# §1.5(i) Partial Derivatives

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 Encodings: TeX, pMML, png

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 Encodings: TeX, pMML, png

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 $\displaystyle\frac{\partial f}{\partial x}$ $\displaystyle=D_{x}f$ $\displaystyle=f_{x}$ $\displaystyle=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h},$ 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 Encodings: TeX, pMML, png 1.5.4 $\displaystyle\frac{\partial f}{\partial y}$ $\displaystyle=D_{y}f$ $\displaystyle=f_{y}$ $\displaystyle=\lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h}.$
 1.5.5 $\displaystyle\frac{{\partial}^{2}f}{\partial x\partial y}$ $\displaystyle=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}% \right),$ $\displaystyle\frac{{\partial}^{2}f}{\partial y\partial x}$ $\displaystyle=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}% \right).$

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 Encodings: TeX, pMML, png

# ¶ Chain Rule

 1.5.7 $\displaystyle\frac{d}{dt}f(x(t),y(t))$ $\displaystyle=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{% \partial y}\frac{dy}{dt},$ 1.5.8 $\displaystyle\frac{\partial}{\partial u}f(x(u,v),y(u,v))$ $\displaystyle=\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac% {\partial f}{\partial y}\frac{\partial y}{\partial u},$ 1.5.9 $\displaystyle\frac{\partial}{\partial v}f(x(u,v),y(u,v),z(u,v))$ $\displaystyle=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac% {\partial f}{\partial y}\frac{\partial y}{\partial v}+\frac{\partial f}{% \partial z}\frac{\partial z}{\partial v}.$

# ¶ Implicit Function Theorem

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 $g^{\prime}(x)=-F_{x}/F_{y}$.

# ¶ 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

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

 1.5.10 $\displaystyle x$ $\displaystyle=r\mathop{\cos\/}\nolimits\phi,$ $\displaystyle y$ $\displaystyle=r\mathop{\sin\/}\nolimits\phi,$
 1.5.11 $\displaystyle\frac{\partial}{\partial x}$ $\displaystyle=\mathop{\cos\/}\nolimits\phi\frac{\partial}{\partial r}-\frac{% \mathop{\sin\/}\nolimits\phi}{r}\frac{\partial}{\partial\phi},$ 1.5.12 $\displaystyle\frac{\partial}{\partial y}$ $\displaystyle=\mathop{\sin\/}\nolimits\phi\frac{\partial}{\partial r}+\frac{% \mathop{\cos\/}\nolimits\phi}{r}\frac{\partial}{\partial\phi}.$

The Laplacian is given by

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

# ¶ Cylindrical Coordinates

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

 1.5.14 $\displaystyle x$ $\displaystyle=r\mathop{\cos\/}\nolimits\phi,$ $\displaystyle y$ $\displaystyle=r\mathop{\sin\/}\nolimits\phi,$ $\displaystyle z$ $\displaystyle=z.$

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

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

# ¶ Spherical Coordinates

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

 1.5.16 $\displaystyle x$ $\displaystyle=\rho\mathop{\sin\/}\nolimits\theta\mathop{\cos\/}\nolimits\phi,$ $\displaystyle y$ $\displaystyle=\rho\mathop{\sin\/}\nolimits\theta\mathop{\sin\/}\nolimits\phi,$ $\displaystyle z$ $\displaystyle=\rho\mathop{\cos\/}\nolimits\theta.$

The Laplacian is given by

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

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

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

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

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 $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0\quad\mbox{ at (% a,b),}$

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

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

and

 1.5.21 $\frac{{\partial}^{2}f}{{\partial x}^{2}}\frac{{\partial}^{2}f}{{\partial y}^{2% }}-\left(\frac{{\partial}^{2}f}{\partial x\partial y}\right)^{2}>0\quad\mbox{ % at (a,b)}.$

# ¶ Finite Integrals

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

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

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

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.

# ¶ Double Integrals

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

 1.5.25 $\displaystyle a$ $\displaystyle=x_{0}$ $\displaystyle $\displaystyle<\dots$ $\displaystyle $\displaystyle=b,$ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.5.E25 Encodings: TeX, pMML, png 1.5.26 $\displaystyle c$ $\displaystyle=y_{0}$ $\displaystyle $\displaystyle<\dots$ $\displaystyle $\displaystyle=d,$ Symbols: $m$: nonnegative integer Permalink: http://dlmf.nist.gov/1.5.E26 Encodings: TeX, pMML, png

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 $\iint_{R}f(x,y)dA={\lim\sum_{j,k}f(\xi_{j},\eta_{k})(x_{j+1}-x_{j})(y_{k+1}-y_% {k})}$

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 $f^{*}(x,y)=\begin{cases}f(x,y),&\mbox{if (x,y)\in D},\\ 0,&\mbox{if (x,y)\in R\setminus D.}\end{cases}$

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 $\displaystyle a$ $\displaystyle\leq x$ $\displaystyle\leq b,$ $\displaystyle\phi_{1}(x)$ $\displaystyle\leq y$ $\displaystyle\leq\phi_{2}(x),$ Symbols: $\phi_{j}(x)$: continuous functions Referenced by: ¶ ‣ §1.5(v), ¶ ‣ §1.5(v) Permalink: http://dlmf.nist.gov/1.5.E30 Encodings: TeX, TeX, pMML, pMML, png, png

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

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

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

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

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

Similarly, if $D$ is the set

 1.5.33 $\displaystyle c$ $\displaystyle\leq y$ $\displaystyle\leq d,$ $\displaystyle\psi_{1}(y)$ $\displaystyle\leq x$ $\displaystyle\leq\psi_{2}(y),$ 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 Encodings: TeX, TeX, pMML, pMML, png, png

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

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

# ¶ Change of Order of Integration

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 $\int^{b}_{a}\int^{\phi_{2}(x)}_{\phi_{1}(x)}f(x,y)dydx=\int^{d}_{c}\int^{\psi_% {2}(y)}_{\psi_{1}(y)}f(x,y)dxdy.$

# ¶ Infinite Double 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

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

 1.5.37 $\displaystyle a$ $\displaystyle\leq x$ $\displaystyle\leq b,$ $\displaystyle\phi_{1}(x)$ $\displaystyle\leq y$ $\displaystyle\leq\phi_{2}(x),$ $\displaystyle\psi_{1}(x,y)$ $\displaystyle\leq z$ $\displaystyle\leq\psi_{2}(x,y).$ Symbols: $z$: variable Permalink: http://dlmf.nist.gov/1.5.E37 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

# ¶ Jacobian

 1.5.38 $\displaystyle\frac{\partial(f,g)}{\partial(x,y)}$ $\displaystyle=\begin{vmatrix}\ifrac{\partial f}{\partial x}&\ifrac{\partial f}% {\partial y}\\ \ifrac{\partial g}{\partial x}&\ifrac{\partial g}{\partial y}\end{vmatrix},$ 1.5.39 $\displaystyle\frac{\partial(x,y)}{\partial(r,\phi)}$ $\displaystyle=r\quad\text{(polar coordinates)}.$ 1.5.40 $\displaystyle\frac{\partial(f,g,h)}{\partial(x,y,z)}$ $\displaystyle=\begin{vmatrix}\ifrac{\partial f}{\partial x}&\ifrac{\partial f}% {\partial y}&\ifrac{\partial f}{\partial z}\\ \ifrac{\partial g}{\partial x}&\ifrac{\partial g}{\partial y}&\ifrac{\partial g% }{\partial z}\\ \ifrac{\partial h}{\partial x}&\ifrac{\partial h}{\partial y}&\ifrac{\partial h% }{\partial z}\end{vmatrix},$ 1.5.41 $\displaystyle\frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}$ $\displaystyle=\rho^{2}\mathop{\sin\/}\nolimits\theta\quad\text{(spherical % coordinates)}.$

# ¶ Change of Variables

 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 $\iiint_{D}f(x,y,z)dxdydz=\iiint_{D^{*}}f(x(u,v,w),y(u,v,w),z(u,v,w))\*\left|% \frac{\partial(x,y,z)}{\partial(u,v,w)}\right|dudvdw.$

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