surface integral hard question Compute $\int\int_\limits{\gamma}\vec{F}\cdot\bar{n} dS$, where
$\vec{F}=[x^{2}yz+xe^{z}]\bar{i}+[x^{2}+y(1-e^{z})]\bar{j}+[2+x^{3}-xyz^{2}]\bar{k}$
$\gamma =x^{2}+y^{2}=(z-1)^{2}  ,0\leq  z\leq 1$
Am not interested only in solving this particular example but in understanding how to work these kind of problems in general so that I can do the rest of problems alone. I need detailed answer , the more detailed the better .
Thanks in advance.
 A: Here is the general methodology for these problems:


*

*Sketch the surface if it is not too complicated: here it is the part of the cone shown below with $0\le z \le 1$:





*Find a parametrization of the surface on which you are integrating: here is one possibility:
\begin{cases}
x=x\\
y=y\quad\quad\quad\quad\quad\quad\quad x,y \; |\; 0 \le x^2+y^2 \le 1 \\
z=\sqrt{x^2+y^2}+1
\end{cases}


By experience, it is often a good idea to use $x$ and $y$ as parameters, as step 3. is always easy this way. It might be tempting to use parameters $r$ and $\theta$, but this may complicate the next step. 


*Compute $\vec{r}_x \times \vec{r}_y$:
$$
\vec{r}_x \times \vec{r}_y = (1,0,\frac{x}{\sqrt{x^2+y^2}}) \times (0,1,\frac{y}{\sqrt{x^2+y^2}}) = (-\frac{x}{\sqrt{x^2+y^2}},-\frac{y}{\sqrt{x^2+y^2}},1)
$$

*Make sure the orientation is correct (it depends on the convention you are using). Here, the third component of $\vec{r}_x \times \vec{r}_y$ is positive, so I will assume the orientation is correct.

*You are ready to integrate:
$$
\iint_{\gamma}\vec{F}\cdot d\vec{S} =\iint_{x,y| 0\le x^2+y^2 \le 1}\vec{F}(x,y)\cdot \vec{r}_x \times \vec{r}_y\; dx dy = \iint_{x,y| 0\le x^2+y^2 \le 1}f(x,y)\; dx dy
$$


*Perform a change of variables if necessary:
$$
\iint_{x,y| 0\le x^2+y^2 \le 1}f(x,y)\; dx dy = \int_{r=0}^1\int_{\theta=0}^{2\pi}f(r,\theta) \; r dr d\theta
$$

*If possible, check your answer with a software, or more interestingly, by comparing it with another method such as the divergence theorem here. As pointed out by @Mattos, the problem is much simpler using the divergence theorem. I suspect your teacher wants you to try it the hard way to fully appreciate the divergence theorem next week :) 
