It all starts from the leap jump accomplished by calculus in the attempt to calculate irregular shape. It does’t take super smartness to go about by dividing the shape into infinitesimal small pieces, then sum up. But the sum-up is at most an approximation, not a finite, accurate value, what makes it accurate?

That leap from approximation to an accurate value is through the concept of differential/integral and empirical summarization of numerous functions of such. for example the differentiation solution for function -f(X^2) is 2x.

Sal has a rigorous mathematical proving video (need to dig out).

two-dimensional scenario too:

Logically straightforward, one could infer it also holds in three-dimensional calculation. The question is what’s the scalar field based on this three-dimensional object. It’s hard to perceive as we live in a three-dimensional world. However, there are concrete use cases, such as density. Density is depended on x, y and z, so to calculate the mass of an object, triple integral can be applied.

Now the fun part, about how to do the integral calculation of three-dimensional surface, starting with the classical torus.

In math language, two parameters s and t are needed to define:

There complicated steps in middle to do the integral of torus, the fina result is neat as

In physics, work done by a vector force through a path, closed or open, is often needed to calculate. It is so called vector line to scalar field integration:

Moreover, it’s important to know both forms in explict dr.fvector and without explicit dr.