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)=⟨M,N⟩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)=⟨M,N,P⟩F(x,y) = \langle M, N, P \rangle is conservative then

∂P∂y=∂N∂z,∂P∂x=∂M∂z,∂N∂x=∂M∂y\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)z=g(x,y) Method 1
∫∫Sf(x,y,z)dS=∫D∫f(x,y,g(x,y))1+(∂g∂x)2+(∂g∂y)2dA\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
∫S∫f(x,y,z)dS=∫D∫f(r⃗(u,v))∣∣ru⃗×rv⃗∣∣dA\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
∫CF⋅Nds=∫R∫(∂M∂x+∂N∂y)dA=∫R∫div F dA\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
∫S∫F⋅NdS=∫∫Q∫divFdV\int_S\int F \cdot N dS = \int \int_Q \int \textrm{div} \textbf{F} dV
  • Stoke's Theorem
∫CF⋅dr=∫S∫(curl F)⋅NdS\int_C F \cdot dr = \int_S \int (\textrm{curl F}) \cdot N dS
  • Curl where F1F_1 is the first component of the vector, F2F_2 as the second, and F3F_3 as the third. Ex. ⟨x,y,z⟩\langle x, y, z \rangle F1=xF_1 = x
curlF=∇×F\textrm{curl} \textbf{\textrm{F}} = \nabla \times \textbf{\textrm{F}}
=⟨∂F3∂y−∂F2∂z,∂F1∂z−∂F3∂x,∂F2∂x−∂F1∂y⟩=\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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