Intuition behind surface integrals

Question

While line integrals derive their intuition from , and are analogous to, the concept of Work in physics, what intuition is there to back up the notion of surface integrals? In the texts I've been using, the authors just extend the notions abstractly from the mathematics of line integrals (basically, generalisations) and do not give a foundational intuition or some logic independent of line integrals. How would one derive surface integrals notions and results by general reasoning, assuming no knowledge of line integrals?

Answer 1

I'm not sure if you're asking for mathematical reasoning or a physical motivation for surface integrals. If it's the latter, you'll find plenty motivation for surface integrals when dealing with flux. Let's start with a simpler example.

Assume you have some metal plate that has a temperature gradient on it. You know the temperature $T(x,y)$ at each point, but you are interested in the average temperature of the metal plate. Of course, the average temperature is just the total temperature, i.e. the surface integral of the temperature over the plate $\int_AT(x,y)\mathrm{d}x\mathrm{d}y$, divided by the area of the plate.

Now for flux: Consider a horizontal pipe with water flowing through it. If we assume that we know the horizontal velocity of the water at each point at a certain perpendicular slice of the pipe, we can take the surface integral over the slice to calculate the total amount of water flowing through that section each second. If we know now not the horizontal velocity but the total velocity at each point, and we are still interested in the water flowing through a certain area, we have to integrate over a vector field.

In physics you're quite often interested in how much of a certain quantity flows through a specific area per unit time, which we call the flux. Surface integrals arise naturally then. If you are interested in another (harder) example, look up Gauss's law.