Evaluate the integral integral ∫ from (1,2,3 ) to (5, 7,-2 ) y dx + x dy + 4 dz by finding parametric equations for the line segment from (1,2,3) to (5,7,- 2) and evaluating the line integral of of f = yi + x j+ 3k along the segment. since f is conservative, the integral is independent of the path.

is conservative if there is a scalar function such that . This would require

(or perhaps the last partial derivative should be 4 to match up with the integral?)

From these equations we find

so is indeed conservative, and the gradient theorem (a.k.a. fundamental theorem of calculus for line integrals) applies. The value of the line integral depends only the endpoints: