# Review of Curve and Surface Integrals

### Definitions

Curve integrals

$C$:   $\mathbf{r}\left(t\right),\phantom{\rule{0.6em}{0ex}}a\le t\le b$

Note:  $d\mathbf{r}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\mathbf{r}}^{\prime }\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\mathbf{T}\phantom{\rule{0.3em}{0ex}}ds$;    $ds=||d\mathbf{r}||=||{\mathbf{r}}^{\prime }\left(t\right)||dt$

Surface integrals

$S$:   $\mathbf{W}\left(u,v\right),\phantom{\rule{0.6em}{0ex}}\left(u,v\right)\in E$     ($E$ a planar region)

Note:  $d\mathbf{W}=\left({\mathbf{W}}_{u}×{\mathbf{W}}_{v}\right)dudv\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\mathbf{N}d\sigma$;    $d\sigma =||d\mathbf{W}||=||{\mathbf{W}}_{u}×{\mathbf{W}}_{v}||dudv$
where ${W}_{u}=\frac{\partial W}{\partial u}\phantom{\rule{0.6em}{0ex}}\text{,}\phantom{\rule{1.8em}{0ex}}{W}_{v}=\frac{\partial W}{\partial v}$

For both curves and surfaces a parameterization determines an orientation. The orientation of a surface in ${\mathbf{R}}^{3}$ is determined by the two-fold choice of a unit normal to that surface, and the boundary of an oriented surface is given the orientation that is related to the chosen unit normal in a right-handed coordinate system by the “right-hand rule”.

### Theorems of the form ${\int }_{\partial G}\phantom{\rule{0.3em}{0ex}}\omega ={\int }_{G}\phantom{\rule{0.3em}{0ex}}d\omega$

 dim $G$
 $\mathbit{\omega }$
 $\mathbit{d}\mathbit{\omega }$
 left side
 right side
 Remarks
$1$ $f$ ${f}^{\prime }$ $f\left(b\right)-f\left(a\right)$ ${\int }_{I}{f}^{\prime }\left(t\right)\phantom{\rule{0.3em}{0ex}}dt$
 Fund. Thm. of Calculus $I$ interval in $\mathbf{R}$ from $a$ to $b$
$1$ $f$ $\nabla f$ $f\left(B\right)-f\left(A\right)$ ${\int }_{C}\mathrm{grad}f\phantom{\rule{0.3em}{0ex}}·\phantom{\rule{0.3em}{0ex}}d\mathbf{r}$
 $C$ path in ${\mathbf{R}}^{n}$ from $A$ to $B$
$2$ $F·d\mathbf{r}$ $\left(\nabla \wedge F\right)\phantom{\rule{0.3em}{0ex}}dA$ ${\int }_{\partial R}F·d\mathbf{r}$ ${\iint }_{R}\mathrm{curl}F\phantom{\rule{0.3em}{0ex}}dA$
 Green's Thm. $R$ region in ${\mathbf{R}}^{2}$ ($\partial R$ anti-clockwise in right-hand coord. system)
$2$ $F·d\mathbf{r}$ $\left(\nabla ×F\right)·d\mathbf{W}$ ${\int }_{\partial S}F·d\mathbf{r}$ ${\iint }_{S}\mathrm{curl}F·d\mathbf{W}$
 Classical Stokes’ Thm. $S$ surface in ${\mathbf{R}}^{3}$ (with r/l hand rule)
$3$ $F·d\mathbf{W}$ $\left(\nabla ·F\right)\phantom{\rule{0.3em}{0ex}}dV$ ${\iint }_{\partial D}F·d\mathbf{W}$ ${\iiint }_{D}\mathrm{div}F\phantom{\rule{0.3em}{0ex}}dV$
 The Divergence Thm. $D$ domain in ${\mathbf{R}}^{3}$ ($\partial D$ with outer normal)