# Surface Integral, Stokes Theorem, Divergence theorem

We were given problems to work out before asked the meaning, but I have done them and am pretty confident with the answers, however I do not know the meaning of what I calculated. Please let me know if you need the entire problem to do this, although I think you can give me an idea of what I calculated for each.

Explain the meaning of what we are calculating for the following;

Integral with Respect to Surface Area:

• My attempt: Clueless

Surface Integral

• My attempt: Measured the flux of a vector field across surface area?

Stoke’s Theorem

• My attempt: Stokes theorem is Green’s theorem with essentially two small changes. Changing the line integral to 3 dimensions and the double integral to a surface integral F.n(ie floating in space). So basically we calculated an area of a surface in 3 dimensional region using stokes theorem.

Divergence Theorem

• My attempt: By using the divergence theorem we calculated a result that relates the flow the flux of a vector field through a surface to the behavior of the vector field inside the surface. Overall, what we calculated is the net flow out of a particular region which was defined by the problem given.

Help extremely appreciated, thank you!

• The surface integral $\iint_S f(x,y,z) \, dS$ gives the total mass on a 2-dimensional surface $S$ if the mass density (per unit area) at the point $(x,y,z)$ on $S$ is given by $f(x,y,z)$. – Michael Joyce May 4 '15 at 3:19
• isn't that the integral with respect to surface area? or would that be both actually except in 3d? – Dangerous Game May 4 '15 at 4:01
• Sorry, yes according to your terminology, the above comment was an interpretation of an integral with respect to surface area. It refers to a surface $S$ in 3-dimensional space. – Michael Joyce May 4 '15 at 11:33