# §1.6 Vectors and Vector-Valued Functions

## §1.6(i) Vectors

 1.6.1 $\displaystyle\mathbf{a}$ $\displaystyle=(a_{1},a_{2},a_{3}),$ $\displaystyle\mathbf{b}$ $\displaystyle=(b_{1},b_{2},b_{3}).$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.6(i), §1.6 and Ch.1

### Dot Product (or Scalar Product)

 1.6.2 $\mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}.$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E2 Encodings: TeX, pMML, png See also: Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1

### Magnitude and Angle of Vector $\mathbf{a}$

 1.6.3 $\|\mathbf{a}\|=\sqrt{\mathbf{a}\cdot\mathbf{a}},$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E3 Encodings: TeX, pMML, png See also: Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1
 1.6.4 $\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\;\|\mathbf{b}\|};$ ⓘ Symbols: $\cos\NVar{z}$: cosine function and $\theta$: angle between $\mathbf{a}$ and $\mathbf{b}$ Permalink: http://dlmf.nist.gov/1.6.E4 Encodings: TeX, pMML, png See also: Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1

$\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.

### Unit Vectors

 1.6.5 $\displaystyle\mathbf{i}$ $\displaystyle=(1,0,0),$ $\displaystyle\mathbf{j}$ $\displaystyle=(0,1,0),$ $\displaystyle\mathbf{k}$ $\displaystyle=(0,0,1),$ ⓘ Defines: $\mathbf{i}$: unit vector (locally), $\mathbf{j}$: unit vector (locally) and $\mathbf{k}$: unit vector (locally) Referenced by: §1.6(ii) Permalink: http://dlmf.nist.gov/1.6.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1
 1.6.6 $\mathbf{a}=a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}.$ ⓘ Symbols: $\mathbf{i}$: unit vector, $\mathbf{j}$: unit vector and $\mathbf{k}$: unit vector Permalink: http://dlmf.nist.gov/1.6.E6 Encodings: TeX, pMML, png See also: Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1

### Cross Product (or Vector Product)

 1.6.7 $\displaystyle\mathbf{i}\times\mathbf{j}$ $\displaystyle=\mathbf{k},$ $\displaystyle\mathbf{j}\times\mathbf{k}$ $\displaystyle=\mathbf{i},$ $\displaystyle\mathbf{k}\times\mathbf{i}$ $\displaystyle=\mathbf{j},$ ⓘ Symbols: $\mathbf{i}$: unit vector, $\mathbf{j}$: unit vector and $\mathbf{k}$: unit vector Referenced by: §1.6(ii) Permalink: http://dlmf.nist.gov/1.6.E7 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1
 1.6.8 $\displaystyle\mathbf{j}\times\mathbf{i}$ $\displaystyle=-\mathbf{k},$ $\displaystyle\mathbf{k}\times\mathbf{j}$ $\displaystyle=-\mathbf{i},$ $\displaystyle\mathbf{i}\times\mathbf{k}$ $\displaystyle=-\mathbf{j}.$ ⓘ Symbols: $\mathbf{i}$: unit vector, $\mathbf{j}$: unit vector and $\mathbf{k}$: unit vector Referenced by: §1.6(ii) Permalink: http://dlmf.nist.gov/1.6.E8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1
 1.6.9 $\mathbf{a}\times\mathbf{b}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\end{vmatrix}\\ =(a_{2}b_{3}-a_{3}b_{2})\mathbf{i}+(a_{3}b_{1}-a_{1}b_{3})\mathbf{j}+(a_{1}b_{% 2}-a_{2}b_{1})\mathbf{k}\\ =\|\mathbf{a}\|\|\mathbf{b}\|(\sin\theta)\mathbf{n},$

where $\mathbf{n}$ is the unit vector normal to $\mathbf{a}$ and $\mathbf{b}$ whose direction is determined by the right-hand rule; see Figure 1.6.1.

Area of parallelogram with vectors $\mathbf{a}$ and $\mathbf{b}$ as sides $=\|\mathbf{a}\times\mathbf{b}\|$.

Volume of a parallelepiped with vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ as edges $=\left|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\right|$.

 1.6.10 $\displaystyle\mathbf{a}\times(\mathbf{b}\times\mathbf{c})$ $\displaystyle=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{c}(\mathbf{a}\cdot% \mathbf{b}),$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E10 Encodings: TeX, pMML, png See also: Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1 1.6.11 $\displaystyle(\mathbf{a}\times\mathbf{b})\times\mathbf{c}$ $\displaystyle=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{a}(\mathbf{b}\cdot% \mathbf{c}).$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E11 Encodings: TeX, pMML, png See also: Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1

## §1.6(ii) Vectors: Alternative Notations

The following notations are often used in the physics literature; see for example Lorentz et al. (1923, pp. 122–123).

### Einstein Summation Convention

Much vector algebra involves summation over suffices of products of vector components. In almost all cases of repeated suffices, we can suppress the summation notation entirely, if it is understood that an implicit sum is to be taken over any repeated suffix. Thus pairs of indefinite suffices in an expression are resolved by being summed over (or “traced” over).

### Example

 1.6.12 $a_{j}b_{j}=\sum_{j=1}^{3}a_{j}b_{j}=\mathbf{a}\cdot\mathbf{b}.$ ⓘ Symbols: $j$: integer Permalink: http://dlmf.nist.gov/1.6.E12 Encodings: TeX, pMML, png See also: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

Next,

 1.6.13 $\displaystyle\mathbf{e}_{1}$ $\displaystyle=(1,0,0),$ $\displaystyle\mathbf{e}_{2}$ $\displaystyle=(0,1,0),$ $\displaystyle\mathbf{e}_{3}$ $\displaystyle=(0,0,1);$ ⓘ Defines: $\mathbf{e}_{j}$: unit vectors (locally) Symbols: $j$: integer Permalink: http://dlmf.nist.gov/1.6.E13 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

compare (1.6.5). Thus $a_{j}\mathbf{e}_{j}=\mathbf{a}$.

### Levi-Civita Symbol

 1.6.14 $\epsilon_{jk\ell}=\begin{cases}+1,&\text{if }j,k,\ell\text{ is even % permutation of }1,2,3,\\ -1,&\text{if }j,k,\ell\text{ is odd permutation of }1,2,3,\\ \phantom{-}0,&\text{otherwise}.\end{cases}$ ⓘ Defines: $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$: Levi-Civita symbol Symbols: $j$: integer and $k$: integer Permalink: http://dlmf.nist.gov/1.6.E14 Encodings: TeX, pMML, png See also: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

### Examples

 1.6.15 $\displaystyle\epsilon_{123}$ $\displaystyle=\epsilon_{312}=1,$ $\displaystyle\epsilon_{213}$ $\displaystyle=\epsilon_{321}=-1,$ $\displaystyle\epsilon_{221}$ $\displaystyle=0.$ ⓘ Symbols: $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$: Levi-Civita symbol Permalink: http://dlmf.nist.gov/1.6.E15 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1
 1.6.16 $\epsilon_{jk\ell}\epsilon_{\ell mn}=\delta_{j,m}\delta_{k,n}-\delta_{j,n}% \delta_{k,m},$

where $\delta_{j,k}$ is the Kronecker delta.

 1.6.17 $\mathbf{e}_{j}\times\mathbf{e}_{k}=\epsilon_{jk\ell}\mathbf{e}_{\ell};$ ⓘ Symbols: $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$: Levi-Civita symbol, $j$: integer, $k$: integer and $\mathbf{e}_{j}$: unit vectors Permalink: http://dlmf.nist.gov/1.6.E17 Encodings: TeX, pMML, png See also: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

compare (1.6.8).

 1.6.18 $a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k}=\epsilon_{jk\ell}a_{j}b_{k}% \mathbf{e}_{\ell};$ ⓘ Symbols: $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$: Levi-Civita symbol, $j$: integer, $k$: integer and $\mathbf{e}_{j}$: unit vectors Permalink: http://dlmf.nist.gov/1.6.E18 Encodings: TeX, pMML, png See also: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

compare (1.6.7)–(1.6.8).

Lastly, the volume of a parallelepiped with vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ as edges is $|\epsilon_{jk\ell}a_{j}b_{k}c_{\ell}|$.

## §1.6(iii) Vector-Valued Functions

### Del Operator

 1.6.19 $\nabla=\mathbf{i}\frac{\partial}{\partial x}+\mathbf{j}\frac{\partial}{% \partial y}+\mathbf{k}\frac{\partial}{\partial z}.$

The gradient of a differentiable scalar function $f(x,y,z)$ is

 1.6.20 $\operatorname{grad}f=\nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{% \partial f}{\partial y}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}.$ ⓘ Defines: $\operatorname{grad}$: gradient of differentiable scalar function Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$: partial derivative, $\partial\NVar{x}$: partial differential, $z$: variable, $\mathbf{i}$: unit vector, $\mathbf{j}$: unit vector and $\mathbf{k}$: unit vector Permalink: http://dlmf.nist.gov/1.6.E20 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

The divergence of a differentiable vector-valued function $\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}$ is

 1.6.21 $\operatorname{div}\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{\partial F_{1}}{% \partial x}+\frac{\partial F_{2}}{\partial y}+\frac{\partial F_{3}}{\partial z}.$ ⓘ Defines: $\operatorname{div}$: divergence of vector-valued function Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$: partial derivative, $\partial\NVar{x}$: partial differential, $z$: variable and $F_{\NVar{j}}$: vector function components Permalink: http://dlmf.nist.gov/1.6.E21 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

The curl of $\mathbf{F}$ is

 1.6.22 $\operatorname{curl}\mathbf{F}=\nabla\times\mathbf{F}=\begin{vmatrix}\mathbf{i}% &\mathbf{j}&\mathbf{k}\\ \displaystyle{\frac{\partial}{\partial x}}&\displaystyle{\frac{\partial}{% \partial y}}&\displaystyle{\frac{\partial}{\partial z}}\\ F_{1}&F_{2}&F_{3}\end{vmatrix}\\ =\left(\frac{\partial F_{3}}{\partial y}-\frac{\partial F_{2}}{\partial z}% \right)\mathbf{i}+\left(\frac{\partial F_{1}}{\partial z}-\frac{\partial F_{3}% }{\partial x}\right)\mathbf{j}+\left(\frac{\partial F_{2}}{\partial x}-\frac{% \partial F_{1}}{\partial y}\right)\mathbf{k}.$ ⓘ Defines: $\operatorname{curl}$: curl of vector-valued function Symbols: $\det$: determinant, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$: partial derivative, $\partial\NVar{x}$: partial differential, $z$: variable, $\mathbf{i}$: unit vector, $\mathbf{j}$: unit vector, $\mathbf{k}$: unit vector and $F_{\NVar{j}}$: vector function components Permalink: http://dlmf.nist.gov/1.6.E22 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.23 $\nabla(fg)=f\nabla g+g\nabla f,$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E23 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.24 $\nabla(f/g)=(g\nabla f-f\nabla g)/g^{2},$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E24 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.25 $\nabla\cdot(f\mathbf{F})=f(\nabla\cdot\mathbf{F})+\mathbf{F}\cdot\nabla f,$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E25 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.26 $\nabla\cdot(\mathbf{F}\times\mathbf{G})=\mathbf{G}\cdot(\nabla\times\mathbf{F}% )-\mathbf{F}\cdot(\nabla\times\mathbf{G}),$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E26 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.27 $\nabla\cdot(\nabla\times\mathbf{F})=\operatorname{div}\operatorname{curl}% \mathbf{F}=0,$ ⓘ Symbols: $\operatorname{curl}$: curl of vector-valued function and $\operatorname{div}$: divergence of vector-valued function Permalink: http://dlmf.nist.gov/1.6.E27 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.28 $\nabla\times(f\mathbf{F})=f(\nabla\times\mathbf{F})+(\nabla f)\times\mathbf{F},$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E28 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.29 $\nabla\times(\nabla f)=\operatorname{curl}\operatorname{grad}f=0,$ ⓘ Symbols: $\operatorname{curl}$: curl of vector-valued function and $\operatorname{grad}$: gradient of differentiable scalar function Permalink: http://dlmf.nist.gov/1.6.E29 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.30 $\nabla^{2}f=\nabla\cdot(\nabla f),$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E30 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.31 $\nabla^{2}(fg)=f\nabla^{2}g+g\nabla^{2}f+2(\nabla f\cdot\nabla g),$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E31 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.32 $\nabla\cdot(\nabla f\times\nabla g)=0,$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E32 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.33 $\nabla\cdot(f\nabla g-g\nabla f)=f\nabla^{2}g-g\nabla^{2}f,$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E33 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1
 1.6.34 $\nabla\times(\nabla\times\mathbf{F})=\operatorname{curl}\operatorname{curl}% \mathbf{F}=\nabla(\nabla\cdot\mathbf{F})-\nabla^{2}\mathbf{F}.$ ⓘ Symbols: $\operatorname{curl}$: curl of vector-valued function Permalink: http://dlmf.nist.gov/1.6.E34 Encodings: TeX, pMML, png See also: Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

## §1.6(iv) Path and Line Integrals

Note: The terminology open and closed sets and boundary points in the $(x,y)$ plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii).

$\mathbf{c}(t)=(x(t),y(t),z(t))$, with $t$ ranging over an interval and $x(t),y(t),z(t)$ differentiable, defines a path.

 1.6.35 $\mathbf{c}^{\prime}(t)=(x^{\prime}(t),y^{\prime}(t),z^{\prime}(t)).$ ⓘ Permalink: http://dlmf.nist.gov/1.6.E35 Encodings: TeX, pMML, png See also: Annotations for §1.6(iv), §1.6 and Ch.1

The length of a path for $a\leq t\leq b$ is

 1.6.36 $\int_{a}^{b}\|\mathbf{c}^{\prime}(t)\|\mathrm{d}t.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.6.E36 Encodings: TeX, pMML, png See also: Annotations for §1.6(iv), §1.6 and Ch.1

The path integral of a continuous function $f(x,y,z)$ is

 1.6.37 $\int_{\mathbf{c}}f\mathrm{d}s=\int^{b}_{a}f(x(t),y(t),z(t))\|\mathbf{c}^{% \prime}(t)\|\mathrm{d}t.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.6.E37 Encodings: TeX, pMML, png See also: Annotations for §1.6(iv), §1.6 and Ch.1

The line integral of a vector-valued function $\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}$ along $\mathbf{c}$ is given by

 1.6.38 $\int_{\mathbf{c}}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=\int^{b}_{a}\mathbf{F}(% \mathbf{c}(t))\cdot\mathbf{c}^{\prime}(t)\mathrm{d}t=\int^{b}_{a}\left(F_{1}% \frac{\mathrm{d}x}{\mathrm{d}t}+F_{2}\frac{\mathrm{d}y}{\mathrm{d}t}+F_{3}% \frac{\mathrm{d}z}{\mathrm{d}t}\right)\mathrm{d}t=\int_{\mathbf{c}}F_{1}% \mathrm{d}x+F_{2}\mathrm{d}y+F_{3}\mathrm{d}z.$

A path $\mathbf{c}_{1}(t)$, $t\in[a,b]$, is a reparametrization of $\mathbf{c}(t^{\prime})$, $t^{\prime}\in[a^{\prime},b^{\prime}]$, if $\mathbf{c}_{1}(t)=\mathbf{c}(t^{\prime})$ and $t^{\prime}=h(t)$ with $h(t)$ differentiable and monotonic. If $h(a)=a^{\prime}$ and $h(b)=b^{\prime}$, then the reparametrization is called orientation-preserving, and

 1.6.39 $\int_{\mathbf{c}}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=\int_{\mathbf{c}_{1}}% \mathbf{F}\cdot\mathrm{d}\mathbf{s}.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $\mathbf{F}(\NVar{x},\NVar{y})$: vector Permalink: http://dlmf.nist.gov/1.6.E39 Encodings: TeX, pMML, png See also: Annotations for §1.6(iv), §1.6 and Ch.1

If $h(a)=b^{\prime}$ and $h(b)=a^{\prime}$, then the reparametrization is orientation-reversing and

 1.6.40 $\int_{\mathbf{c}}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=-\int_{\mathbf{c}_{1}}% \mathbf{F}\cdot\mathrm{d}\mathbf{s}.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $\mathbf{F}(\NVar{x},\NVar{y})$: vector Permalink: http://dlmf.nist.gov/1.6.E40 Encodings: TeX, pMML, png See also: Annotations for §1.6(iv), §1.6 and Ch.1

In either case

 1.6.41 $\int_{\mathbf{c}}f\mathrm{d}s=\int_{\mathbf{c}_{1}}f\mathrm{d}s,$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.6.E41 Encodings: TeX, pMML, png See also: Annotations for §1.6(iv), §1.6 and Ch.1

when $f$ is continuous, and

 1.6.42 $\int_{\mathbf{c}}\nabla f\cdot\mathrm{d}\mathbf{s}=f(\mathbf{c}(b))-f(\mathbf{% c}(a)),$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.6.E42 Encodings: TeX, pMML, png See also: Annotations for §1.6(iv), §1.6 and Ch.1

when $f$ is continuously differentiable.

The geometrical image $C$ of a path $\mathbf{c}$ is called a simple closed curve if $\mathbf{c}$ is one-to-one, with the exception $\mathbf{c}(a)=\mathbf{c}(b)$. The curve $C$ is piecewise differentiable if $\mathbf{c}$ is piecewise differentiable. Note that $C$ can be given an orientation by means of $\mathbf{c}$.

### Green’s Theorem

Let

 1.6.43 $\mathbf{F}(x,y)=F_{1}(x,y)\mathbf{i}+F_{2}(x,y)\mathbf{j}$ ⓘ Defines: $\mathbf{F}(\NVar{x},\NVar{y})$: vector (locally) Symbols: $\mathbf{i}$: unit vector, $\mathbf{j}$: unit vector and $F_{j}$: vector function components Permalink: http://dlmf.nist.gov/1.6.E43 Encodings: TeX, pMML, png See also: Annotations for §1.6(iv), §1.6(iv), §1.6 and Ch.1

and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. If $C$ is oriented in the positive (anticlockwise) sense, then

 1.6.44 $\iint_{S}\left(\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{% \partial y}\right)\mathrm{d}A=\int_{C}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=\int% _{C}F_{1}\mathrm{d}x+F_{2}\mathrm{d}y.$

Sufficient conditions for this result to hold are that $F_{1}(x,y)$ and $F_{2}(x,y)$ are continuously differentiable on $S$, and $C$ is piecewise differentiable.

The area of $S$ can be found from (1.6.44) by taking $\mathbf{F}(x,y)=-y\mathbf{i}$, $x\mathbf{j}$, or $-\frac{1}{2}y\mathbf{i}+\frac{1}{2}x\mathbf{j}$.

## §1.6(v) Surfaces and Integrals over Surfaces

A parametrized surface $S$ is defined by

 1.6.45 $\boldsymbol{{\Phi}}(u,v)=(x(u,v),y(u,v),z(u,v))$ ⓘ Defines: $\boldsymbol{{\Phi}}(x,y,z)$: parameterization (locally) Symbols: $(\NVar{a},\NVar{b})$: open interval Permalink: http://dlmf.nist.gov/1.6.E45 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

with $(u,v)\in D$, an open set in the plane.

For $x$, $y$, and $z$ continuously differentiable, the vectors

 1.6.46 $\mathbf{T}_{u}=\frac{\partial x}{\partial u}(u_{0},v_{0})\mathbf{i}+\frac{% \partial y}{\partial u}(u_{0},v_{0})\mathbf{j}+\frac{\partial z}{\partial u}(u% _{0},v_{0})\mathbf{k}$

and

 1.6.47 $\mathbf{T}_{v}=\frac{\partial x}{\partial v}(u_{0},v_{0})\mathbf{i}+\frac{% \partial y}{\partial v}(u_{0},v_{0})\mathbf{j}+\frac{\partial z}{\partial v}(u% _{0},v_{0})\mathbf{k}$

are tangent to the surface at $\boldsymbol{{\Phi}}(u_{0},v_{0})$. The surface is smooth at this point if $\mathbf{T}_{u}\times\mathbf{T}_{v}\not=0$. A surface is smooth if it is smooth at every point. The vector $\mathbf{T}_{u}\times\mathbf{T}_{v}$ at $(u_{0},v_{0})$ is normal to the surface at $\boldsymbol{{\Phi}}(u_{0},v_{0})$.

The area $A(S)$ of a parametrized smooth surface is given by

 1.6.48 $A(S)=\iint_{D}\|\mathbf{T}_{u}\times\mathbf{T}_{v}\|\mathrm{d}u\mathrm{d}v,$ ⓘ Defines: $A(\NVar{S})$: area of a parameterized smooth surface $S$ (locally) Symbols: $\mathrm{d}\NVar{x}$: differential, $S$: parameterized surface and $D$: open set in the plane Permalink: http://dlmf.nist.gov/1.6.E48 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

and

 1.6.49 $\|\mathbf{T}_{u}\times\mathbf{T}_{v}\|=\sqrt{\left(\frac{\partial(x,y)}{% \partial(u,v)}\right)^{2}+\left(\frac{\partial(y,z)}{\partial(u,v)}\right)^{2}% +\left(\frac{\partial(x,z)}{\partial(u,v)}\right)^{2}}.$ ⓘ Symbols: $(\NVar{a},\NVar{b})$: open interval, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$: partial derivative and $\partial\NVar{x}$: partial differential Permalink: http://dlmf.nist.gov/1.6.E49 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

The area is independent of the parametrizations.

For a sphere $x=\rho\sin\theta\cos\phi$, $y=\rho\sin\theta\sin\phi$, $z=\rho\cos\theta$,

 1.6.50 $\|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\|=\rho^{2}\left|\sin\theta\right|.$ ⓘ Symbols: $\sin\NVar{z}$: sine function, $\rho$: radius, $\theta$: angle and $\phi$: angle Permalink: http://dlmf.nist.gov/1.6.E50 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

For a surface $z=f(x,y)$,

 1.6.51 $A(S)=\iint_{D}\sqrt{1+\left(\frac{\partial f}{\partial x}\right)^{2}+\left(% \frac{\partial f}{\partial y}\right)^{2}}\mathrm{d}A.$

For a surface of revolution, $y=f(x)$, $x\in[a,b]$, about the $x$-axis,

 1.6.52 $A(S)=2\pi\int^{b}_{a}|f(x)|\sqrt{1+(f^{\prime}(x))^{2}}\mathrm{d}x,$

and about the $y$-axis,

 1.6.53 $A(S)=2\pi\int^{b}_{a}|x|\sqrt{1+(f^{\prime}(x))^{2}}\mathrm{d}x.$

The integral of a continuous function $f(x,y,z)$ over a surface $S$ is

 1.6.54 $\iint_{S}f(x,y,z)\mathrm{d}S=\iint_{D}f(\boldsymbol{{\Phi}}(u,v))\|\mathbf{T}_% {u}\times\mathbf{T}_{v}\|\mathrm{d}u\mathrm{d}v.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $S$: parameterized surface, $\boldsymbol{{\Phi}}(x,y,z)$: parameterization and $D$: open set in the plane Permalink: http://dlmf.nist.gov/1.6.E54 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

For a vector-valued function $\mathbf{F}$,

 1.6.55 $\iint_{S}\mathbf{F}\cdot\mathrm{d}\mathbf{S}=\iint_{D}\mathbf{F}\cdot(\mathbf{% T}_{u}\times\mathbf{T}_{v})\mathrm{d}u\mathrm{d}v,$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $S$: parameterized surface and $D$: open set in the plane Permalink: http://dlmf.nist.gov/1.6.E55 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

where $\mathrm{d}\mathbf{S}$ is the surface element with an attached normal direction $\mathbf{T}_{u}\times\mathbf{T}_{v}$.

A surface is orientable if a continuously varying normal can be defined at all points of the surface. An orientable surface is oriented if suitable normals have been chosen. A parametrization $\boldsymbol{{\Phi}}(u,v)$ of an oriented surface $S$ is orientation preserving if $\mathbf{T}_{u}\times\mathbf{T}_{v}$ has the same direction as the chosen normal at each point of $S$, otherwise it is orientation reversing.

If $\boldsymbol{{\Phi}}_{1}$ and $\boldsymbol{{\Phi}}_{2}$ are both orientation preserving or both orientation reversing parametrizations of $S$ defined on open sets $D_{1}$ and $D_{2}$ respectively, then

 1.6.56 $\iint_{\boldsymbol{{\Phi}}_{1}(D_{1})}\mathbf{F}\cdot\mathrm{d}\mathbf{S}=% \iint_{\boldsymbol{{\Phi}}_{2}(D_{2})}\mathbf{F}\cdot\mathrm{d}\mathbf{S};$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $\boldsymbol{{\Phi}}(x,y,z)$: parameterization and $D$: open set in the plane Permalink: http://dlmf.nist.gov/1.6.E56 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

otherwise, one is the negative of the other.

### Stokes’s Theorem

Suppose $S$ is an oriented surface with boundary $\partial S$ which is oriented so that its direction is clockwise relative to the normals of $S$. Then

 1.6.57 $\iint_{S}(\nabla\times\mathbf{F})\cdot\mathrm{d}\mathbf{S}=\int_{\partial S}% \mathbf{F}\cdot\mathrm{d}\mathbf{s},$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $\int$: integral, $\partial\NVar{x}$: partial differential and $S$: parameterized surface Permalink: http://dlmf.nist.gov/1.6.E57 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

when $\mathbf{F}$ is a continuously differentiable vector-valued function.

### Gauss’s (or Divergence) Theorem

Suppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. Then

 1.6.58 $\iiint_{V}(\nabla\cdot\mathbf{F})\mathrm{d}V=\iint_{S}\mathbf{F}\cdot\mathrm{d% }\mathbf{S},$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $S$: parameterized surface and $V$: closed region Permalink: http://dlmf.nist.gov/1.6.E58 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

when $\mathbf{F}$ is a continuously differentiable vector-valued function.

### Green’s Theorem (for Volume)

For $f$ and $g$ twice-continuously differentiable functions

 1.6.59 $\iiint_{V}(f\nabla^{2}g+\nabla f\cdot\nabla g)\mathrm{d}V=\iint_{S}f\frac{% \partial g}{\partial n}\mathrm{d}A,$

and

 1.6.60 $\iiint_{V}(f\nabla^{2}g-g\nabla^{2}f)\mathrm{d}V=\iint_{S}\left(f\frac{% \partial g}{\partial n}-g\frac{\partial f}{\partial n}\right)\mathrm{d}A,$

where $\ifrac{\partial g}{\partial n}=\nabla g\cdot\mathbf{n}$ is the derivative of $g$ normal to the surface outwards from $V$ and $\mathbf{n}$ is the unit outer normal vector.