For example, I know that the physical meaning behind a standard, single integral is the area under the curve (with respect to the x or y axes).
Likewise, the a line integral can be physically visualized as a "wall" with the base of the wall bordering along the line and the top bordering the surface of interest–the line integral is the area of that wall.
A double integral is the volume under the surface of interest (with respect to the xy/xz/yz plane).
What is the surface integral then? If the surface integral is the 3d analog of the line integral, is it then the volume under one surface with respect to another surface, instead of the xy/xz/yz plane?
If anybody could help me physically visualize the surface integral, I would be extremely grateful!!
Best Answer
A big one is thinking of the surface integral as the amount of flux (i.e. flow) through a surface.
For example, take a pool of water. Suppose $f(x, y, z) :: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ is a vector field that, for each point in the pool, tells you the strength and direction of the water flow at that point.
Then you could integrate $f$ over some surface (let's say the membrane of a jellyfish $J$) to find the total amount of water flowing through $J$.
(However, it's a weird kind of "total" where flow in one direction cancels out flow in the opposite direction.)