§1.5 Calculus of Two or More Variables

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

The Laplacian is given by

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

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

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

The Laplacian is given by

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).

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 $ϵ$, however small, we can find a number $c_0\in [c,d)$ that is independent of $x$ and is such that

for all $c_1\in [c_0,d)$ and all $x\in [a,b]$. Then

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

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

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

Then

provided the latter integral exists.

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

with ${\varphi }_1(x)$ and ${\varphi }_2(x)$ continuous, then

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

In particular, ${\varphi }_1(x)$ and ${\varphi }_2(x)$ can be constants.

Similarly, if $D$ is the set

with ${\psi }_1(y)$ and ${\psi }_2(y)$ continuous, then

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

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

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

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

A more general concept of integrability (both finite and infinite) for functions on domains in ${ℝ}^n$ is Lebesgue integrability. See Rudin (1966).

where $D$ is the image of $D^{\ast }$ 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.

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