Summary of Line and Surface Integrals

Integration

• Line Integrals

$${\int }_c(x,y,z)ds$$

Where the curve C is parametrized as r(t)

and we solve this as..

$${\int }_a^b(F_{\text{oncurve}}||\frac{dr}{dt}||dt)$$

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• 2D Surface Integrals

$${\iint }_{s}f(x,y,z)ds$$

Where S could be S = r(t,k) or H (x,y,z) =0

What’s important here, is that you get the normal, N (Usually the gradient)

and we solve this as…

$${\int }_a^b(F_{\text{onsurface}}||\vec{n}||dA)$$

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• 3D Surface Integrals

$${\iiint }_{vol}f(x,y,z)ds$$

We solve this one is a similar way!

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Theorems (Special cases that give alternative ways to calculate)

• 1D

FOTOC

$${\int }_{x=a}^b(\frac{dg}{dx})=g(b)-g(a)$$

b) Path Independent Special Case

$${\int }_c\nabla g\cdot \hat{T}ds=g(b)-g(a)$$

another form of that

$${\int }_c(\nabla g\times \hat{k})\hat{N}ds=g(b)-g(a)$$

• 2D

Green’s Theorem

Normal/Flux Form, also known as 2D Divergence Theorem.

$${\iint }_R\nabla \cdot \vec{F}dA={\oint }_c\vec{F}\cdot \hat{N}ds$$

Tangential/ Circular form, also known as 2D Stokes theorem

$${\iint }_R(\nabla \times \vec{F})\cdot \hat{k}dA={\oint }_c\vec{F}\cdot \hat{T}ds$$

• 3D

Stokes Theorem

$${\int }_c(\nabla \times \vec{F})\hat{N}ds={\oint }_c\vec{F}\cdot \hat{T}ds$$

When Stokes is a flat region, S turns into R< and we just get Green’s theorem again.

Divergence Theorem

$${\iiint }_{Vol}(\nabla \cdot F)dVol=\oint {\oint }_S\vec{F}\cdot \hat{N}ds$$