The function has a gradient vector field. The gradient vector field does not have continuous 1st order partial derivatives. Therefore, 1) is the gradient vector field a conservative vector field? and 2) Is a line integral of this vector field independent of path?
EDIT: After further research, I've come to the conclusion that (1) a vector field is conservative if and only if it is a gradient of a function. (2) If a vector field is on a simply connected region, and P and Q have continuous first-order partial derivatives, and throughout D, then the vector field is conservative. From (2), if all the conditions are not met, we cannot definitively say that the vector field is non-conservative.
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I have another inquiry. Why must we look at all first-order partial derivatives if only ∂y∂Pand∂x∂Q matter?