@Alejandro Jakubi Thanks!, I am still don't attend to the factor until you prompt me,I re-do it with down-half sphere,and it -Pi,like this:

So ,it must be zero,and as you word, I need use polar coordinate,thanks!

And there is also other queation related to the integral, the image is follow:

g is the vector same as above mentioned, and \{sigma}_{2} is a sphere embracing the coordinate orgin point (0,0,0),{e}_{ijk} is Levi-Civita symbol or say 3-order anti-symmetrical tensor,the value of it is 1 or -1 base on the permutation of the i,j,k, i run over 1,2,3, in short ,it is a Einstein sum rule with 3*2*1=6 terms.From above discussion, I gauss the value of it also zero! If it is, it's interesting!Let's discuss it!

@Alejandro Jakubi what do you think about it?

@Alejandro Jakubi look as your proposal is right, So I do it step by step,the result also Pi ,and upload the image,please check where is wrong?

@Carl Love I can wait,please don't forget show your answer.

come on!,we can discuss it together, I need help!

y/(p^3) is the result of the vector multiple 
if I do it like above, it's zero,is it ture?

@Carl Love 

I upload image success, and re-write a new post as link:http://www.mapleprimes.com/questions/153412-How-To-Do-This-Surface-IntegralPlease

@Carl Love the expression is complicate, but i can't upload the image, so I describe it by a new

reply, please check,thanks!

Hello,every people,

I re-write my problem, my problem about this sphere integral:

 ∫g*((∂g)/(∂ x)* (∂g)/(∂ y))ⅆS

the express is difficult to  identify, I describe it as follow:

g is a vector defined as (1/p)(x,y,z), p is the radius of the spherical surface, the integrated function is the vector multiple of the  derivative g.((dg/dx) cross multiple (dg/dy)),the sphere surface function is x^2+y^2+z^2=p^2

I do it by my hand, the result is Pi, but I suspect this result, So I do it with Maple, the Maple tell me it's zero.

How to do it fairly in maple? and please demonstrate.