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