Evaluate the line integral c 2xyz dx + x2z dy + x2y dz where c is an oriented simple curve connecting (1, 2, 4) to (1, 4, 16).

The fact that is not described exactly is a clue to try to show the vector field is conservative. That means you need to show there is some scalar function for which ; that is, the following set of PDEs is satisfied:





Integrating both sides of the first PDE with respect to gives



Differentiating with respect to gives



Differentiating with respect to gives



So the potential scalar function is



By the fundamental theorem of calculus, we have for any oriented simple curve parameterized by a vector-valued function , starting at and terminating at ,



In this case, and , so the line integral has a value of