Advanced Vector Calculus: Cheat Sheet

Essential Concepts

Divergence and Curl

The divergence of a vector field is a scalar function. Divergence measures the “outflowing-ness” of a vector field. If [latex]{\bf{v}}[/latex] is the velocity field of a fluid, then the divergence of [latex]{\bf{v}}[/latex] at a point is the outflow of the fluid less the inflow at the point.
The curl of a vector field is a vector field. The curl of a vector field at point [latex]P[/latex] measures the tendency of particles at [latex]P[/latex] to rotate about the axis that points in the direction of the curl at [latex]P[/latex].
A vector field with a simply connected domain is conservative if and only if its curl is zero.

Surface Integrals

Surfaces can be parameterized, just as curves can be parameterized. In general, surfaces must be parameterized with two parameters.
Surfaces can sometimes be oriented, just as curves can be oriented. Some surfaces, such as a Möbius strip, cannot be oriented.
A surface integral is like a line integral in one higher dimension. The domain of integration of a surface integral is a surface in a plane or space, rather than a curve in a plane or space.
The integrand of a surface integral can be a scalar function or a vector field. To calculate a surface integral with an integrand that is a function, use [latex]\displaystyle\iint_{S} f(x,y,z)dS=\displaystyle\iint_{D}f ({\bf{r}}(u,v)) \Arrowvert{{\bf{t}}_{u}\times{\bf{t}}_{v}} \Arrowvert{dA}[/latex]. To calculate a surface integral with an integrand that is a vector field, use [latex]\displaystyle\iint_{S} {\bf{F}}\cdot {d{\bf{S}}}=\displaystyle\iint_{S} {\bf{F}}\cdot {\bf{N}}dS[/latex]
If [latex]S[/latex] is a surface, then the area of [latex]S[/latex] is [latex]\displaystyle\iint_{S}dS[/latex]

Stokes’ Theorem

Stokes’ theorem relates a flux integral over a surface to a line integral around the boundary of the surface. Stokes’ theorem is a higher dimensional version of Green’s theorem, and therefore is another version of the Fundamental Theorem of Calculus in higher dimensions.
Stokes’ theorem can be used to transform a difficult surface integral into an easier line integral, or a difficult line integral into an easier surface integral.
Through Stokes’ theorem, line integrals can be evaluated using the simplest surface with boundary [latex]C[/latex].
Faraday’s law relates the curl of an electric field to the rate of change of the corresponding magnetic field. Stokes’ theorem can be used to derive Faraday’s law.

The Divergence Theorem

The divergence theorem relates a surface integral across closed surface [latex]S[/latex] to a triple integral over the solid enclosed by [latex]S[/latex]. The divergence theorem is a higher dimensional version of the flux form of Green’s theorem, and is therefore a higher dimensional version of the Fundamental Theorem of Calculus.
The divergence theorem can be used to transform a difficult flux integral into an easier triple integral and vice versa.
The divergence theorem can be used to derive Gauss’ law, a fundamental law in electrostatics.

Curl
[latex]\nabla\times{\bf{F}}=(R_{y}-Q_{z}){\bf{i}}+(P_{z}-R_{x}){\bf{j}}+(Q_{x}+P_{y}){\bf{k}}[/latex]
Divergence
[latex]\nabla\cdot{\bf{F}}=P_{x}+Q_{y}+R_{z}[/latex]
Divergence fo curl is zero
[latex]\nabla\cdot(\nabla\times{\bf{F}})=0[/latex]
Curl of a gradient is the zero vector
[latex]\nabla\times(\nabla{f})=0[/latex]
Scalar surface integral
[latex]\displaystyle\iint_{S} f(x,y,z)dS=\displaystyle\iint_{D}f ({\bf{r}}(u,v)) \Arrowvert{{\bf{t}}_{u}\times{\bf{t}}_{v}} \Arrowvert{dA}[/latex]
Flux integral
[latex]\displaystyle\iint_{S} {\bf{F}}\cdot {\bf{N}}dS=\displaystyle\iint_{S} {\bf{F}}\cdot {d{\bf{S}}}=\displaystyle\iint_{S} {\bf{F}}({\bf{r}}(u,v))\cdot\left({\bf{t}}_{u}\times{\bf{t}}_{v}\right)dA[/latex]
Stokes’ theorem
[latex]\displaystyle\int_{C} {\bf{F}}\cdot{d{\bf{r}}}=\displaystyle\iint_{S}\text{curl } {\bf{F}}\cdot{d{\bf{S}}}[/latex]
Divergence theorem
[latex]\displaystyle\iiint_{E} \text{div }{\bf{F}}dV=\underset{S}{\displaystyle\iint}{\bf{F}}\cdot{d{\bf{S}}}[/latex]

curl: the curl of vector field [latex]{\bf{F}}=\langle{P,Q,R}\rangle[/latex], denoted [latex]\nabla\times{\bf{F}}[/latex] is the “determinant” of the matrix [latex]\begin{vmatrix}{\bf{i}} & {\bf{j}} & {\bf{k}}\\ \frac{d}{dx} & \frac{d}{dy} & \frac{d}{dz}\\P & Q & R\end{vmatrix}[/latex] and is given by the expression [latex](R_{y}-Q_{z}){\bf{i}}+(P_{z}-R_{x}){\bf{j}}+(Q_{x}+P_{y}){\bf{k}}[/latex]; it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point

divergence: the divergence of a vector field [latex]{\bf{F}}=\langle{P,Q,R}\rangle[/latex], denoted [latex]\nabla\times{\bf{F}}[/latex] is [latex]P_{x}+Q_{y}+R_{z}[/latex]; it measures the “outflowing-ness” of a vector field

divergence theorem: a theorem used to transform a difficult flux integral into an easier triple integral and vice versa

flux integral: another name for a surface integral of a vector field; the preferred term in physics and engineering

Gauss’ law: if [latex]S[/latex] is a piecewise, smooth closed surface in a vacuum and [latex]Q[/latex] is the total stationary charge inside of [latex]S[/latex], then the flux of electrostatic field [latex]\bf{E}[/latex] across [latex]S[/latex] is [latex]Q|{\varepsilon}_{0}[/latex]

grid curves: curves on a surface that are parallel to grid lines in a coordinate plane

heat flow: a vector field proportional to the negative temperature gradient in an object

inverse-square law: the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge

mass flux: the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area

orientation of a surface: if a surface has an “inner” side and an “outer” side, then an orientation is a choice of the inner or the outer side; the surface could also have “upward” and “downward” orientations

parameter domain (parameter space): the region of the uv plane over which the parameters u and v vary for parameterization [latex]{\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}[/latex]

parameterized surface: a surface given by a description of the form [latex]{\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}[/latex] , where the parameters u and v vary over a parameter domain in the uv-plane

regular parameterization: parameterization [latex]{\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}[/latex] such that [latex]{\bf{r}}_{u}{\times}{\bf{r}}_{v}[/latex] is not zero for point [latex](u, v)[/latex] in the parameter domain

Stokes’ theorem: relates the flux integral over a surface [latex]S[/latex] to a line integral around the boundary [latex]C[/latex] of the surface [latex]S[/latex]

surface area: the area of surface S given by the surface integral [latex]\displaystyle{\int_{} {\int_{S} d{\bf{S}}}}[/latex]

surface independent: flux integrals of curl vector fields are surface independent if their evaluation does not depend on the surface but only on the boundary of the surface

surface integral of a scalar-valued function: a surface integral in which the integrand is a scalar function

surface integral of a vector field: a surface integral in which the integrand is a vector field

The Fundamental Theorem for Line Integrals: the value of the line integral [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}[/latex] depends only on the value of [latex]f[/latex] at the endpoints of [latex]C[/latex]: [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}=f({\bf{r}}(b)))-f({\bf{r}}(a))[/latex]