is diﬀerentiable to all orders on an open interval or connected open set and is locally the sum of its own convergent Taylor series. Then, (5.2.2) I = ∫ C f ( z) z 4 d z = 2 π i 3! They are given by: So$\displaystyle{\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}}$everywhere. Cauchy Theorem when internal efforts are bounded, and for fixed normal n (at point M), the linear mapping n ↦ t ( M ; n ) is continuous, then t ( M ; n ) is a linear function of n , so that there exists a second order spatial tensor called Cauchy stress σ such that Theorem 2says thatitisnecessary for u(x,y)and v(x,y)toobey the Cauchy–Riemann equations in order for f(x+iy) = u(x+iy)+v(x+iy) to be diﬀerentiable. Something does not work as expected? With Cauchy’s formula for derivatives this is easy. Wikidot.com Terms of Service - what you can, what you should not etc. Cauchy’s formula 4. then completeness Theorem (Cauchy’s integral theorem 2): Let Dbe a simply connected region in C and let Cbe a closed curve (not necessarily simple) contained in D. Let f(z) be analytic in D. Then Z C f(z)dz= 0: Example: let D= C and let f(z) be the function z2 + z+ 1. The first order partial derivatives of$u$and$v$clearly exist and are continuous. How to use Cayley's theorem to prove the following? Also: So$\displaystyle{\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}}$everywhere as well. Notify administrators if there is objectionable content in this page. f ‴ ( 0) = 8 3 π i. Power series expansions, Morera’s theorem 5. This means that we can replace Example 13.9 and Proposition 16.2 with the following. We have, by the mean value theorem, , for some such that . Liouville’s theorem: bounded entire functions are constant 7. Find out what you can do. In Figure 11 (a) and (b) the shaded grey area is the region and a typical closed Addeddate 2006-11-11 01:04:08 Call number 29801 Digitalpublicationdate 2005/06/21 Identifier complexintegrati029801mbp Identifier-ark ark:/13960/t0rr1q351 Stã|þtÇÁ²vfÀ& Iæó>@dÛ8.ËÕ2?hm]ÞùJõ:³@ØFÃ¦ÄÔç¯3³$W°¤hxÔIÇç/ úÕØØ¥¢££`ÿ3 If f(z)=u(z)+iv(z)=u(x,y)+iv(x,y) is analytic in a … General Wikidot.com documentation and help section. Now, having found suitable substitutions for the notions in Theorem 2.2, we are prepared to state the Generalized Cauchy’s Theorem. Let V be a region and let Ube a bounded open subset whose boundary is the nite union of continuous piecewise smooth paths such that U[@UˆV. 1. Theorem 23.7. Cauchy's Integral theorem concept with solved examples Subject: Engineering Mathematics /GATE maths. Examples. If is a finite group, and is a prime number dividing the order of , then has a subgroup of order exactly . 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text:
## Oliver Johns

Print publication date: 2005

Print ISBN-13: 9780198567264

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198567264.001.0001

# (p.540) Appendix D THE CALCULUS OF MANY VARIABLES

Source:
Analytical Mechanics for Relativity and Quantum Mechanics
Publisher:
Oxford University Press

We summarize here some standard results from the calculus of many variables. These theorems will be invoked frequently throughout the text. In particular, the reader should review this material before beginning the chapter on Lagrangian methods. More background on this topic can be found in many standard calculus texts; a particularly accessible source is the two volumes of Courant (1936a,b). The mathematical appendices of Desloge (1982) are also very useful.

# D.1 Basic Properties of Functions

Afunction y of the set of variables x 1, x 2,…, xN will be written in either of the two equivalent forms,

(D.1)

In the second form, the single unsubscripted variable x denotes the whole set, x = x 1, x 2,…, xN. Note that the same letter y is used to denote both quantity and function. Whereas a calculus book might write y = f(x), wewrite y = y(x). This notation has the advantage of clarity and alphabetic economy. It will be clear from context when letters like y denote the function and when they denote the value of the function. Usually, a letter on the left of an equal sign denotes the value, the same letter before the (x) denotes the function that produces that value.

# D.2 Regions of Definition of Functions

Functions are assumed to be single valued for all x 1, x 2,…, xN in a region R. A simple example of such a region is an open rectangle R defined by ak < xk < bk for k = 1,…, N, wherethe ak < bk are some constants.

The region R is connected if any two points in R can be joined by a curve lying entirely in R. The region is simply connected if it is connected, and if every closed curve lying entirely in R can be continuously shrunk to a point without leaving R. For example, the interior of a ball is simply connected, the interior of a doughnut is not.

Region R is called open if, for every point in R, there is some (possibly quite small) open rectangle lying entirely in R and containing the point. Such an open rectangle will be called an open neighborhood of the point x and will be denoted Nx. See Chapter I of Spivak (1965) for more definitions.

# (p.541) D.3 Continuity of Functions

The definition of continuity is similar to that for functions of one variable.128 Define an arbitrary increment hk for each variable xk for k = 1,…, N. Then function y is continuous at point x if, for every possible choice of the hk,

(D.2)

For example, there is no assigned value of y(0, 0) for which $y = y ( x 1 , x 2 ) = ( x 1 2 - x 2 2 ) / ( x 1 2 + x 2 2 )$ would be continuous at the point x 1 = 0, x 2 = 0, since the left side of eqn (D.2) would have the limit h 1 = 1, h 2 = 0 but the limit −1 when h 1 = 0, h 2 = 1.

# D.4 Compound Functions

Suppose we have a function

(D.3)

and each of the x 1, x 2,…, xN is a function of another set of variables r 1, r 2,…rM as in

(D.4)

Then y is a compound function of the variables r. This compound function y = y(r 1, r 2,…, rM) is defined by direct substitution of the xk of eqn (D.4) into eqn (D.3). This substitution will be denoted by

(D.5)

or, in a somewhat shorter notation,

(D.6)

or, even more simply,

(D.7)

# D.5 The Same Function in Different Coordinates

A scalar field T is often represented by functions of different kinds of coordinates. For example,

(D.8)

where the subscripts refer to Cartesian, cylindrical polar, and spherical polar coor dinates, respectively. Of course, eqn (D.8) contains the implicit assumption that the various coordinates in it must be linked by the relations in Section A.8, and so represent the same point r. The function denoted T C(x 1, x 2, x 3) is plainly not the same (p.542) function of its arguments x 1, x 2, x 3 as T s(r, θ, ϕ) is of its arguments r, θ, ϕ. For example, suppose T C(x 1, x 2, x 3) = ax 2/x 1 which leads to T s(r, θ, ϕ) = a tanϕ. But they both represent the same underlying scalar field function T = T(r). The almost universal custom in physics texts is to omit the subscripts “C”, “c”, and “s” in eqn (D.8) and write simply

(D.9)

The argument list of a function is taken as an adequate label for it.

The various functions in eqn (D.9) are thought of as the same function. The value T is really a function of the underlying field point r and is only being represented in Cartesian, cylindrical polar, or spherical polar form. This physics custom is especially evident in Lagrangian and Hamiltonian mechanics where one sees expressions such as L (s, ṡ, t) for $L ( q , q ˙ , t )$ functions expressing the same underlying physical quantity in terms of different variable sets.

We will follow the physics custom in this book, and in fact have already done so by using the same letter y to denote both y(x 1, x 2,…, xN) and the compound function y(r 1, r 2,…, rM) in Section D.4 above. Note that the different functional forms in eqn (D.9) can be defined as compound functions. For example, T(r, θ, ϕ) may be derived from T(x 1, x 2, x 3) by

(D.10)

# D.6 Partial Derivatives

Lagrangian mechanics makes extensive use of partial derivatives. Given a function y = y(x 1, x 2,…, xN), a partial derivative with respect to variable xk is defined as

(D.11)

which says to take an ordinary derivative with respect to xk as if variables x 1, x 2,…, x k−1, x k+1,…, xN were constants. It does not say that they are constants, only that they are to be treated as such in calculating the derivative.

Partial derivatives thus depend on the list of variables as well as on the variable being differentiated with respect to, since only that list tells what to hold constant as y is differentiated. The often-seen notation ∂y/∂xk is inherently ambiguous, unless one happens, as here, to know from context what list of variables is intended. In this text we will give the list of variables in all partial derivatives whenever there is any cause for doubt as to what that list might be, often using the shorthand form like ∂y(x)/∂xk in which x stands for the whole list x 1, x 2,…, xN.

One should note that the list of variables x = x 1, x 2,…, xN in a partial derivative is just to indicate those variables to be treated as constants when a derivative is taken. It does not indicate that y necessarily depends on each member of the list in every (p.543) case. Thus, we might have y = 2x 1 + x 3 as the function, even though the list is x 1, x 2,…, x 5. In this case ∂y(x)/∂x 2 = 0, ∂y(x)/∂x 4 = 0, and ∂y(x)/∂x 5 = 0 for all x values.

Conversely, as discussed in Corollary D.10.2 in Section D.10 below, if ∂y(x)/∂xn = 0 for all x values in region R, then y does not depend on xn, and we may choose to expunge xn from the list of variables and write y = y(x 1, x 2,…, x N −1, x N+1,…, xN). For example, the list of variables for the above function might be shortened to x 1, x 2, x 3, with x 4 and x 5 dropped but the x 2 retained.

Partial derivatives are themselves functions of the same variable list x 1, x 2,…, xN as was the function y being differentiated. Thus second and higher derivatives can be taken by repeated application of rules like eqn (D.11). If they exist, these higher derivatives are denoted by expressions like

(D.12)

# D.7 Continuously Differentiable Functions

If all first partial derivatives of a function y = y(x 1, x 2,…, xN) exist and are continuous functions of x, then the function y itself is continuous. Such functions are called continuously differentiable. If all partial derivatives up to and including the nth order exist and are continuous functions of x, the function is called continuously differentiable to nth order. Unless specifically stated otherwise, we will assume that all functions used in the present text are continuously differentiable to any order.

# D.8 Order of Differentiation

If the second partial derivatives exist and are continuous functions for all x in R (i.e. if y(x) is continuously differentiable to second order), then the order of the secondorder partial derivatives is unimportant since, for all i, j,

(D.13)

Generalizing, if y is continuously differentiable to nth order, then all partial derivatives of that order or less are also independent of the order in which they are taken.

# D.9 Chain Rule

Let y be a compound function of r as defined in eqn (D.5). Assume that all partial derivatives of the form ∂y(x)/∂xk and ∂xk(r)/∂rj exist. Then the partial derivatives of y with respect to the r variables exist and may be written using what is called the (p.544) chain rule of partial differentiation,

(D.14)

or, in shorter but equivalent notation,

(D.15)

# D.10 Mean Values

## Theorem D.10.1: The Mean Value Theorem

Suppose that a function y = y(x 1, x 2,…, xN) is continuously differentiable in a region R and has partial derivativesy(x)/∂xk = gk(x). Let hk be increments added to xk and assume that point (x 1 + ηh 1, x 2 + ηh 2,…, xN + ηhN) lies in region R for all η in the range 0 ≤ η ≤ 1. Then

(D.16)

for some θ in the range 0 < θ < 1.

## Corollary D.10.2: Constancy of Functions

Ify(x)/∂xk = 0 for all k = 1,…, N and for all x in R, then function y(x) is constant in that region.

## Corollary D.10.3: Non-dependence on Variables

If a function y = y(x 1, x 2,…, xN) hasy(x)/∂xn = 0 for some n and for all x in R, then function y(x) does not depend on the variable xn and so can be written as y = y(x 1, x 2,…, x N −1, x N+1,…, xN) in that region.

# D.11 Orders of Smallness

We often want to compare two functions as a variable approaches some limit L. A useful notation is found in Chapter I of Titchmarsh (1939). It is

(D.17)

In words, “Function f is of smaller order than ϕ as x approaches L.” The following notation is also used.

(D.18)

In words, “The difference between f and g is of smaller order than ϕ as x approaches L.”

(p.545) For example, with L = 0,

(D.19)

and, with L=∞,

(D.20)

# D.12 Differentials

If y = y(x) is a function of one variable, the change in y as the independent variable is incremented from x to x + dx is y = y(x + dx) − y(x). The increment dx in this expression is not assumed to be small. It may take any value. Assuming that the function is differentiable and has a finite derivative at x, the differential dy at point x is defined as the linear approximation to y based on the tangent line to the curve at point x,

(D.21)

The approximation of y by dy may be good or bad, of course. But we are guaranteed that the difference between the two vanishes for small enough dx. Moreprecisely, it follows from the definition of the derivative that limdx→0 {(Δydy)/dx} = 0, orequivalently, that y = dy + o(dx) as dx → 0. In other words, the difference between Δy and dy is of smaller order than dx, as dx approaches zero.

# D.13 Differential of a Function of Several Variables

The definition of differential can be extended to functions of more than one variable. Our definition of this differential follows that of pages 66–69 of Courant (1936b). Note that he calls it the total differential.

If y = y(x 1, x 2,…, xN) and each variable is independently incremented from xk to xk + dxk for all k = 1,…, N, then the change in y is

(D.22)

and the differential dy is defined as as the linear approximation to y given by

(D.23)

As in the one variable case, the increments dxk may be large or small. If the set x 1, x 2,…, xN is a set of independent variables, then the dxk are independent and may take any values.

(p.546) In the many-variable case, just as for the functions of one-variable discussed above, the approximation of Δy by dy may be good or bad. Define h = maxk{|dxk|}. If y(x) is a continuously differentiable function of x, then the mean value theorem, Theorem D.10.1, implies that

(D.24)

where Δy and dy are defined in eqns (D.22, D.23).

The differential in eqn (D.23) is well defined for any values of dxk. Nonetheless, since eqn (D.24) says that, as h = maxk{|dxk|} goes to zero, the difference between Δy and dy if of smaller order than h, it is also legitimate to think of the differential as the small change in the value of y as the independent variables are incremented by small amounts. The differential is often used heuristically in this way.

# D.14 Differentials and the Chain Rule

The differential of a compound function may be constructed by direct substitution. As in Section D.4, suppose we have a function y = y(x 1, x 2,…, xN) whose differential is

(D.25)

and suppose that each xk is in turn a function of r so that, for for k = 1,…, N,

(D.26)

Substituting eqn (D.26) into eqn (D.25), gives

(D.27)

where the chain rule eqn (D.15) was used to obtain the last equality. But the last expression in eqn (D.27) is precisely the differential of the compound function y = y(r 1, r 2,…, rm). This example illustrates that differentials provide a clear and correct notation for manipulating the chain rule of partial differentiation.

# D.15 Differentials of Second and Higher Orders

The first order differential defined in eqn (D.23) is itself a function of the variables x 1,…, xN, dx 1,…, dxN. Thus, the second-order differential may be defined as the differential of the first-order differential, using the same increments. Writing eqn (p.547) (D.23) using an operator formalism as

(D.28)

we may define

(D.29)

assuming that the second-order partial differentials exist. Generalizing to nth order gives

(D.30)

where we assume that the partial differentials to nth order exist.

# D.16 Taylor Series

The Taylor theorem may be thought of as a generalization of the mean value theorem of Section D.10.

## Theorem D.16.1: Taylor Series

If function y(x 1,…, xN) is continuously differentiable to (n + 1)st order, then

(D.31)

where all of the differentials dy, d 2 y, etc., are to be evaluated using the same increments dxk and where the remainder Rn is

(D.32)

where

(D.33)

and θ is some number in the range 0 < θ < 1.

# (p.548) D.17 Higher-Order Differential as a Difference

Suppose that the conditions of the Taylor theorem of Section D.16 apply, and define the difference y as

(D.34)

Then inspection of eqns (D.31, D.32) shows that

(D.35)

where h = maxk{|dxk|} for 1 ≤ kN.

# D.18 Differential Expressions

Given N functions ak(x) of the set of independent variables x 1, x 2,…, xN, we may form differential expressions like

(D.36)

which may or may not be the actual differential of some function. These expressions may be manipulated by the usual rules of algebra (think of the dxk simply as finite increments).

We adopt the usual convention that an equality involving differential expressions includes the implicit assumption that it holds for all possible dxk values. Since the increments dxk of independent variables x 1, x 2,…, xN can take any values, it follows that the dxk may be set nonzero one at a time, which leads to the following Lemmas.

## Lemma D.18.1: Zero Differential Expressions

The differential expression is zero

(D.37)

if and only if ak(x) = 0 for all k = 1,…, N.

## Lemma D.18.2: Equal Differential Expressions

Two differential expressions are equal,

(D.38)

if and only if ak(x) = bk(x) for all k = 1,…, N.

## (p.549) Lemma D.18.3: Differential Expression and Differential

It follows from eqn (D.38) together with the definition of the differential in eqn (D.23) that, if we are given a function f = f(x 1, x 2,…, xN) of the independent variables x 1, x 2,…, xN and a differential expression $∑ k = 1 N a k ( x ) d x k$, the equality

(D.39)

holds if and only iff(x)/∂xk = ak(x) for all k = 1,…, N.

## Lemma D.18.4: Zero Differential

It follows from Lemma D.18.1 that a function f = f(x 1, x 2,…, xN) of the independent variables x 1, x 2,…, xN will have df = 0 at a point x 1, x 2,…, xN if and only iff(x)/∂xk = 0 for all k = 1,…, N at that point.

In some cases of interest, the variables x will be functions of another set of variables r = r 1, r 2,…, rM as in the discussion of compound functions in Section D.4. In that case, using the chain rule from Section D.14, the differential expression becomes

(D.40)

where

(D.41)

In some applications, it is important to know if the Lemmas above still apply to the differential expression $∑ k = 1 N a k ( x ) d x k$ when one assumes only that the r variables are independent.

## Theorem D.18.5: Compound Differential Expressions

Let the variables r be assumed to be independent so that the increments drj can be set equal to zero one at a time in eqn (D.40). Then Lemmas D.18.1 through D.18.4 will continue to hold for the differential expression $∑ k = 1 N a k ( x ) d x k$ if and only if M = N and the determinant condition

(D.42)

is satisfied.

Proof: Theorem D.24.1 below shows that the determinant condition in eqn (D.42) is the necessary and sufficient condition for the transformation xr to be invertible.

(p.550) Thus, one can use the inverse matrix from Section D.25 to write

(D.43)

from which a choice of the independent increments drj can be found that will make the dxk have any value desired. Thus the dxk are also arbitrary and independent, and can be set nonzero one at a time, which is the condition needed.

# D.19 Line Integral of a Differential Expression

A curve can be defined in region R by making each xk be a function of some monotonically varying parameter β, sothat xk = xk(β) for k = 1,…, N. The integral

(D.44)

is called a line integral of the differential expression $∑ k = 1 N a k ( x ) d x k$ along a portion of that curve.

The line integral in eqn (D.44) is often denoted more simply, as just the integral of a differential expression with integration along a particular curve being understood. Thus, one often sees I 01 denoted as

(D.45)

where the latter form treats the differential expression as a dot product of two vectors in an N-dimensional Cartesian space with the xk as its coordinates.

# D.20 Perfect Differentials

Differential expressions $∑ k = 1 N a k ( x ) d x k$ for which a function f exists satisfying $d f = ∑ k = 1 N a k ( x ) d x k$ are called perfect differentials. Line integrals and perfect differentials. are treated in Chapter V of Courant (1936b) and Appendix 11 in Volume I of Desloge (1982).

The function f is sometimes called a potential function since the ak(x) can be derived from it by partial differentiation in a way analogous to the derivation of the electric field components from the electric potential. A condition for $∑ k = 1 N a k ( x ) d x k$ to be a perfect differential is given by the following theorem.

## Theorem D.20.1: Condition for a Perfect Differential

Assume the variables x 1, x 2,…, xN to lie in an open rectangle R . Given a set of continuously differentiable functions ak(x) for k = 1,…, N, there exists a potential function (p.551) f = f(x 1, x 2,…, xN) such that the following two equivalent conditions are satisfied,

(D.46)

if and only if, for all x in R and all pairs of indices i, j = 1,…, N,

(D.47)

Proof: First we prove that eqn (D.46) implies eqn (D.47). For if an f exists satisfying eqn (D.46), then, using eqn (D.13) gives

(D.48)

To prove that eqn (D.47) implies eqn (D.46) we construct a suitable f explicitly. Starting from some arbitrary point $x 1 ( 0 ) , x 2 ( 0 ) , … , x N ( 0 )$, we perform a line integral along a series of straight line segments, first along x 1, then along x 2, etc., until the final point 1 at x 1, x 2,…, xN is reached. Such a path will lie entirely in R , and the integral I 10 will be the sum of the integrals along its segments. Along the jth segment, all xi for ij are held constant but xj varies as xj = β, giving dxk(β)/d β = δjk. Inserting this result into eqn (D.44), and setting f(x 1, x 2,…, xN) equal to the integral I 10 gives

(D.49)

Since any point x can be reached by this integration, the function f is defined for all x in R .

The partial derivatives of this f may be written as

(D.50)

With xj temporarily replaced by β, the assumption stated in eqn (D.47) implies that

(D.51)

(p.552) Thus

(D.52)

which shows that the f constructed in eqn (D.49) does have the required property eqn (D.46).

The line integral eqn (D.49) used in this proof is of great practical use. It will be used, for example, to compute generating functions for canonical transformations.

The theorem proved in this section (but not, of course, the proof of it given here) remains true even when open rectangle R is replaced by a more general region R which is only assumed to be open and simply connected. See Chapter V of Courant (1936b) for details.

# D.21 Perfect Differential and Path Independence

An alternate, and equivalent, condition for a differential expression to be a perfect differential is that its line integral between two end points be independent of the particular choice of the path between them.

## Theorem D.21.1: Path Independence

Assume a given set of functions ak(x) for k = 1,…, N, continuously differentiable in an open and simply connected region R. The differential expression $∑ k = 1 N a k ( x ) d x k$ is a perfect differential with a potential function satisfying the equivalent conditions eqn (D.46),

(D.53)

if and only if the line integral between any two points in R

(D.54)

depends only on x at the endpoints of the integration x 10), x 20),…, xN0) (p.553) and x 11), x 21),…, xN1) and is therefore independent of the path xk(β) taken between those endpoints.

Proof: First assume eqn (D.53) and prove the path independence of eqn (D.54). If a potential function f exists, then eqn (D.54) becomes

(D.55)

which depends only on the value of f at the end points and hence is independent of the path, as was to be proved. For proof of the converse, see Chapter V of Courant (1936b).

It follow from Theorem D.21.1 that integration along any path between the points $x 1 ( 0 ) , x 2 ( 0 ) , … , x N ( 0 )$ would x 1, x ,,…, x N have given the same integral as the particular path used in the proof of Theorem D.20.1.

The path independence of line integrals between any two points is equivalent to the vanishing of line integrals around closed paths. The following Corollary is given without proof.

## Corollary D.21.2: Closed Paths

The line integral I 01 between any two points is independent of the path if and only if the line integral around any closed path is zero.

# D.22 Jacobians

Suppose that we are given yk = yk(x 1, x 2,…, xN, z 1, z 2,…, zP) = yk(x, z) for k = 1,…, N, where P may have any non-negative value, including the value zero (which would indicate that the extra z variables are absent). Then we may form an N × N matrix of partial derivatives denoted (∂y(x, z)/∂x) and defined by

(D.56)

This matrix is called the Jacobian matrix, and its determinant is called the Jacobian determinant, or simply the Jacobian. It is variously denoted. We give here the two forms we will use, and the determinant itself,

(D.57)

The first form, which is the traditional one, does not specify the list of variables to be held constant when the partial derivatives are taken. It is implicit in the notation that (p.554) all of the variables listed in the denominator are on that list, but there may be others, as here. Usually the intended list is obvious, but in cases of doubt we will modify the traditional notation and use expressions like

(D.58)

in which the list is explicitly stated. Jacobians are treated in Volume I, Appendix 3 of Desloge (1982) and in Chapter III of Courant (1936b).

## Lemma D.22.1: Jacobian of a Compound Function

If the variables xi are themselves functions of another set of variables r, as well as of the same extra variables z as above, xi = xi(r 1, r 2,…, rN, z 1, z 2,…, zP) for i = 1,…, N, then the compound functions yk(r 1, r 2,…, rN, z 1, z 2,…, zP) may be defined by

(D.59)

Then the Jacobians obey the relation

(D.60)

(D.61)

Proof: The chain rule of partial differentiation gives

(D.62)

which may be written as a matrix equation

(D.63)

Equating the determinant of both sides of eqn (D.63) gives eqn (D.60), as was to be proved.

## Lemma D.22.2: Jacobian of an Augmented Variable Set

If yk = yk(x 1, x 2,…, xN, z 1, z 2,…, zP) = yk(x, z) for k = 1,…, N as before, the following identity holds

(D.64)

Proof: The left expression in eqn (D.64) is the determinant of an (N + P) × (N + P) matrix whose last P rows consist of N zeroes followed by an element of the P × P identity matrix. Its determinant is thus the determinant of the N × N upper left-hand block, which is the right expression in eqn (D.64).

## (p.555) Lemma D.22.3: Jacobian of an Inverse Function

If a set of functions yk = yk(x 1, x 2,…, xN, z 1, z 2,…, zP) can be solved for x, giving xi = xi(y 1, y 2,…, yN, z 1, z 2,…, zP) then

(D.65)

(D.66)

Proof: As is proved in Theorem D.24.1 below, the necessary and sufficient condition for yk = yk(x, z) to be solved for xi = xi(y, z) is the determinant condition

(D.67)

Then, as discussed in Section D.25, the following matrix equation holds

(D.68)

where U is the N × N unit matrix. Taking the determinant of both sides gives eqn (D.65).

## Lemma D.22.4: Change of Variable in an Integral

Given a set of continuously differentiable functions yk = yk(x 1, x 2,…, xN) for k = 1,…, N, whose Jacobian does not vanish in the range of integration,

(D.69)

the multiple integral

(D.70)

may be transformed into an integral over the variables x

(D.71)

where the compound function f(x 1, x 2,…, xN) is

(D.72)

and the limits of integration in eqn (D.71) are chosen so that x ranges over the inverse image of the range of y.

In practice, these limits of integration in eqn (D.71) are usually chosen so that the two integrals would have the same value, including the same sign, for the case in which f is replaced by the number 1.

# (p.556) D.23 Global Inverse Function Theorem

We often need to invert a set of functions like yi = yi(x 1, x 2,…, xN, z 1, z 2,…, zP) where i = 1,…, N, that is, to solve them for the variables xj = xj(y 1, y 2,…, yN, z 1, z 2,…, zP). The following global and local inverse function theorems are of great importance.

The inverses proved by the present theorem are called global, because the same, unique inverse functions apply to the whole of an open rectangle. This open rectangle may be indefinitely large. For example, in the transformation from plane polar coordinates ρ, ϕ to plane Cartesian coordinates x, y, the open rectangle might be 0 < ρ < ∞ and −π < ϕ < π.

## Theorem D.23.1: The Global Inverse Function Theorem

Assume that all points x 1, x 2,…, xN, z 1, z 2,…, zP lie in an open rectangle R , and that

(D.73)

for i = 1,…, N, are a set of continuously differentiable functions of the stated variables. If for all x, z in R , the Jacobian determinant

(D.74)

is nonzero and has a persistent, nested set of critical minors,129 then functions yi = yi(x 1, x 2,…, xN, z 1, z 2,…, zP) can be solved for the inverse functions xj = xj(y 1, y 2,…, yN, z 1, z 2,…, zP) for j = 1,…, N so that

(D.75)

These inverse functions will be unique and continuously differentiable in the range covered by variables y, z as variables x, z range over R .

Proof: This proof is adapted from Volume II, Appendix 18 of Desloge (1982). The proof is by induction. First prove the theorem for N = 1. Then prove that, if the theorem is true for N = k − 1, it must be true for N = K. It follows that the theorem must be true for any integer N.

For N = 1, (∂y(x, z)/∂x) = ∂y 1(x 1, z)/∂x 1. By assumption, this partial derivative is nonzero in a range of values a 1 < x 1 < b 1. Considering the z variables as fixed parameters, the inverse function theorem of ordinary one-variable calculus applies. Since ∂y 1(x 1, z)/∂x 1 is continuous and nonzero, it must have the same sign throughout the range. Thus y 1(x 1, z) is monotonic and has a unique inverse x 1 = x 1(y 1, z). Since (p.557) x 1(x 1, z)/∂y 1 = (∂y 1(x 1, z)/∂x 1)−1 it follows that x 1 = x 1(y 1, z) is also a continuously differentiable function.

For N = K, the Jacobian determinant is

(D.76)

Since by assumption this determinant is nonzero, it must have a critical (k − 1) rowed minor. By assumption, the same minor is nonzero for all x, z in R . For simplicity (and without loss of generality since the functions and variables can be relabeled in any way) we assume that this is the minor with the first row and first column removed. Then

(D.77)

By the induction assumption, the theorem is true for N = k − 1. So eqn (D.77), and its assumed persistent nested critical minors, imply that inverse functions exist of the form

(D.78)

where the set z 1, z 2,…, zP is now being represented by the single unsubscripted letter z. Substitute eqn (D.78) into y 1 to obtain the compound function y 1 = y 1(x 1, y 2,…, yK, z) defined by

(D.79)

We now show that this function can be solved for x 1. Using the chain rule and eqn (p.558) (D.79), the partial derivative ∂y 1(x 1, y 2,…, yK, z)/∂x 1 can be expanded as

(D.80)

where eqn (D.64) has been used. Now multiply each term in eqn (D.80) by the Jacobian determinant

(D.81)

and use eqn (D.61) to obtain

(D.82)

But eqn (D.64) and the usual rules for sign change when rows or columns of a determinant are exchanged, show the last Jacobian determinant in eqn (D.82) to be

(D.83)

Thus

(D.84)

The right side of eqn (D.84) is just the Jacobian determinant ∂(y 1, y 2,…, yK)/∂(x 1, x 2,…, xK) evaluated using expansion by cofactors along its first row. Thus eqn (D.82) becomes

(D.85)

Since both

(D.86)

(p.559) by assumption, this proves that the partial derivative

(D.87)

Since R is an open rectangle, eqn (D.87) will hold for any fixed values of y 2,…, yK, z 1, z 2,…, zP and for all a 1 < x 1 < b 1 where a 1 and b 1 are the least and greatest value of x 1 in the open rectangle. Thus, by the same reasoning as was used for the case N = 1 above, y 1 = y 1(x 1, y 2,…, yK, z) may be inverted to yield the unique, continuously differentiable inverse function x 1 = x 1(y 1, y 2,…, yK, z). Substituting that equation into eqn (D.78) gives the desired result: xi = xi(y 1, y 2,…, yK, z) for all i = 1,… K.

Since the truth of the theorem for N = K − 1 proves its truth for N = K, the theorem must be true for any N.

# D.24 Local Inverse Function Theorem

In many cases, global inverses provided by Theorem D.23.1 are not needed, and indeed may not be available. In these cases, we can still define local inverse functions. These local inverses will be proved to exist only in some open neighborhood N x, z surrounding point x, z.

Local inverses are important because they may exist at points of a wider class of regions than the R assumed in Theorem D.23.1. And, of course, if a global inverse does exist in a region, then local inverses will exist at each point of that region also.

## Theorem D.24.1: The Local Inverse Function Theorem

Assume that point x 1, x 2,…, xN, z 1, z 2,…, zP lies in an open region R, and that

(D.88)

are continuously differentiable functions of the stated variables. If the Jacobian determinant

(D.89)

is nonzero at the point x, z, then there is some open neighborhood Nx, z of this point in which functions yi = yi(x 1, x 2,…, xN, z 1, z 2,…, zP) can be solved for the inverse functions

(D.90)

for j = 1,…, N. These inverse function will be unique and continuously differentiable in the range covered by variables y, z as variables x, z vary over Nx, z.

Proof: Since R is open, every point x, z is in some open neighborhood N x, z which is contained entirely in R. Since the function y is assumed to be continuously differentiable, the Jacobian eqn (D.74), and all of its minors, are continuous functions of x, z. Thus, since the Jacobian (and hence a set of nested minors) are nonzero at x, z by assumption, we may shrink the open neighborhood N x, z until these determinants (p.560) are nonzero over the whole of N x, z. Since, by definition, the open neighborhood is a (possibly small) open rectangle, the conditions of Theorem D.23.1 are satisfied, and the local inverse functions exist, as was to be proved. A direct proof of this theorem, not based on the global inverse function theorem, is given on page 152 of Courant (1936b).

# D.25 Derivatives of the Inverse Functions

If inverse functions, from either the global or local inverse function theorems, exist at some point x, z, then the partial derivatives of the inverse functions xj can be expressed in terms of the partial derivatives of the original functions yi at that point.

Substituting eqn (D.90) into eqn (D.88) gives the compound function

(D.91)

The chain rule then gives

(D.92)

which in matrix form is

(D.93)

where U denotes the N × N identity matrix. It follows that both product matrices are nonsingular and that

(D.94)

and hence that, for any i, j

(D.95)

which expresses the partials of xj with respect to the yi as functions of the partials of the original functions yi(x, z).

Similarly, applying the chain rule to the differentiation of eqn (D.91) with respect to zn gives

(D.96)

which leads to the matrix equation

(D.97)

and hence

(D.98)

which expresses the partials of xj with respect to the zn in terms of the partials of the (p.561) original functions yi(x, z).

# D.26 Implicit Function Theorem

It often happens that a set of functions yj = yj(x 1,x 2,…,xP), for j = 1,…,N, is not given directly, but rather in implicit form. One defines other functions fi(x, y), for i = 1,…,N, and requires all x,y values to be those that will make these fi identically zero,

(D.99)

The following theorem gives the conditions under which such identities actually specify the implicit functions yj.

## Theorem D.26.1: Implicit Function Theorem

Assume that fi(x 1, x 2,…, xP, y 1, y 2,…, yN) for i = 1,…, N, are continuously differentiable functions. If the Jacobian determinant

(D.100)

is nonzero at point x, y, then there is an open neighborhood Nxy of the point x, y in which the identities

(D.101)

for i = 1,…, N, can be solved for the implicit functions yj = yj(x 1, x 2,…, xP) for j = 1,…, N. These functions will be unique and continuously differentiable in the open neighborhood.

Proof: Apply the local inverse function theorem, Theorem D.24.1, to solve fi = fi(x 1, x 2,…, x p, y 1, y 2,…, y N) for y j = y j(f 1,…, f N, x 1,…, x p). Then apply the identity to set fi = 0 for each i, and so obtain

(D.102)

which are the desired functions.

# D.27 Derivatives of Implicit Functions

Applying eqn (D.98) of Section D.25 with the replacements yf, xy, zx, and f then set equal to zero, gives the partial derivatives of the implicit functions in terms of the partial derivatives of the fi. Thus, for all j = 1,…, N and n = 1,…, P,

(D.103)

# (p.562) D.28 Functional Independence

Consider the M continuously differentiable functions of N variables fk(x 1, x 2,…, xN) for k = 1,…, M. These functions are functionally dependent and have a dependency relation at point x if there is a continuously differentiable function F with at least one nonzero partial derivative ∂ F(f)/∂ fi ≠ 0 for which

(D.104)

holds identically in an open neighborhood Nx of point x.

If the functions have no dependency relation at point x, they are said to be functionally independent at that point. We now give a condition for a set of functions fk to be functionally independent.

## Theorem D.28.1: Condition for Functional Independence

Consider M continuously differentiable functions of N variables, fk(x 1, x 2,…, xN) for k = 1,…, M. Letthe M × N matrix (∂ f(x)/∂x) be defined by its matrix elements

(D.105)

If, for some x, therank r of this matrix is r = M (which is possible only if MN since r cannot be greater than N), then the functions fk are functionally independent at x.

Proof: We show that the existence of a dependency relation implies that r < M, and therefore that r = M implies functional independence. Differentiating an assumed dependency relation eqn (D.104) with respect to xi using the chain rule gives

(D.106)

Defining an M × 1 column vector[∂ F(f)/∂ f] by

(D.107)

we may rewrite eqn (D.106) as the matrix equation

(D.108)

By assumption the column vector [∂ F(f)/∂f] will have at least one nonzero element. Thus Corollary B.19.2 requires that the matrix in eqn (D.108), and hence its transpose (∂f(x)/∂x), must have rank less than M. Since the existence of a dependency relation implies r < M, it follows that r = M at point x implies the non-existence of a dependency relation, as was to be proved.

# (p.563) D.29 Dependency Relations

When functions are functionally dependent, there may be one or more dependency relations among them. It is often important to know how many dependency relations there are, as specified in the following theorem.

## Theorem D.29.1: Dependency Relations

Consider M continuously differentiable functions,

(D.109)

of N variables x = x 1, x 2,…, xN. Consider again the M × N matrix (∂f(x)/∂x) defined by its matrix elements in eqn (D.105). If at the point x the rank r of this matrix is r < M, then there are Mr functionally independent dependency relations among the fk which hold in an open neighborhood Nx of point x,

(D.110)

For a proof of this theorem, see Volume I, Appendix 14 of Desloge (1982).

# D.30 Legendre Transformations

We often have a function like f(x 1,…, xM, y 1,…, yN) = f(x, y) that is important mainly because its partial derivatives yield desired functions ui = ui(x, y) and wj = wj(x, y), as in

(D.111)

for i = 1,…, M and j = 1,…, N.

It is sometimes useful to have a different, but related, function

(D.112)

whose partial derivatives are

(D.113)

again for i = 1,…, M and j = 1,…, N. Notice that the roles of yj and wj are interchanged in the last partial derivatives in eqns (D.111, D.113), while the partials with respect to xi yield the same function ui, except for the minus sign and its expression in the new x, w variable set. This transformation from f to g is called a Legendre transformation.

(p.564) The Legendre transformation is effected by defining g to be

(D.114)

But the g in eqn (D.114) is not yet expressed in the correct set of variables x, w. In order to complete the Legendre transformation, we must prove that the functions wj = wj(x, y) defined in the second of eqn (D.111) can be inverted to give yj = yj(x, w). By Section D.24, the condition for this inversion to be possible is that

(D.115)

where the matrix (∂w(x, y)/∂y) is defined by

(D.116)

In general, eqn (D.115) must be proved in each individual case to which the Legendre transformation is to be applied. Assume now that this has been done.

The inverse function yj = yj(x, w) then allows us to write g as a compound function of the variable set x, w as

(D.117)

Now consider the differential of function g. Equation (D.114) gives

(D.118)

where the second of eqn (D.111) has been used to cancel the dyj terms, and the first of eqn (D.111) has been used to get the ui. We know from eqn (D.117) that g(x, w) exists and is well defined, and hence has the differential

(D.119)

Equations (D.118, D.119) are two expressions for the same differential dg and hence equal to each other. Since|∂w(x, y)/∂y|≠ 0 by assumption, Theorem D.18.5 (p.565) shows that the differentials dw, dx may be considered as arbitrary and independent. Thus the equality of eqns (D.118, D.119) implies the equality of each of the coefficients of the differentials dw j and dxi in the two equations. Thus eqn (D.113) holds, as desired.

# D.31 Homogeneous Functions

Let f (x 1, x 2, …, xN, z 1, z 2, …, zP) = f(x, z) be a continuously differentiable function of the stated variables, defined over a region R. Function f(x, z) is homogeneous of degree k in the set of variables x 1, x 2, …, xN if and only if, for some k and any positive number λ > 0,

(D.120)

An alternate, and equivalent, definition of homogeneous functions is given in the following theorem, whose proof can be found in Chapter II of Courant (1936b).

## Theorem D.31.1: Euler Condition

Function f(x, z) is homogeneous of degree k in the set of variables x1, x2, …, xN as defined in eqn (D.120) if and only if

(D.121)

# D.32 Derivatives of Homogeneous Functions

In Lagrangian mechanics, it is important also to consider the homogeneity of the partial derivatives of homogeneous functions.

## Theorem D.32.1: Derivatives of Homogeneous Functions

Function f is homogeneous of degree k + 1 in the set of variables x1, x2, …, xN, that is

(D.122)

if and only if all partial derivativesf(x, z)/∂xi = gi(x, z) for i = 1, …, N are homogeneous of degree k, that is

(D.123)

Proof: First, we assume eqn (D.122) and prove eqn (D.123). Taking partial derivatives, and using eqn (D.122) gives

(D.124)

which establishes eqn (D.123).

(p.566) Conversely, assume eqn (D.123) and prove eqn (D.122). Equation (D.123) can be written

(D.125)

Hence

(D.126)

and so

(D.127)

It follows from Corollary D.10.2 that fx, z) − λ(k+1) f(x, z) = C. Setting λ = 1 proves that the constant C = 0, which is equivalent to eqn (D.122). ⎸

# D.33 Stationary Points

For functions of one variable f = f(x), maxima, minima, and points of inflection (collectively called stationary points) are those points at which df(x)/dx = 0.

For functions of many variables, stationary points may be defined similarly as points at which, for all k = 1, …, N,

(D.128)

Since the variables x and hence their differentials dxk for k = 1, …, N are assumed independent, Lemma D.18.4 implies that this definition may be stated more simply, and equivalently, as

(D.129)

Since we know that this differential approximates the difference Δf when the differentials dxj are small, it is also valid to say that the extremum is a point such that f is constant to first order for small excursions from it in any direction.

In one sense, the many variable case is much more complex than the single variable one. A function of many variables may have maxima in some variables, minima in others, etc. We avoid this complexity here by assuming that it will be clear from context in most practical examples whether the stationary point is a maximum, minimum, or some more complicated mixture of the two.

# D.34 Lagrange Multipliers

Often, we want to find the stationary points of a function f(x) given certain constraints on the allowed values of x 1, x 2, …, xN and their differentials. These constraints are expressed by defining a functionally independent set of functions Ga and setting them equal to zero, 0= Ga(x 1, x 2, …, xN) for a = 1, …, C.

For example, with

(D.130)

defined to be the distance from the origin in a three-dimensional Cartesian space, eqn (D.128) gives a stationary point at (0, 0, 0), obviously a minimum. But suppose that (p.567) we want the stationary points of f subject to the constraint that x lie on a plane at distance Λ from the origin, which can be expressed by

(D.131)

where constants α, β, γ obeyα2 + β2 + γ2 = 1. The Lagrange multiplier theorem gives an elegant method for solving such problems.

## Theorem D.34.1: Lagrange Multiplier Theorem

Let the values of the independent variables x 1, x 2,…, xN in some open region R be constrained by equations of the form

(D.132)

for a = 1,…, C, where the Ga are continuously differentiable functions. Assume that the C × N Jacobian matrix g defined by

(D.133)

has rank C so that the functions Ga are functionally independent and represent C independent constraints.

A continuously differentiable function f(x 1, x 2,…, xN) has a stationary point at x, subject to these constraints, if and only if there exist Lagrange multipliers λa = λa(x) such that, at the stationary point,

(D.134)

for all k = 1,…, N.

Proof: The necessary and sufficient condition for a stationary point is that df = 0 subject to the constraints on the possible dxk values given by the vanishing of the differentials of eqn (D.132)

(D.135)

for all a = 1,…, C. Since the matrix g defined in eqn (D.133) has rank C, it must have an C-rowed critical minor. The variables x can be relabeled in any way, so we lose no generality by assuming that this critical minor is the determinant containing the C rows and the last C columns. Call the corresponding matrix g(b) so that $g a j ( b ) = g a ( N - C + j )$ for a, j = 1,…, C. By construction, the determinant of this matrix is a critical minor, and so |g(b)| ≠ 0. In the following, we will also use the shorthand notations x(f) for what will be called the free variables x 1,…, x NC and x(b) for the bound variables x NC+1,…, xN. And x will continue to denote the full set of variables x1, x2,…, xN.

(p.568) Using these notations, eqn (D.135) may be written with separate sums over the free and bound variables

(D.136)

Since g(b) is nonsingular, it has an inverse g(b)−1, which may be used with eqn (D.136) to write the bound differentials dx (b) in terms of the free ones.

(D.137)

The differential df can also be written with separate sums over the free and bound variables

(D.138)

Substituting eqn (D.137) into this expression gives

(D.139)

where the λa in the last expression are defined, for a = 1,…,C, as

(D.140)

The solution, eqn (D.137), for the bound differentials dx (b) reduces eqn (D.135) to an identity. Hence no constraint is placed on the free differentials dx(f). Setting these free differentials nonzero one at a time in eqn (D.139), the condition df = 0 for x to be a stationary point implies and is implied by

(D.141)

for all i = 1,…,(NC), which establishes the theorem for those values of the index.

For indices in the range (NC + 1),…, N, with the same choice of λa as defined in eqn (D.140), the eqn (D.134) is satisfied identically at all points, including (p.569) the stationary one. To demonstrate this, let j = 1,…, C and use eqn (D.140) to write

(D.142)

Thus eqn (D.134) holds for all index values if and only if x is a stationary point, as was to be proved.

Note that the N equations of eqn (D.134) together with the C equations of eqn (D.132), are N + C equations in the N + C unknowns x 1, x 2,…, xN, λ1,…, λC, and so can be solved to find the stationary points.

# D.35 Geometry of the Lagrange Multiplier Theorem

When applied to the simple example in eqns (D.130, D.131) of Section D.34, the three equations of the Lagrange multiplier condition eqn (D.134) can be written as a single vector equation, the equality of two gradient vectors,

(D.143)

Since, as described in Section D.37, vector ∇G 1 = αê 1 + βê 2 + γ ê 3 is perpendicular to the surface of constraint G 1 = 0, eqn (D.143) says that ∇ f must be in the same or opposite direction, and so also perpendicular to that constraint surface.

If we denote by d r = dx 1 ê 1 + dx 2 ê 2 + dx 3 ê 3 the differential displacement vector whose Cartesian components are the differentials dxi, the chain rule gives

(D.144)

The constraint G 1 = 0, and hence dG 1 = 0, constrains the vector d r to have a zero dot product with ∇G 1, and so to be perpendicular to the perpendicular to the surface of constraint. Thus d r must lie in the surface of constraint.

If there were no constraint, the condition for f to have a stationary point would simply be that

(D.145)

Since without a constraint the displacement d r can take any direction, eqn (D.145) implies that ∇ f = 0, which is equivalent to the three equations of eqn (D.128).

However, with the constraint, eqn (D.145) does not imply that ∇ f = 0, butonly that ∇ f must have no components in possible d r directions. Since d r can lie anywhere (p.570) in the surface of constraint, this means that ∇ f must be perpendicular to that surface, in other words that ∇ f must be parallel or anti-parallel to ∇G 1, as eqn (D.143) states.

If another constraint were added, then eqn (D.143) would become

(D.146)

Adding the second constraint further restricts the possible directions of d r and hence increases the possible directions that ∇ f can have while still maintaining the condition d r · ∇ f = 0.

# D.36 Coupled Differential Equations

A basic theorem of one-variable calculus is that a first-order differential equation dx/dβ = f(β, x), with the initial condition that x = b when the independent variable is β = β0, has a unique solution x = x(β, b). That result is generalized to a set of N functions of β by the following theorem.

## Theorem D.36.1: Coupled Differential Equations

Consider a set of N unknown functions xi for i = 1, …, N obeying the coupled, first order differential equations

(D.147)

and the initial conditions that xi = bi when the independent variable is β = β0 (where bi for i = 1, …, N are arbitrarily chosen constants). Assume that the functions fi are continuously differentiable. These equations have a unique solution depending on β and the set of constants b 1, …, bN. The solution is the set of equations

(D.148)

for i = 1, …, N. The same solution can also be written in implicit form by solving eqn (D.148) for the b and writing

(D.149)

where the ϕi are functionally independent, and are called integrals of the set of differential equations.

The proof of this standard theorem can be found, for example, in Chapter 6 of Ford (1955). Note that the initial value of the independent variable β0 is not included in the list of constants upon which the xi depend. It is simply the value of β at which the integration constants bi are specified.

The special case in which the functions fi in eqn (D.147) do not depend explicitly on β is of great importance in analytical mechanics.

## (p.571) Theorem D.36.2: Sets of Equations Without the Independent Variable

Consider the set of differential equations which are the same as eqn (D.147) but with the independent variable β not appearing explicitly in the functions fi

(D.150)

for i = 1, …, N. Assume that N ≥ 2 and that there is some index l for which fl = 0 in some region of interest. For simplicity, but without loss of generality, assume the variables relabeled so that l = 1. Given the initial conditions that, for j = 2, …, N, xj = bj when x 1 = b 1 , these equations have a unique solution

(D.151)

for j = 2, …, N, in which the role of independent variable has been assumed by x 1 and the number of arbitrary integration constants has been reduced by one.

This solution also can be written in implicit form by solving eqn (D.151) for the b and writing

(D.152)

for j = 2, …, N. The ϕj are the functionally independent integrals of eqn (D.150). There are only N − 1 integrals ϕj, and they do not depend explicitly on β.

Proof: Assuming f 1 ≠ 0, divide the last N − 1 of eqn (D.150) by the first one, giving

(D.153)

for j = 2, …, N. The functions Aj defined in eqn (D.153) are continuously differentiable, and hence Theorem D.36.1 can be applied with fA, the replacement β → x 1 for the independent variable, and the range of unknown functions now running from x 2 to xN. The theorem follows immediately.

Note that the initial value x 1 = b 1 for the new independent variable is not included among the arbitrary integration constants b 2, …, bN. As in the original theorem with β0, it is simply the value of the new independent variable x 1 at which the arbitrary integration constants xj = bj, for j = 2, …, N, are specified.

The variable β has disappeared from the solutions, eqns (D.151, D.152). Instead of having all the xi as functions of some β, we have solutions with all of the xj for j = 1 given as functions of one of them, the x 1.

But sometimes, even under the conditions of Theorem D.36.2, it is convenient to have a solution with all of the unknowns written as functions of β. Such a parametric expression can always be found, as is shown in the following Corollary.

## Corollary D.36.3: Recovery of the Parameter

Assume that the conditions in Theorem D.36.2 hold. Using solutions eqn (D.151), a set of functions

(D.154)

for i = 1, …, N, can always be found that are solutions of the original equations, eqn (D.150). They depend only on the difference (β − β0) as indicated. Thus no generality is (p.572) lost by simply taking β0 = 0, as is frequently done. The solutions in eqn (D.154) can be constructed to obey the initial conditions that β = β0 implies xi = bi for all i = 1, …, N.

Proof: The first of eqn (D.150) is

(D.155)

where eqn (D.151) has been used to get the last equality. Thus we have a differential equation with one unknown function x 1

(D.156)

Integrating, with x 1 assigned the value b 1 at β = β0,

(D.157)

Denoting by F −1 the inverse to function F for fixed values of the constants b then gives

(D.158)

which has, by construction, the value x 1 = b 1 when β = β0. Substituting this x 1 into the solutions eqn (D.151) then gives

(D.159)

for j = 2, …, N. Since the solutions eqn (D.151) have the property that x 1 = b 1 implies xj = bj, the xj defined in eqn (D.159) must taken those same values bj when β = β0. Equations (D.158, D.159) are the desired eqn (D.154) and the corollary is proved.

A note of caution: Even though eqn (D.154) does write the solution to eqn (D.150) in a form that appears to have N arbitrary integration constants b 1, …, bN, the b 1 is not actually an integration constant. It is the initial value of the independent variable x 1 in solution eqn (D.151).

# D.37 Surfaces and Envelopes

A two-dimensional surface in three dimensions may be written either as

(D.160)

Any surface in the first form can be written in the second form as F (x, y, z) = ϕ(x, y) − z = 0. The second form is preferable since it often leads to simpler expressions. For example, writing the unit sphere as F(x, y, z) = x 2 + y 2 + z 2 −1 = 0 avoids (p.573) the need to use different signs of a square root for the upper and lower hemispheres as the first form would require.

The differential change in F resulting from differential displacement d r is given by the chain rule as dF = d r · ∇F, which is maximum when d r∥∇F and zero when d r⊥∇F. Since a small displacement along the surface keeps F = 0 and so must have dF = 0, it follows that vector ∇F is perpendicular to the surface at point x, y, z. The tangent plane touching the surface at r thus consists of the points r′ where

(D.161)

and ∇F is normal to the tangent plane.

Consider now what is called a one-parameter family of surfaces, defined by

(D.162)

where each different value of parameter a in general gives a different surface. Often (but not always) the two surfaces with parameter values a+da and ada in the limit da→0 will intersect in a curved line called a curve of intersection. This is also called a characteristic curve by some authors, but we use the term “curve of intersection” from Courant and Hilbert (1962) to avoid confusion with other curves in the theory of partial differential equations that are also called characteristic curves.

The curve of intersection lies in both surfaces and hence obeys both F(x, y, z, a + da) = 0 and F(x, y, z, ada) = 0. But these two equations are satisfied if and only if both

(D.163)

Taking the limit da→0, the curve of intersection can therefore be defined as the solution of the two equations

(D.164)

It is the intersection of the surfaces F = 0 and G = 0 where

(D.165)

As an example, consider the family of spheres F = x 2 + y 2 + z 2a 2 = 0. Since the surfaces in this family are concentric spheres, the surfaces with a+da and ada never intersect and so there is no curve of intersection.

A better example for our purposes is the family of unit spheres with center at point a on the z-axis, F = x 2 + y 2+(za)2 − 1 = 0. Then ∂ F/∂a = 2(za)= 0 shows that the curve of intersection for a given a is the intersection of the plane z=a with the sphere x 2 + y 2+(za)2 − 1 = 0. It is a unit circle lying in a plane parallel to the x-y plane and at height z=a. This circle is also the equator of the sphere for that a value.

(p.574) A suitable one-parameter family of surfaces has a curve of intersection for every value of a. The envelope of the one-parameter family is a surface defined as the set of all points of all possible curves of intersection. It may be thought of as the surface swept out by the curve of intersection as a is varied. It is found by solving the second of eqn (D.164) for a as a function of x, y, z and substituting this a(x, y, z) into the first of eqn (D.164), thus eliminating a between the two equations. For a given value of a, thesurface F = 0 and the envelope are in contact along the curve of intersection defined by that a value.

In the above example, the envelope is got by using ∂ F/∂a = 2(za)= 0 to get a(x, y, z) = z and then substituting that a into F = x 2 + y 2+(za)2 −1 = 0 to obtain x 2 + y 2 − 1 = 0. The envelope is thus a right circular cylinder of unit radius with its symmetry line along the z-axis. This envelope is the surface swept out by all of the curves of intersections (circles of unit radius at height a)as a is varied. For any value of a, the sphere is in contact with the envelope along the equator of the sphere. The sphere thus contacts the envelope along the curve of intersection for that a value.

## Notes:

(128) Definitions, and proofs of theorems stated here without proof, may be found in Chapter II of Courant (1936b).

(129) “Persistent” here means that the nonzero Jacobian determinant in eqn (D.74) must have the same N − 1 rowed critical (i.e. nonzero) minor throughout R . “Nested” means that this persistent N − 1 rowed critical minor must, in turn, have the same N − 2 rowed critical minor throughout R , etc.


text: dc.contributor.author Halabi, GM dc.contributor.author Izzard, Robert dc.contributor.author Tout, Christopher dc.date.accessioned 2019-01-11T00:32:28Z dc.date.available 2019-01-11T00:32:28Z dc.date.issued 2018-11 dc.identifier.issn 0035-8711 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/287857 dc.description.abstract We evolve stellar models to study the common envelope (CE) interaction of an early asymptotic giant branch star of initial mass $5\,\rm M_{\odot}$ with a companion star of mass ranging from $0.1$ to $2\,\rm M_{\odot}$. We model the CE as a fast stripping phase in which the primary experiences rapid mass loss and loses about 80 per cent of its mass. The post-CE remnant is then allowed to thermally readjust during a Roche-lobe overflow (RLOF) phase and the final binary system and its orbital period are investigated. We find that the post-CE RLOF phase is long enough to allow nuclear burning to proceed in the helium shell. By the end of this phase, the donor is stripped of both its hydrogen and helium and ends up as carbon-oxygen white dwarf of mass about $0.8\,\rm M_{\odot}$. We study the sensitivity of our results to initial conditions of different companion masses and orbital separations at which the stripping phase begins. We find that the companion mass affects the final binary separation and that helium-shell burning causes the star to refill its Roche lobe leading to post-CE RLOF. Our results show that double mass transfer in such a binary interaction is able to strip the helium and hydrogen layers from the donor star without the need for any special conditions or fine tuning of the binary parameters. dc.publisher Oxford University Press (OUP) dc.title Post-common envelope binary systems experiencing helium-shell-driven stable mass transfer dc.type Article prism.endingPage 5183 prism.issueIdentifier 4 prism.publicationDate 2018 prism.publicationName Monthly Notices of the Royal Astronomical Society prism.startingPage 5176 prism.volume 480 dc.identifier.doi 10.17863/CAM.35172 dcterms.dateAccepted 2018-08-10 rioxxterms.versionofrecord 10.1093/mnras/sty2243 rioxxterms.version VoR rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved rioxxterms.licenseref.startdate 2018-01-01 dc.contributor.orcid Izzard, Robert [0000-0003-0378-4843] dc.contributor.orcid Tout, Christopher [0000-0002-1556-9449] dc.identifier.eissn 1365-2966 dc.publisher.url http://dx.doi.org/10.1093/mnras/sty2243 rioxxterms.type Journal Article/Review pubs.funder-project-id Science and Technology Facilities Council (ST/M003892/1) cam.issuedOnline 2018-08-16 cam.orpheus.success Thu Jan 30 10:52:26 GMT 2020 - The item has an open VoR version. rioxxterms.freetoread.startdate 2100-01-01