§1.5 Calculus of Two or More Variables

Keywords:: calculus
Referenced by:: §1.1
Permalink:: http://dlmf.nist.gov/1.5

§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

§1.5(i) Partial Derivatives

Notes:: See Marsden and Tromba (1996, Chapters 2, 3).
Keywords:: continuous function, derivatives, differentiation, functions, limits of functions, partial derivative, partial differentiation
Permalink:: http://dlmf.nist.gov/1.5.i

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

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

Symbols:: $(a,b)$ : open interval
Referenced by:: ¶ ‣ §1.9(ii)
Permalink:: http://dlmf.nist.gov/1.5.E1
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that is, for every arbitrarily small positive constant $\epsilon$ there exists $\delta$ ( $>0$ ) such that

1.5.2 $|f(a+\alpha,b+\beta)-f(a,b)|<\epsilon,$

Referenced by:: ¶ ‣ §1.9(ii)
Permalink:: http://dlmf.nist.gov/1.5.E2
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for all $\alpha$ and $\beta$ that satisfy $|\alpha|,|\beta|<\delta$ .

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 $\frac{\partial f}{\partial x}=D_{x}f=f_{x}=\lim _{{h\to 0}}\frac{f(x+h,y)-f(x,y)}{h},$

Defines:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $\partial x$ : partial differential of $x$
Symbols:: $D_{x}$ : differential operator
Permalink:: http://dlmf.nist.gov/1.5.E3
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1.5.4 $\frac{\partial f}{\partial y}=D_{y}f=f_{y}=\lim _{{h\to 0}}\frac{f(x,y+h)-f(x,y)}{h}.$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $D_{x}$ : differential operator
Permalink:: http://dlmf.nist.gov/1.5.E4
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1.5.5

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

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

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $\partial x$ : partial differential of $x$
Permalink:: http://dlmf.nist.gov/1.5.E5
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The function $f(x,y)$ is continuously differentiable if $f$ , $\ifrac{\partial f}{\partial x}$ , and $\ifrac{\partial f}{\partial y}$ are continuous, and twice-continuously differentiable if also $\ifrac{{\partial}^{2}f}{{\partial x}^{2}}$ , $\ifrac{{\partial}^{2}f}{{\partial y}^{2}}$ , ${\partial}^{2}f/\partial x\partial y$ , and ${\partial}^{2}f/\partial y\partial x$ are continuous. In the latter event

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

Symbols:: $\partial x$ : partial differential of $x$
Permalink:: http://dlmf.nist.gov/1.5.E6
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¶ Chain Rule

Keywords:: chain rule, derivatives

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ and $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$
Permalink:: http://dlmf.nist.gov/1.5.E7
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1.5.8 $\frac{\partial}{\partial u}f(x(u,v),y(u,v))=\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial u},$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$
Permalink:: http://dlmf.nist.gov/1.5.E8
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1.5.9 $\frac{\partial}{\partial v}f(x(u,v),y(u,v),z(u,v))=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial v}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial v}.$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$
Permalink:: http://dlmf.nist.gov/1.5.E9
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¶ Implicit Function Theorem

Keywords:: implicit function theorem, neighborhood

If $F(x,y)$ is continuously differentiable, $F(a,b)=0$ , and $\ifrac{\partial F}{\partial y}\not=0$ 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 $g^{{\prime}}(x)=-F_{x}/F_{y}$ .

§1.5(ii) Coordinate Systems

Notes:: See Davis and Snider (1987, Chapter 5).
Referenced by:: §10.73(i), §10.73(ii), §12.17, §14.30(iv), §18.39(i), §23.21(iii), §30.13(i), §30.14(i), Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/1.5.ii
Clarification (effective with 1.0.5):: The paragraph about notations was added at the beginning of this subsection.
Reported 2012-08-30

¶ Notations

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 $\phi$ now are usually interchanged.

¶ Polar Coordinates

Keywords:: Laplacian, coordinate systems, plane polar coordinates, polar coordinates

With $0\leq r<\infty$ , $0\leq\phi\leq 2\pi$ ,

1.5.10

$x=r\mathop{\cos\/}\nolimits\phi,$

$y=r\mathop{\sin\/}\nolimits\phi,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $r$ : radius and $\phi$ : angle
Permalink:: http://dlmf.nist.gov/1.5.E10
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1.5.11 $\frac{\partial}{\partial x}=\mathop{\cos\/}\nolimits\phi\frac{\partial}{\partial r}-\frac{\mathop{\sin\/}\nolimits\phi}{r}\frac{\partial}{\partial\phi},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $\mathop{\sin\/}\nolimits z$ : sine function, $r$ : radius and $\phi$ : angle
Referenced by:: ¶ ‣ §1.5(ii)
Permalink:: http://dlmf.nist.gov/1.5.E11
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1.5.12 $\frac{\partial}{\partial y}=\mathop{\sin\/}\nolimits\phi\frac{\partial}{\partial r}+\frac{\mathop{\cos\/}\nolimits\phi}{r}\frac{\partial}{\partial\phi}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $\mathop{\sin\/}\nolimits z$ : sine function, $r$ : radius and $\phi$ : angle
Referenced by:: ¶ ‣ §1.5(ii)
Permalink:: http://dlmf.nist.gov/1.5.E12
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The Laplacian is given by

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

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $r$ : radius and $\phi$ : angle
Permalink:: http://dlmf.nist.gov/1.5.E13
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¶ Cylindrical Coordinates

Keywords:: Laplacian, coordinate systems, cylindrical coordinates, cylindrical polar coordinates

With $0\leq r<\infty$ , $0\leq\phi\leq 2\pi$ , $-\infty<z<\infty$ ,

1.5.14

$x=r\mathop{\cos\/}\nolimits\phi,$

$y=r\mathop{\sin\/}\nolimits\phi,$

$z=z.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : variable, $r$ : radius and $\phi$ : angle
Permalink:: http://dlmf.nist.gov/1.5.E14
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Equations (1.5.11) and (1.5.12) still apply, but

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

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $z$ : variable, $r$ : radius and $\phi$ : angle
Permalink:: http://dlmf.nist.gov/1.5.E15
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¶ Spherical Coordinates

Keywords:: Laplacian, coordinate systems, spherical coordinates, spherical polar coordinates

With $0\leq\rho<\infty$ , $0\leq\phi\leq 2\pi$ , $0\leq\theta\leq\pi$ ,

1.5.16

$x=\rho\mathop{\sin\/}\nolimits\theta\mathop{\cos\/}\nolimits\phi,$

$y=\rho\mathop{\sin\/}\nolimits\theta\mathop{\sin\/}\nolimits\phi,$

$z=\rho\mathop{\cos\/}\nolimits\theta.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : variable, $\phi$ : longitude, $\rho$ : radius and $\theta$ : azimuth
Permalink:: http://dlmf.nist.gov/1.5.E16
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The Laplacian is given by

1.5.17 $\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{\partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}={\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial f}{\partial\rho}\right)+\frac{1}{\rho^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta}\frac{{\partial}^{2}f}{{\partial\phi}^{2}}}+\frac{1}{\rho^{2}\mathop{\sin\/}\nolimits\theta}\frac{\partial}{\partial\theta}\left(\mathop{\sin\/}\nolimits\theta\frac{\partial f}{\partial\theta}\right).$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : variable, $\phi$ : longitude, $\rho$ : radius and $\theta$ : azimuth
Permalink:: http://dlmf.nist.gov/1.5.E17
Encodings:: TeX, pMML, png

For applications and other coordinate systems see §§12.17, 14.19(i), 14.30(iv), 28.32, 29.18, 30.13, 30.14. See also Morse and Feshbach (1953a, pp. 655-666).

§1.5(iii) Taylor’s Theorem; Maxima and Minima

Notes:: See Marsden and Tromba (1996, Chapter 3).
Keywords:: Taylor series, Taylor’s theorem, maximum, minimum, negative definite, positive definite
Permalink:: http://dlmf.nist.gov/1.5.iii

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

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

Symbols:: $!$ : $n!$ : factorial, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $R_{n}$ : remainder
Referenced by:: §1.5(iii)
Permalink:: http://dlmf.nist.gov/1.5.E18
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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\quad\mbox{ at $(a,b)$,}$

Symbols:: $(a,b)$ : open interval and $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$
Permalink:: http://dlmf.nist.gov/1.5.E19
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and the second-order term in (1.5.18) is positive definite (negative definite), that is,

1.5.20 $\frac{{\partial}^{2}f}{{\partial x}^{2}}>0\;\;\;\mbox{$(<0)$}\quad\mbox{ at $(a,b)$,}$

Symbols:: $(a,b)$ : open interval and $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$
Permalink:: http://dlmf.nist.gov/1.5.E20
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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\quad\mbox{ at $(a,b)$}.$

Symbols:: $(a,b)$ : open interval, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $\partial x$ : partial differential of $x$
Permalink:: http://dlmf.nist.gov/1.5.E21
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§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals

Notes:: See Protter and Morrey (1991, pp. 288, 298).
Keywords:: differentiation, integrals
Referenced by:: §1.10(viii), ¶ ‣ §1.8(i)
Permalink:: http://dlmf.nist.gov/1.5.iv

¶ Finite Integrals

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $dx$ : differential of $x$ , $\int$ : integral and $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$
A&S Ref:: 3.3.7
Permalink:: http://dlmf.nist.gov/1.5.E22
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Sufficient conditions for validity are: (a) $f$ and $\ifrac{\partial f}{\partial x}$ are continuous on a rectangle $a\leq x\leq b$ , $c\leq y\leq d$ ; (b) when $x\in[a,b]$ both $\alpha(x)$ and $\beta(x)$ are continuously differentiable and lie in $[c,d]$ .

¶ Infinite Integrals

Keywords:: convergence, differentiation, integrals

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

1.5.23 $\left|\int _{{c_{1}}}^{d}(\ifrac{\partial f}{\partial x})dy\right|<\epsilon,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $\epsilon$ : positive number
Permalink:: http://dlmf.nist.gov/1.5.E23
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for all $c_{1}\in[c_{0},d)$ and all $x\in[a,b]$ . Then

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $dx$ : differential of $x$ , $\int$ : integral and $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$
Permalink:: http://dlmf.nist.gov/1.5.E24
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§1.5(v) Multiple Integrals

Notes:: See Marsden and Tromba (1996, Chapters 5, 6). For (1.5.36) see Love (1970, 1972a).
Keywords:: double integrals, integrals
Referenced by:: §9.12(viii)
Permalink:: http://dlmf.nist.gov/1.5.v

¶ Double Integrals

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

1.5.25 $a=x_{0}<x_{1}<\dots<x_{n}=b,$

Symbols:: $n$ : nonnegative integer
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1.5.26 $c=y_{0}<y_{1}<\dots<y_{m}=d,$

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

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

Symbols:: $dx$ : differential of $x$ , $j$ : integer, $k$ : integer and $R$ : closed rectangle
Permalink:: http://dlmf.nist.gov/1.5.E27
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as $\max((x_{{j+1}}-x_{j})+(y_{{k+1}}-y_{k}))\to 0$ . 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{cases}f(x,y),&\mbox{if $(x,y)\in D$},\\ 0,&\mbox{if $(x,y)\in R\setminus D$.}\end{cases}$

Symbols:: $\in$ : element of, $(a,b)$ : open interval, $\setminus$ : set subtraction, $R$ : closed rectangle and $D$ : region contained in $R$
Permalink:: http://dlmf.nist.gov/1.5.E28
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Then

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

Symbols:: $dx$ : differential of $x$ , $R$ : closed rectangle and $D$ : region contained in $R$
Permalink:: http://dlmf.nist.gov/1.5.E29
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provided the latter integral exists.

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

1.5.30

$a\leq x\leq b,$

$\phi _{1}(x)\leq y\leq\phi _{2}(x),$

Symbols:: $\phi _{j}(x)$ : continuous functions
Referenced by:: ¶ ‣ §1.5(v), ¶ ‣ §1.5(v)
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with $\phi _{1}(x)$ and $\phi _{2}(x)$ continuous, then

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $D$ : region contained in $R$ and $\phi _{j}(x)$ : continuous functions
Permalink:: http://dlmf.nist.gov/1.5.E31
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where the right-hand side is interpreted as the repeated integral

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $\phi _{j}(x)$ : continuous functions
Referenced by:: ¶ ‣ §1.5(v)
Permalink:: http://dlmf.nist.gov/1.5.E32
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In particular, $\phi _{1}(x)$ and $\phi _{2}(x)$ can be constants.

Similarly, if $D$ is the set

1.5.33

$c\leq y\leq d,$

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

Symbols:: $\psi _{1}(y)$ , $\psi _{2}(y)$ : continuous functions
Referenced by:: ¶ ‣ §1.5(v), ¶ ‣ §1.5(v)
Permalink:: http://dlmf.nist.gov/1.5.E33
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with $\psi _{1}(y)$ and $\psi _{2}(y)$ continuous, then

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $D$ : region contained in $R$ and $\psi _{1}(y)$ , $\psi _{2}(y)$ : continuous functions
Referenced by:: ¶ ‣ §1.5(v)
Permalink:: http://dlmf.nist.gov/1.5.E34
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¶ Change of Order of Integration

Keywords:: double integrals

If $D$ can be represented in both forms (1.5.30) and (1.5.33), and $f(x,y)$ is continuous on $D$ , then

1.5.35 $\int^{b}_{a}\int^{{\phi _{2}(x)}}_{{\phi _{1}(x)}}f(x,y)dydx=\int^{d}_{c}\int^{{\psi _{2}(y)}}_{{\psi _{1}(y)}}f(x,y)dxdy.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\phi _{j}(x)$ : continuous functions and $\psi _{1}(y)$ , $\psi _{2}(y)$ : continuous functions
Permalink:: http://dlmf.nist.gov/1.5.E35
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¶ Infinite Double Integrals

Keywords:: double integrals, integrals

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

Symbols:: $dx$ : differential of $x$ and $\int$ : integral
Referenced by:: §1.5(v)
Permalink:: http://dlmf.nist.gov/1.5.E36
Encodings:: TeX, pMML, png

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

¶ Triple Integrals

Keywords:: triple integrals

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

1.5.37

$a\leq x\leq b,$

$\phi _{1}(x)\leq y\leq\phi _{2}(x),$

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

Symbols:: $z$ : variable
Permalink:: http://dlmf.nist.gov/1.5.E37
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§1.5(vi) Jacobians and Change of Variables

Notes:: See Marsden and Tromba (1996, pp. 358–371).
Keywords:: Jacobian, derivatives
Permalink:: http://dlmf.nist.gov/1.5.vi

¶ Jacobian

1.5.38 $\frac{\partial(f,g)}{\partial(x,y)}=\begin{vmatrix}\ifrac{\partial f}{\partial x}&\ifrac{\partial f}{\partial y}\\ \ifrac{\partial g}{\partial x}&\ifrac{\partial g}{\partial y}\end{vmatrix},$

Symbols:: $(a,b)$ : open interval and $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$
Permalink:: http://dlmf.nist.gov/1.5.E38
Encodings:: TeX, pMML, png

1.5.39 $\frac{\partial(x,y)}{\partial(r,\phi)}=r\quad\text{(polar coordinates)}.$

Symbols:: $(a,b)$ : open interval, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $r$ : radius and $\phi$ : angle
Permalink:: http://dlmf.nist.gov/1.5.E39
Encodings:: TeX, pMML, png

1.5.40 $\frac{\partial(f,g,h)}{\partial(x,y,z)}=\begin{vmatrix}\ifrac{\partial f}{\partial x}&\ifrac{\partial f}{\partial y}&\ifrac{\partial f}{\partial z}\\ \ifrac{\partial g}{\partial x}&\ifrac{\partial g}{\partial y}&\ifrac{\partial g}{\partial z}\\ \ifrac{\partial h}{\partial x}&\ifrac{\partial h}{\partial y}&\ifrac{\partial h}{\partial z}\end{vmatrix},$