Vector Fields

Neo Wang / March 15, 2021

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• Vector fields that can be represented as the gradients of differentiable functions are called conservative vector fields

• Vector fields are given by $F(x,y)=\langle M, N\rangle$

• To find the potential function of a vector field use partial integration.

• In three dimensions, if a vector field $F(x,y) = \langle M, N, P \rangle$ is conservative then

$\frac{\partial P}{\partial y}=\frac{\partial N}{\partial z},\frac{\partial P}{\partial x}=\frac{\partial M}{\partial z},\frac{\partial N}{\partial x} = \frac{\partial M}{\partial y}$
• Surface Integral for $z=g(x,y)$ Method 1
$\int \int_S f(x,y,z)dS = \int_D\int f(x,y,g(x,y)) \sqrt{1 + (\frac{\partial g}{\partial x})^2 + (\frac{\partial g}{\partial y})^2}dA$
• Surface Integral for parametrics
$\int_S\int f(x,y,z)dS = \int_D \int f(\vec{r}(u,v))||\vec{r_u}\times \vec{r_v}||dA$
• Divergence theorem
$\int_C F\cdot N ds = \int_R\int (\frac{\partial M}{\partial x}+\frac{\partial N}{\partial y})dA = \int_R\int \textrm{div F d}A$
$\int_S\int F \cdot N dS = \int \int_Q \int \textrm{div} \textbf{F} dV$
• Stoke's Theorem
$\int_C F \cdot dr = \int_S \int (\textrm{curl F}) \cdot N dS$
• Curl where $F_1$ is the first component of the vector, $F_2$ as the second, and $F_3$ as the third. Ex. $\langle x, y, z \rangle$ $F_1 = x$
$\textrm{curl} \textbf{\textrm{F}} = \nabla \times \textbf{\textrm{F}}$
$=\langle \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \rangle$
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