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

$\int _{{\vec {r}}=A}^{B}{\vec {\nabla }}f\cdot \mathrm {d} {\vec {s}}=f(B)-f(A)$

Green's Theorem

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

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

2D Stokes' (Curl) Theorem

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

2D Gauss' (Divergence) Theorem

If ${\vec {G}}=\left\langle Q(x,y),-P(x,y),0\right\rangle$ is a nice vector field on $R$, then

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

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 ${\vec {F}}$ is a nice vector field on $R$, then

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

Gauss' (Divergence) Theorem

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

$\iiint _{V}{\vec {\nabla }}\cdot {\vec {F}}\,\mathrm {d} V=\iint _{\partial V}{\vec {F}}\cdot \mathrm {d} {\vec {S}}$