Review of Curve and Surface Integrals

Definitions

Curve integrals

$C$:   $\mathbf{r}\left(t\right),a\le t\le b$

(a) scalar function $f$: $${\int }_{C}fds={\int }_a^{b}f\left(\mathbf{r}\left(t\right)\right)||{\mathbf{r}}^{\prime }\left(t\right)||dt$$

(b) vector function $F$: $${\int }_{C}F·d\mathbf{r}={\int }_a^{b}F\left(\mathbf{r}\left(t\right)\right)·{\mathbf{r}}^{\prime }\left(t\right)dt$$

   Note:  $d\mathbf{r}={\mathbf{r}}^{\prime }\left(t\right)dt=\mathbf{T}ds$;    $ds=||d\mathbf{r}||=||{\mathbf{r}}^{\prime }\left(t\right)||dt$

Surface integrals

$S$:   $\mathbf{W}\left(u,v\right),\left(u,v\right)\in E$     ($E$ a planar region)

(a) scalar function $f$: $${\iint }_{S}fd\sigma ={\iint }_{E}f\left(\mathbf{W}\left(u,v\right)\right)||{\mathbf{W}}_u\times {\mathbf{W}}_v||dudv$$

(b) vector function $F$: $${\iint }_{S}F·d\mathbf{W}={\iint }_{E}F\left(\mathbf{W}\left(u,v\right)\right)·\left({\mathbf{W}}_u\times {\mathbf{W}}_v\right)dudv$$

   Note:  $d\mathbf{W}=\left({\mathbf{W}}_u\times {\mathbf{W}}_v\right)dudv=\mathbf{N}d\sigma$;    $d\sigma =||d\mathbf{W}||=||{\mathbf{W}}_u\times {\mathbf{W}}_v||dudv$
where $$W_u=\frac{\partial W}{\partial u}\text{,}{W}_v=\frac{\partial W}{\partial v}$$

Special case 1: graph of a function of two variables $$S:z=f\left(x,y\right),\left(u,v\right)=\left(x,y\right),\mathbf{W}\left(x,y\right)=\left(x,y,f\left(x,y\right)\right),d\mathbf{W}=\left(-\frac{\partial f}{\partial x},-\frac{\partial f}{\partial y},1\right)dxdy$$

Special case 2: a sphere parameterized with spherical coordinates $$\mathbf{W}\left(u,v\right)=a\left(\cos u\sin v,\sin u\sin v,\cos v\right)d\mathbf{W}=-a^2\sin v\left(\cos u\sin v,\sin u\sin v,\cos v\right)dudv$$

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”.

The Fundamental Theorem of Calculus

Theorems of the form ${\int }_{\partial G}\omega ={\int }_{G}d\omega$