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 $F:V \subset \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ , 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 $F:S \subset \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ 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 $F$.