§1.6 Vectors and Vector-Valued Functions

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).

A path is defined by $𝐜(t)=(x(t),y(t),z(t))$, with $t$ ranging over an interval and $x(t),y(t),z(t)$ differentiable. Letting

then the length of a path for $a\le t\le b$ is

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

The line integral of a vector-valued function $𝐅=F_1𝐢+F_2𝐣+F_3𝐤$ along $𝐜$ is given by

A path ${𝐜}_1(t)$, $t\in [a,b]$, is a reparametrization of $𝐜(t^{\prime })$, $t^{\prime }\in [a^{\prime },b^{\prime }]$, if ${𝐜}_1(t)=𝐜(t^{\prime })$ 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

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

In either case

when $f$ is continuous, and

when $f$ is continuously differentiable.

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

A parametrized surface $S$ is defined by

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

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

and

are tangent to the surface at $𝚽(u_0,v_0)$. The surface is smooth at this point if ${𝐓}_u\times {𝐓}_v\ne 0$. A surface is smooth if it is smooth at every point. The vector ${𝐓}_u\times {𝐓}_v$ at $(u_0,v_0)$ is normal to the surface at $𝚽(u_0,v_0)$.

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

and

The area is independent of the parametrizations.

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

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

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

and about the $y$-axis,

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

For a vector-valued function $𝐅$,

where $d𝐒$ is the surface element with an attached normal direction ${𝐓}_u\times {𝐓}_v$.

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 $𝚽(u,v)$ of an oriented surface $S$ is orientation preserving if ${𝐓}_u\times {𝐓}_v$ has the same direction as the chosen normal at each point of $S$, otherwise it is orientation reversing.

If ${𝚽}_1$ and ${𝚽}_2$ are both orientation preserving or both orientation reversing parametrizations of $S$ defined on open sets $D_1$ and $D_2$ respectively, then

otherwise, one is the negative of the other.