Vector Analysis Theorems

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This page is ©2002 P. Yasskin and is available on his web page.

Fundamental Theorem of Calculus for Curves

If $\overrightarrow{r}\left(t\right)$ is a nice curve in ${\mathbb{ℝ}}^n$ and $f$ is a nice function in ${\mathbb{ℝ}}^n$, then

$\underset{\overrightarrow{r}=A}{\overset{B}{\int }}\overrightarrow{\mathrm{\nabla }}f\cdot \mathrm{d}\overrightarrow{s}=f(B)-f(A)$

Green's Theorem

If $R$ is a nice region in ${\mathbb{ℝ}}^2$ and $\partial R$ is its boundary curve traversed counter-clockwise, and $\overrightarrow{F}=⟨P(x,y),Q(x,y),0⟩$ is a nice vector field on $R$, then

$\underset{R}{\iint }(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})\mathrm{d}x\mathrm{d}y={\oint }_{\partial R}P\mathrm{d}x+Q\mathrm{d}y$

2D Stokes' (Curl) Theorem

$\underset{R}{\iint }\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{F}\cdot \hat{k}\mathrm{d}x\mathrm{d}y={\oint }_{\partial R}\overrightarrow{F}\cdot \mathrm{d}\overrightarrow{s}$

2D Gauss' (Divergence) Theorem

If $\overrightarrow{G}=⟨Q(x,y),-P(x,y),0⟩$ is a nice vector field on $R$, then

$\underset{R}{\iint }\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{G}\mathrm{d}x\mathrm{d}y={\oint }_{\partial R}\overrightarrow{G}\cdot \mathrm{d}\overrightarrow{n}$

Stokes' (Curl) Theorem

If $S$ is a nice surface in ${\mathbb{ℝ}}^3$ and $\partial S$ is its boundary curve traversed counter-clockwise as seen from the tip of the normal to $S$, and $\overrightarrow{F}$ is a nice vector field on $R$, then

$\underset{S}{\iint }\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{F}\cdot \mathrm{d}\overrightarrow{S}={\oint }_{\partial S}\overrightarrow{F}\cdot \mathrm{d}\overrightarrow{s}$

Gauss' (Divergence) Theorem

If $V$ is a volume in ${\mathbb{ℝ}}^3$ and $\partial V$ is its boundary surface oriented outward from $V$, and $\overrightarrow{F}$ is a nice vector field on $V$, then

$\underset{V}{\iiint }\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{F}\mathrm{d}V=\underset{\partial V}{\iint }\overrightarrow{F}\cdot \mathrm{d}\overrightarrow{S}$