How can we tie this back to conservative fields and such? heres where the fundamental theorem of line integrals comes into play.
remember a conservative field is just the gradient of a potential field. so the left hand side of the equation isnt really saying much. the important part lies in the fact that you can evaluate the integral by evaluating the potential function at the end points! so if you know you have a conservative field dont bother going through the line integral, use the fundamental theorem. this is once again analogous to the fundamental theorem of calculus, funny how math repeats form.
Is there a quick and painless way to check if a field is conservative? yes!
if you have a function F = Pi + Qj
and you take the partial derivative P with respect to y and Q with respect to x
they should be equal. this is true for a simply connected region.
