Fundamental Theorem of Line Integrals

Calculus 3 · Axiom Academy

LESSON Fundamental Theorem of Line Integrals A gradient field's line integral depends only on the endpoints — , the vector-field twin of the FTC. If — that is, is conservative with potential function f — and C is a smooth curve running from a point A to a point B , then the line integral is just a difference of potential values: Only the endpoints A and B enter — never the route between them. 2. Conservative Fields & Path Independence A vector field is conservative when it is the gradient of a scalar potential f . For such a field, these four statements all say the same thing: (in a simply connected region) 3. Closed Curves & Circulation Here is the sharpest consequence: for a conservative field, the line integral around any closed curve is zero — because a closed curve starts and ends at the same point. Why? Start at A and return to A , so the answer is f(A) - f(A) = 0 . Think of gravity: walk up a hill and back down to where you started, and gravity does zero net work. Compute where and C is any path from (0,0) to (1,1) . Solution. Because with f(x,y) = x^2 + y^2 , the Fundamental Theorem gives the answer straight from the endpoints: You've seen why integrating a gradient field is just a difference of potential values — the multivariable echo of the Fundamental Theorem of Calculus. Scroll up to revisit any step.

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