Vector Calculus (15) - Vector Integration

50. Vector Integration

 

$ \vec{A}=\vec{A}(x_{i}) $

 

1) Volume Integral

$ \int_{V}^{}\vec{A}dv=(\int_{V}^{}A_{1}dv, \int_{V}^{}A_{2}dv, \int_{V}^{}A_{3}dv) $

 

2) Surface Integral

$ \int_{S}^{}\vec{A}\cdot d\vec{a}=\int_{S}^{}\vec{A}\cdot \hat{n}da=\int_{S}^{}\sum_{i}^{}A_{i}da_{i} $

(Outward Normal is positive.)

 

3) Line Integral

$ \int_{BC}^{}\vec{A}\cdot d\vec{s}=\int_{BC}^{}\sum_{i}^{}A_{i}dx_{i} $

 

 

51. Some Theorems

 

1) Green's Theorem (Double Integral ↔ Line Integral)

$ \iint_{R}^{}(\frac{\partial A_{2}}{\partial x}-\frac{\partial A_{1}}{\partial y})dxdy=\oint_{C}^{}(A_{1}dx+A_{2}dy) $

 

2) Stokes' Theorem (Line Integral ↔ Surface Integral)

$ \int_{C}^{}\vec{A}\cdot d\vec{s}=\int_{S}^{}(\vec{\bigtriangledown }\times \vec{A})\cdot d\hat{a} $

 

3) Divergence Theorem (Gauss' Theorem) (Surface Integral ↔ Volume Integral)

$ \int_{S}^{}\vec{A}\cdot d\vec{a}=\int_{S}^{}(\vec{\bigtriangledown }\cdot  \vec{A})dv $