Question

integral (1-y*e^-x)dx + (e^-x)dy on C where C is any path from (0,1) to (1,2). Show that line integral is independent of path and evaluate the integral.

Answer 1

I don't tme to do this problem but this is how to do it. A line integral is independent of path if and only if the force F is conservative. Thus, Fdr=0 where F is the force, r is a radius vector and dr is the derivative of r. Hence, take the partial derivative of M with respect ot y and equate it with partial derivative of N with respect to x. Take the partial derivative of M with respect ot z and equate it with partial derivative of P with respect to x. take the partial derivative of M with respect ot y and equate it with partial derivative of N with respect to x. Take the partial derivative of N with respect ot z and equate it with partial derivative of P with respect to y. If these partial derivatives are equal, the force F is conservative and therefore independent of path. If F is conservative and therefore independent of path, all you need do to evaluate the integral is evaluate the integral at the end points after you find some function "f' whose gradient if "F". If these is not clear to you consult your calculus book.