Circulation-Curl Form
$$ \oint F\cdot T ds=\oint_C Mdx+Ndy=\iint_R({\partial N\over \partial x}-{\partial M\over \partial y})dxdy $$
Flux-Divergence Form
$$ \oint F\cdot nds=\oint_C Mdy-Ndx=\iint_R({\partial M\over \partial x}+{\partial N\over \partial y})dxdy $$
Surface area is key.
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Parameterized Surfaces(Explicit)
$$ d\sigma=|r_u\times r_v|dudv $$
Parameterizing a surface is often challenging, except for well-known, symmetrical shapes like sphere, cylinder, cone.
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Level Surfaces(Implicit)
$$ d\sigma={|\nabla f|\over |\nabla f\cdot \hat p|}dA $$
Which is why such complex surfaces are defined by a scalar function $f(x,y,z)=c$ where $c$ is constant.
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Flux of the field’s curl
$$ \oint_R F\cdot dr=\iint_S (\nabla\times F)\cdot nd\sigma $$
$$ \iint_S F\cdot nd\sigma=\iiint_D \nabla\cdot FdV $$
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Green’s Theorem(2D) → Stoke’s and Divergence Theorem(3D)
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Order → $dy\over dx$(first order), $dy^2\over dx$(second order)
Linear: dependent variable and their derivatives appear linearly→$a_n(x){d^ny\over dx^n}=b(x)$
nonlinear→${\partial y\over \partial t}+y{\partial y\over \partial x}=v{\partial^2y\over \partial x^2}$
Homogeneous: no term is a function of the independent variables alone → $f(x)=0$
$$ \begin{aligned} {y'\over g(y)}&=f(x) \\\int {dy\over g(y)}&=\int f(x)dx \end{aligned} $$
linear
$$ \begin{aligned} y'+p(x)y&=q(x) \\y&=r(x)^{-1}[\int r(x)q(x)dx+C] \\\text{where, }r(x)&=e^{\int p(x)dx} \end{aligned} $$
non-linear → linear
$$ \begin{aligned} y'+p(x)y=q(x)y^n \\\text{(Bernoulli Equations)} \\v=y^{1-n} \end{aligned} $$
homogeneous
$$ \begin{aligned} y'&=F({y\over x}) \\v&={y\over x} \\\int{1\over F(v)-v}dv&=ln|x|+C \end{aligned} $$