Integral on a surface

Let $S$ be a surface parametrized by the function $\varphi (u,v)=(x(u,v),y(u,v),z(u,v))$, and $f$ a function defined in all the points of the surface, then:

If $f$ is a scalar field (so, if $f (x, y, z)$ belongs to the real line)

Then,$$\displaystyle \int_S f\cdot dL=\int_D f(\varphi(u,v)) \cdot ||T_u \times T_v|| \ dudv$$

where, $$\displaystyle \begin{array}{l} T_u=\Big( \frac{d}{du} x(u,v), \frac{d}{du}y(u,v), \frac{d}{du}z(u,v) \\ T_v=\Big( \frac{d}{dv} x(u,v), \frac{d}{dv}y(u,v), \frac{d}{dv}z(u,v) \Big)\end{array}$$

and $D$ is a region of the real plane where $\varphi$ is defined.

If $F$ is a vector field, $F(x,y,z)=(F_1(x,y,z),F_2(x,y,z),F_3(x,y,z))$

Then, $$\displaystyle \int_S F \cdot dS=\int_Df(\varphi(u,v)) \cdot (T_u \times T_v) \ dudv$$ Namely, the integral of $F$ in the surface $S$ is the scalar product of the composed function with the parametrization times the vector product of $T_u$ and $T_v$.

Procedure:

1. Take the parametrization of the surface $S$, and compute its vectors $T_u$, $T_v$. Therefore, compute the vector product.
2. Substitute $x$, $y$ and $z$ by $x (u, v)$, $y(u, v)$ and $z (u, v)$ in the function $F$, in accordance with the given parametrization.
3. Calculate the scalar product with the results of steps 1 and 2.
4. Calculate the final integral.