1 Algebraic and Analytic MethodsAreas1.4 Calculus of One Variable1.6 Vectors and Vector-Valued Functions

- §1.5(i) Partial Derivatives
- §1.5(ii) Coordinate Systems
- §1.5(iii) Taylor’s Theorem; Maxima and Minima
- §1.5(iv) Leibniz’s Theorem for Differentiation of Integrals
- §1.5(v) Multiple Integrals
- §1.5(vi) Jacobians and Change of Variables

A function $f(x,y)$ is *continuous at a point* $(a,b)$ if

1.5.1 | $$\underset{(x,y)\to (a,b)}{lim}f(x,y)=f(a,b),$$ | ||

that is, for every arbitrarily small positive constant $\u03f5$ there exists $\delta $ ($>0$) such that

1.5.2 | $$ | ||

for all $\alpha $ and $\beta $ that satisfy $$.

A function is *continuous on a point set* $D$ if it is continuous at all
points of $D$. A function $f(x,y)$ is *piecewise continuous* on
${I}_{1}\times {I}_{2}$, where ${I}_{1}$ and ${I}_{2}$ are intervals, if it is piecewise
continuous in $x$ for each $y\in {I}_{2}$ and piecewise continuous in $y$ for each
$x\in {I}_{1}$.

1.5.3 | $\frac{\partial f}{\partial x}$ | $={\mathit{D}}_{x}f={f}_{x}=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h,y)-f(x,y)}{h}},$ | ||

1.5.4 | $\frac{\partial f}{\partial y}$ | $={\mathit{D}}_{y}f={f}_{y}=\underset{h\to 0}{lim}{\displaystyle \frac{f(x,y+h)-f(x,y)}{h}}.$ | ||

1.5.5 | $\frac{{\partial}^{2}f}{\partial x\partial y}$ | $={\displaystyle \frac{\partial}{\partial x}}\left({\displaystyle \frac{\partial f}{\partial y}}\right),$ | ||

$\frac{{\partial}^{2}f}{\partial y\partial x}$ | $={\displaystyle \frac{\partial}{\partial y}}\left({\displaystyle \frac{\partial f}{\partial x}}\right).$ | |||

The function $f(x,y)$ is *continuously differentiable* if $f$,
$\partial f/\partial x$, and $\partial f/\partial y$ are continuous, *and
twice-continuously differentiable* if also ${\partial}^{2}f/{\partial x}^{2}$,
${\partial}^{2}f/{\partial y}^{2}$, ${\partial}^{2}f/\partial x\partial y$, and
${\partial}^{2}f/\partial y\partial x$ are continuous. In the latter event

1.5.6 | $$\frac{{\partial}^{2}f}{\partial x\partial y}=\frac{{\partial}^{2}f}{\partial y\partial x}.$$ | ||

1.5.7 | $\frac{d}{dt}}f(x(t),y(t))$ | $={\displaystyle \frac{\partial f}{\partial x}}{\displaystyle \frac{dx}{dt}}+{\displaystyle \frac{\partial f}{\partial y}}{\displaystyle \frac{dy}{dt}},$ | ||

1.5.8 | $\frac{\partial}{\partial u}}f(x(u,v),y(u,v))$ | $={\displaystyle \frac{\partial f}{\partial x}}{\displaystyle \frac{\partial x}{\partial u}}+{\displaystyle \frac{\partial f}{\partial y}}{\displaystyle \frac{\partial y}{\partial u}},$ | ||

1.5.9 | $\frac{\partial}{\partial v}}f(x(u,v),y(u,v),z(u,v))$ | $={\displaystyle \frac{\partial f}{\partial x}}{\displaystyle \frac{\partial x}{\partial v}}+{\displaystyle \frac{\partial f}{\partial y}}{\displaystyle \frac{\partial y}{\partial v}}+{\displaystyle \frac{\partial f}{\partial z}}{\displaystyle \frac{\partial z}{\partial v}}.$ | ||

If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and
$\partial F/\partial y\ne 0$ at
$(a,b)$, then in a *neighborhood* of $(a,b)$, that is, an open disk
centered at $a,b$, the equation $F(x,y)=0$ defines a continuously
differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and
${g}^{\prime}(x)=-{F}_{x}/{F}_{y}$.

The notations given in this subsection, and also in other coordinate systems in the DLMF, are those generally used by physicists. For mathematicians the symbols $\theta $ and $\varphi $ now are usually interchanged.

With $$, $0\le \varphi \le 2\pi $,

1.5.10 | $x$ | $=r\mathrm{cos}\varphi ,$ | ||

$y$ | $=r\mathrm{sin}\varphi ,$ | |||

1.5.11 | $\frac{\partial}{\partial x}$ | $=\mathrm{cos}\varphi {\displaystyle \frac{\partial}{\partial r}}-{\displaystyle \frac{\mathrm{sin}\varphi}{r}}{\displaystyle \frac{\partial}{\partial \varphi}},$ | ||

1.5.12 | $\frac{\partial}{\partial y}$ | $=\mathrm{sin}\varphi {\displaystyle \frac{\partial}{\partial r}}+{\displaystyle \frac{\mathrm{cos}\varphi}{r}}{\displaystyle \frac{\partial}{\partial \varphi}}.$ | ||

The *Laplacian* is given by

1.5.13 | $${\nabla}^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{\partial y}^{2}}=\frac{{\partial}^{2}f}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial}^{2}f}{{\partial \varphi}^{2}}.$$ | ||

With $$, $0\le \varphi \le 2\pi $, $$,

1.5.14 | $x$ | $=r\mathrm{cos}\varphi ,$ | ||

$y$ | $=r\mathrm{sin}\varphi ,$ | |||

$z$ | $=z.$ | |||

Equations (1.5.11) and (1.5.12) still apply, but

1.5.15 | $${\nabla}^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{\partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}=\frac{{\partial}^{2}f}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial}^{2}f}{{\partial \varphi}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}.$$ | ||

With $$, $0\le \varphi \le 2\pi $, $0\le \theta \le \pi $,

1.5.16 | $x$ | $=\rho \mathrm{sin}\theta \mathrm{cos}\varphi ,$ | ||

$y$ | $=\rho \mathrm{sin}\theta \mathrm{sin}\varphi ,$ | |||

$z$ | $=\rho \mathrm{cos}\theta .$ | |||

The Laplacian is given by

1.5.17 | $$\begin{array}{ll}{\nabla}^{2}f& =\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{\partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}\\ & =\frac{1}{{\rho}^{2}}\frac{\partial}{\partial \rho}\left({\rho}^{2}\frac{\partial f}{\partial \rho}\right)+\frac{1}{{\rho}^{2}{\mathrm{sin}}^{2}\theta}\frac{{\partial}^{2}f}{{\partial \varphi}^{2}}+\frac{1}{{\rho}^{2}\mathrm{sin}\theta}\frac{\partial}{\partial \theta}\left(\mathrm{sin}\theta \frac{\partial f}{\partial \theta}\right).\end{array}$$ | ||

If $f$ is $n+1$ times continuously differentiable, then

1.5.18 | $$f(a+\lambda ,b+\mu )=f+\left(\lambda \frac{\partial}{\partial x}+\mu \frac{\partial}{\partial y}\right)f+\mathrm{\dots}+\frac{1}{n!}{\left(\lambda \frac{\partial}{\partial x}+\mu \frac{\partial}{\partial y}\right)}^{n}f+{R}_{n},$$ | ||

where $f$ and its partial derivatives on the right-hand side are evaluated at $(a,b)$, and ${R}_{n}/{({\lambda}^{2}+{\mu}^{2})}^{n/2}\to 0$ as $(\lambda ,\mu )\to (0,0)$.

$f(x,y)$ has a *local minimum* (*maximum*)
at $(a,b)$ if

1.5.19 | $$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0\mathit{\hspace{1em}}\text{at}(a,b)\text{,}$$ | ||

and the second-order term in (1.5.18) is *positive definite
(negative definite)*, that is,

1.5.20 | $$ | ||

and

1.5.21 | $$\frac{{\partial}^{2}f}{{\partial x}^{2}}\frac{{\partial}^{2}f}{{\partial y}^{2}}-{\left(\frac{{\partial}^{2}f}{\partial x\partial y}\right)}^{2}>0\mathit{\hspace{1em}}\text{at}(a,b).$$ | ||

1.5.22 | $$\frac{d}{dx}{\int}_{\alpha (x)}^{\beta (x)}f(x,y)dy=f(x,\beta (x)){\beta}^{\prime}(x)-f(x,\alpha (x)){\alpha}^{\prime}(x)+{\int}_{\alpha (x)}^{\beta (x)}\frac{\partial f}{\partial x}dy.$$ | ||

Sufficient conditions for validity are: (a) $f$ and $\partial f/\partial x$ are continuous on a rectangle $a\le x\le b$, $c\le y\le d$; (b) when $x\in [a,b]$ both $\alpha (x)$ and $\beta (x)$ are continuously differentiable and lie in $[c,d]$.

Suppose that $a,b,c$ are finite, $d$ is finite or $+\mathrm{\infty}$, and $f(x,y)$,
$\partial f/\partial x$ are continuous on the partly-closed rectangle or infinite
strip $[a,b]\times [c,d)$. Suppose also that ${\int}_{c}^{d}f(x,y)dy$
converges and ${\int}_{c}^{d}(\partial f/\partial x)dy$ *converges uniformly*
on $a\le x\le b$, that is, given any positive number $\u03f5$, however
small, we can find a number ${c}_{0}\in [c,d)$ that is independent of $x$ and is
such that

1.5.23 | $$ | ||

for all ${c}_{1}\in [{c}_{0},d)$ and all $x\in [a,b]$. Then

1.5.24 | $$\frac{d}{dx}{\int}_{c}^{d}f(x,y)dy={\int}_{c}^{d}\frac{\partial f}{\partial x}dy,$$ | ||

$$. | |||

Let $f(x,y)$ be defined on a closed rectangle $R=[a,b]\times [c,d]$. For

1.5.25 | $a$ | $$ | ||

1.5.26 | $c$ | $$ | ||

let $({\xi}_{j},{\eta}_{k})$ denote any point in the rectangle
$[{x}_{j},{x}_{j+1}]\times [{y}_{k},{y}_{k+1}]$, $j=0,\mathrm{\dots},n-1$, $k=0,\mathrm{\dots},m-1$. Then
the *double integral* of $f(x,y)$ over $R$ is defined by

1.5.27 | $${\iint}_{R}f(x,y)dA=lim\sum _{j,k}f({\xi}_{j},{\eta}_{k})({x}_{j+1}-{x}_{j})({y}_{k+1}-{y}_{k})$$ | ||

as $\mathrm{max}(({x}_{j+1}-{x}_{j})+({y}_{k+1}-{y}_{k}))\to 0$. Sufficient conditions for the limit to exist are that $f(x,y)$ is continuous, or piecewise continuous, on $R$.

For $f(x,y)$ defined on a point set $D$ contained in a rectangle $R$, let

1.5.28 | $${f}^{*}(x,y)=\{\begin{array}{cc}f(x,y),\hfill & \text{if}(x,y)\in D,\hfill \\ 0,\hfill & \text{if}(x,y)\in R\backslash D\text{.}\hfill \end{array}$$ | ||

Then

1.5.29 | $${\iint}_{D}f(x,y)dA={\iint}_{R}{f}^{*}(x,y)dA,$$ | ||

provided the latter integral exists.

If $f(x,y)$ is continuous, and $D$ is the set

1.5.30 | $a$ | $\le x\le b,$ | ||

${\varphi}_{1}(x)$ | $\le y\le {\varphi}_{2}(x),$ | |||

with ${\varphi}_{1}(x)$ and ${\varphi}_{2}(x)$ continuous, then

1.5.31 | $${\iint}_{D}f(x,y)dA={\int}_{a}^{b}{\int}_{{\varphi}_{1}(x)}^{{\varphi}_{2}(x)}f(x,y)dydx,$$ | ||

where the right-hand side is interpreted as the repeated integral

1.5.32 | $${\int}_{a}^{b}\left({\int}_{{\varphi}_{1}(x)}^{{\varphi}_{2}(x)}f(x,y)dy\right)dx.$$ | ||

In particular, ${\varphi}_{1}(x)$ and ${\varphi}_{2}(x)$ can be constants.

Similarly, if $D$ is the set

1.5.33 | $c$ | $\le y\le d,$ | ||

${\psi}_{1}(y)$ | $\le x\le {\psi}_{2}(y),$ | |||

with ${\psi}_{1}(y)$ and ${\psi}_{2}(y)$ continuous, then

1.5.34 | $${\iint}_{D}f(x,y)dA={\int}_{c}^{d}{\int}_{{\psi}_{1}(y)}^{{\psi}_{2}(y)}f(x,y)dxdy.$$ | ||

Infinite double integrals occur when $f(x,y)$ becomes infinite at points in $D$ or when $D$ is unbounded. In the cases (1.5.30) and (1.5.33) they are defined by taking limits in the repeated integrals (1.5.32) and (1.5.34) in an analogous manner to (1.4.22)–(1.4.23).

Moreover, if $a,b,c,d$ are finite or infinite constants and $f(x,y)$ is piecewise continuous on the set $(a,b)\times (c,d)$, then

1.5.36 | $${\int}_{a}^{b}{\int}_{c}^{d}f(x,y)dydx={\int}_{c}^{d}{\int}_{a}^{b}f(x,y)dxdy,$$ | ||

whenever both repeated integrals exist and at least one is absolutely convergent.

Finite and infinite integrals can be defined in a similar way. Often the $(x,y,z)$ sets are of the form

1.5.37 | $a$ | $\le x\le b,$ | ||

${\varphi}_{1}(x)$ | $\le y\le {\varphi}_{2}(x),$ | |||

${\psi}_{1}(x,y)$ | $\le z\le {\psi}_{2}(x,y).$ | |||

1.5.38 | $\frac{\partial (f,g)}{\partial (x,y)}$ | $=\left|\begin{array}{cc}\hfill \partial f/\partial x\hfill & \hfill \partial f/\partial y\hfill \\ \hfill \partial g/\partial x\hfill & \hfill \partial g/\partial y\hfill \end{array}\right|,$ | ||

1.5.39 | $\frac{\partial (x,y)}{\partial (r,\varphi )}$ | $=r\hspace{1em}\text{(polar coordinates)}.$ | ||

1.5.40 | $\frac{\partial (f,g,h)}{\partial (x,y,z)}$ | $=\left|\begin{array}{ccc}\hfill \partial f/\partial x\hfill & \hfill \partial f/\partial y\hfill & \hfill \partial f/\partial z\hfill \\ \hfill \partial g/\partial x\hfill & \hfill \partial g/\partial y\hfill & \hfill \partial g/\partial z\hfill \\ \hfill \partial h/\partial x\hfill & \hfill \partial h/\partial y\hfill & \hfill \partial h/\partial z\hfill \end{array}\right|,$ | ||

1.5.41 | $\frac{\partial (x,y,z)}{\partial (\rho ,\theta ,\varphi )}$ | $={\rho}^{2}\mathrm{sin}\theta \hspace{1em}\text{(spherical coordinates)}.$ | ||

1.5.42 | $${\iint}_{D}f(x,y)dxdy={\iint}_{{D}^{*}}f(x(u,v),y(u,v))\left|\frac{\partial (x,y)}{\partial (u,v)}\right|dudv,$$ | ||

where $D$ is the image of ${D}^{*}$ under a mapping $(u,v)\to (x(u,v),y(u,v))$ which is one-to-one except perhaps for a set of points of area zero.

1.5.43 | $$\begin{array}{l}{\iiint}_{D}f(x,y,z)dxdydz\\ \phantom{\rule{2em}{0ex}}={\iiint}_{{D}^{*}}f(x(u,v,w),y(u,v,w),z(u,v,w))\left|\frac{\partial (x,y,z)}{\partial (u,v,w)}\right|dudvdw.\end{array}$$ | ||

Again the mapping is one-to-one except perhaps for a set of points of volume zero.