If is any curve from to and . Compute the line integral
Here's a different way to do it: The question implies that the path does not matter and a little checking of "mixed derivatives" verifies that. That means that this is an "exact differential"- there exist a function, F(x,y,z), such that . Find that "anti-derivative" and evaluate at (1,1,1) and (0,0,0).
From we get F(x,y,z)= xz+ 4xy+ g(y,z). Notice that the "constant of integration" may be a function of y and z since they are treated as constants in differentiation with respect to x.
Differentiating that with respect to y, . The "4x" terms cancel (as they had to do- otherwise the integral would depend on the path) so now we have . Taking the anti-derivative of that with respect to y, g(y,z)= 2yz+ h(z) where now the "constant of integration" may depend on z. That is, we now have [tex]F(x,y,z)= xz+ 4xy+ 2yz+ h(z).
Differentiating that with respect to z, , so - it really is a constant.
F(x,y,z)= xz+ 4xy+ 2yz+ C.