1.
What is similar between this theorem and the Fundamental Theorem of Calculus II from back in 1st year calculus? What is different?
Section 3.2 Fundamental Theorem of Line Integrals
Back in 1st year calculus we have seen the Fundamental Theorem of Calculus II, which loosely said that integrating the derivative of a function just gave the difference of the function at the endpoints. That is, what happened in the middle did not matter. In this video we upgrade to the Fundamental Theorem of Line Integrals, which is a generalization of the Fundamental Theorem of Calculus. It says that when you take the line integral of a conservative vector field (ie one where the field can be written as the gradient of a scalar potential function), then this line integral is similarly just the difference of the function at the endpoints and is thus path independent - only the endpoints matter.
Learning Objectives:
- State the Fundamental Theorem of Line Integrals
- Prove the Fundamental Theorem of Line Integral
