Theorem of Green, theorem of Gauss and theorem of Stokes

Theorem of Green

Let $F(x,y)=(F_x(x,y),F_y(x,y))$ be a differentiable function of two variables in the plane, and let $D$ be a region of the real plane. The border of $D$ is $C$.

Therefore:$${\displaystyle {\int }_{C}f\cdot dL={\int }_D(\frac{d}{dx}{F}_y-\frac{d}{dy}{F}_x)dxdy}$$

Theorem of Gauss

$V$ is a closed volume in space, and $S$ is its border parametrized (its "skin"), therefore, if $F:V\subset {\mathbb{R}}^3⟶{\mathbb{R}}^3$ , it is a differentiable function in $V$, $${\displaystyle {\int }_{S}F\cdot dS={\int }_{V}div(F)\cdot dxdydz}$$ With this theorem, we can convert complicated surface integrals into volume integrals.

Procedure

1. Calculate $div(F)$
2. Find the integration region $V$ (a volume, so $3$ variables)
3. Calculate the integral with $3$ variables.

Theorem of Stokes

A surface of space is $S$ and $C$ is its border (or limits), and let $F:S\subset {\mathbb{R}}^3⟶{\mathbb{R}}^3$ be a differentiable function in $S$, then $${\displaystyle {\int }_{C}F\cdot dL={\int }_{S}rot(F)\cdot dS}$$

This theorem can be useful in solving problems of integration when the curve in which we have to integrate is complicated.

It also shows that if $F$ has rotational $0$ in $S$, then its integral along the curve $C$ is zero.

Procedure

1. Find the parametrized integration region $S$ (a surface, so $2$ variables).
2. Calculate $rot(F)$.
3. Calculate the integral of $2$ variables of the rotacional of $F$.