Vector Integrals

Topics

• Scalar Line Integral
• Vector Line Integral
• Flux Integral

Scalar, Vector, and Flux Integrals
- $\text{2D Flux:}{\int }_{\mathcal{C}}\vec{F}\cdot \hat{n}ds={\int }_{t_0}^{t_f}\vec{F}(\vec{r}(t))\cdot \vec{N}(t)dt$
$\hat{n}$ is obtained by rotating $\hat{t}$ by $\frac{\pi }{2}$ clockwise
$\vec{t}$ can be simplified to just ${\vec{r}}^{\prime }(t)$
- $\text{Vortex Field}$
$\vec{F}(x,y)=⟨-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2}⟩\text{ if }(x,y)\ne (0,0)$
Satisfies the cross-partials condition
Not conservative on paths encircling the origin
$\vec{G}(u,v)=(x(u,v),y(u,v),z(u,v))$
- ${\vec{T}}_u(P)=\frac{\partial \vec{G}(u,v)}{\partial u},{\vec{T}}_v(P)=\frac{\partial \vec{G}(u,v)}{\partial v}$
${\vec{T}}_u$ points toward increasing $u$,
${\vec{T}}_v$ points toward increasing $v$,
$\vec{N}(P)={\vec{T}}_u(P)\times {\vec{T}}_v(P)$
${\vec{N}}_{\text{sphere}}(\theta ,\varphi )=⟨-\cos \theta {\sin }^2\varphi ,-\sin \theta {\sin }^2\varphi ,-\cos \varphi \sin \varphi ⟩$
$(⟨x,y,z⟩-\vec{G}(u,v))\cdot \vec{N}=0\text{ (Equation for tangent plane)}$
- $\text{Area}(S)={\int }_{v_0}^{v_f}{\int }_{u_0}^{u_f}||\vec{N}(u,v)||dudv$
$\vec{N}(u,v)$ is the same as $||{\vec{T}}_u\times {\vec{T}}_v||$, which represents the area of a small paralleleogram on the surface $S$. This representation happens because $||{\vec{T}}_u\times {\vec{T}}_v||$ is $||{\vec{T}}_u||||{\vec{T}}_v||\sin (\theta )$, where $\theta$ is the angle between the two vectors that make up the parallelogram. ${\vec{T}}_u$ for example, is $\frac{\partial \vec{G}}{\partial u}\Delta u=\vec{G}(u+\Delta u,v)-\vec{G}(u,v)=\vec{u}$
- $\text{Scalar surface integral:}{\iint }_{S}f(x,y,z)dS={\int }_{v_0}^{v_f}{\int }_{u_0}^{u_f}f(\vec{G}(u,v))||\vec{N}(u,v)||dudv$
${\iint }_{C}dS=\text{Area}(s)$
Is akin to scalar line integral in that it is the mass of the surface if $f$ is mass density
- $\text{Vector surface integral/3D Flux: }{\iint }_S\vec{F}\cdot \hat{n}dS={\iint }_S\vec{F}\cdot d\vec{S}={\int }_{v_0}^{v_f}{\int }_{u_0}^{u_f}\vec{F}(\vec{G}(u,v))\cdot \vec{N}(u,v)dudv$
Physically measures the amount of “stuff” (say electric field or water current) that passes through a surface
Is akin to the vector line integral and 2D flux, but for surfaces.
Heuristically, $d\vec{S}=\vec{N}dudv,dS=||\vec{N}||dudv$