§1.6 Vectors and Vector-Valued Functions

1.6.1		$\mathbf{a}$	$=(a_1,a_2,a_3),$
		$\mathbf{b}$	$=(b_1,b_2,b_3).$
ⓘ 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

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

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 $a\le t\le b$ is

1.6.36		$${\int }_a^b\parallel {\mathbf{c}}^{\prime }(t)\parallel dt.$$
ⓘ Symbols: $dx$: differential of $x$ 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		$${\int }_{\mathbf{c}}fds={\int }_a^{b}f(x(t),y(t),z(t))\parallel {\mathbf{c}}^{\prime }(t)\parallel dt.$$
ⓘ Symbols: $dx$: differential of $x$ 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		$${\int }_{\mathbf{c}}\mathbf{F}\cdot d\mathbf{s}={\int }_a^b\mathbf{F}(\mathbf{c}(t))\cdot {\mathbf{c}}^{\prime }(t)dt={\int }_a^b\left(F_1\frac{dx}{dt}+F_2\frac{dy}{dt}+F_3\frac{dz}{dt}\right)dt={\int }_{\mathbf{c}}{F}_{1}dx+F_{2}dy+F_{3}dz.$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $dx$: differential of $x$, $\int$: integral and $F_j$: vector function components Permalink: http://dlmf.nist.gov/1.6.E38 Encodings: TeX, pMML, png See also: Annotations for §1.6(iv), §1.6 and Ch.1

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		$${\int }_{\mathbf{c}}\mathbf{F}\cdot d\mathbf{s}={\int }_{{\mathbf{c}}_1}\mathbf{F}\cdot d\mathbf{s}.$$
ⓘ Symbols: $dx$: differential of $x$ and $\int$: integral 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		$${\int }_{\mathbf{c}}\mathbf{F}\cdot d\mathbf{s}=-{\int }_{{\mathbf{c}}_1}\mathbf{F}\cdot d\mathbf{s}.$$
ⓘ Symbols: $dx$: differential of $x$ and $\int$: integral 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		$${\int }_{\mathbf{c}}fds={\int }_{{\mathbf{c}}_1}fds,$$
ⓘ Symbols: $dx$: differential of $x$ 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		$${\int }_{\mathbf{c}}\nabla f\cdot d\mathbf{s}=f(\mathbf{c}(b))-f(\mathbf{c}(a)),$$
ⓘ Symbols: $dx$: differential of $x$ 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 $\mathbf{c}$.

A parametrized surface $S$ is defined by

1.6.45		$$\mathbf{\Phi }(u,v)=(x(u,v),y(u,v),z(u,v))$$
ⓘ Defines: $\mathbf{\Phi }(x,y,z)$: parameterization (locally) Symbols: $(a,b)$: 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		$${\mathbf{T}}_u=\frac{\partial x}{\partial u}(u_0,v_0)\mathbf{i}+\frac{\partial y}{\partial u}(u_0,v_0)\mathbf{j}+\frac{\partial z}{\partial u}(u_0,v_0)\mathbf{k}$$
ⓘ Symbols: $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $\mathbf{i}$: unit vector, $\mathbf{j}$: unit vector and $\mathbf{k}$: unit vector Permalink: http://dlmf.nist.gov/1.6.E46 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

and

1.6.47		$${\mathbf{T}}_v=\frac{\partial x}{\partial v}(u_0,v_0)\mathbf{i}+\frac{\partial y}{\partial v}(u_0,v_0)\mathbf{j}+\frac{\partial z}{\partial v}(u_0,v_0)\mathbf{k}$$
ⓘ Symbols: $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $\mathbf{i}$: unit vector, $\mathbf{j}$: unit vector and $\mathbf{k}$: unit vector Permalink: http://dlmf.nist.gov/1.6.E47 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

are tangent to the surface at $\mathbf{\Phi }(u_0,v_0)$. The surface is smooth at this point if ${\mathbf{T}}_u\times {\mathbf{T}}_v\ne 0$. 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 $\mathbf{\Phi }(u_0,v_0)$.

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

1.6.48		$$A(S)={\iint }_D\parallel {\mathbf{T}}_u\times {\mathbf{T}}_v\parallel dudv,$$
ⓘ Defines: $A(S)$: area of a parameterized smooth surface $S$ (locally) Symbols: $dx$: differential of $x$, $S$: 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		$$\parallel {\mathbf{T}}_u\times {\mathbf{T}}_v\parallel =\sqrt{{\left(\frac{\partial (x,y)}{\partial (u,v)}\right)}^2+{\left(\frac{\partial (y,z)}{\partial (u,v)}\right)}^2+{\left(\frac{\partial (x,z)}{\partial (u,v)}\right)}^2}.$$
ⓘ Symbols: $(a,b)$: 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/1.6.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 $x=\rho \sin \theta \cos \varphi$, $y=\rho \sin \theta \sin \varphi$, $z=\rho \cos \theta$,

1.6.50		$$\parallel {\mathbf{T}}_{\theta }\times {\mathbf{T}}_{\varphi }\parallel ={\rho }^2\left|\sin \theta \right|.$$
ⓘ Symbols: $\sin z$: sine function, $\rho$: radius, $\theta$: angle and $\varphi$: 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		$$A(S)={\iint }_D\sqrt{1+{\left(\frac{\partial f}{\partial x}\right)}^2+{\left(\frac{\partial f}{\partial y}\right)}^2}dA.$$
ⓘ Symbols: $dx$: differential of $x$, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $S$: parameterized surface, $D$: open set in the plane and $A(S)$: area of a parameterized smooth surface $S$ Permalink: http://dlmf.nist.gov/1.6.E51 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

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

1.6.52		$$A(S)=2\pi {\int }_a^b|f(x)|\sqrt{1+{(f^{\prime }(x))}^2}dx,$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $S$: parameterized surface and $A(S)$: area of a parameterized smooth surface $S$ Permalink: http://dlmf.nist.gov/1.6.E52 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

and about the $y$-axis,

1.6.53		$$A(S)=2\pi {\int }_a^b|x|\sqrt{1+{(f^{\prime }(x))}^2}dx.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $S$: parameterized surface and $A(S)$: area of a parameterized smooth surface $S$ Permalink: http://dlmf.nist.gov/1.6.E53 Encodings: TeX, pMML, png See also: Annotations for §1.6(v), §1.6 and Ch.1

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

1.6.54		$${\iint }_{S}f(x,y,z)dS={\iint }_{D}f(\mathbf{\Phi }(u,v))\parallel {\mathbf{T}}_u\times {\mathbf{T}}_v\parallel dudv.$$
ⓘ Symbols: $dx$: differential of $x$, $S$: parameterized surface, $\mathbf{\Phi }(x,y,z)$: 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		$${\iint }_S\mathbf{F}\cdot d\mathbf{S}={\iint }_D\mathbf{F}\cdot ({\mathbf{T}}_u\times {\mathbf{T}}_v)dudv,$$
ⓘ Symbols: $dx$: differential of $x$, $S$: 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 $d\mathbf{S}$ 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 $\mathbf{\Phi }(u,v)$ 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 ${\mathbf{\Phi }}_1$ and ${\mathbf{\Phi }}_2$ 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		$${\iint }_{{\mathbf{\Phi }}_1(D_1)}\mathbf{F}\cdot d\mathbf{S}={\iint }_{{\mathbf{\Phi }}_2(D_2)}\mathbf{F}\cdot d\mathbf{S};$$
ⓘ Symbols: $dx$: differential of $x$, $\mathbf{\Phi }(x,y,z)$: 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.