Suppose that \nabla f(x, y,z) = 2xyze^{x^2}\mathbf{i} + ze^{x^2}\mathbf{j} + ye^{x^2}\mathbf{k}. if f(0,0,0) = 2, find f(1,1,1). hint: as a first step, define a path from (0,0,0) to (1, 1, 1) and compute a line integral.

The simplest path from (0, 0, 0) to (1, 1, 1) is a straight line, denoted , which we can parameterize by the vector-valued function,

for , which has differential

Then with , we have

Complete the square in the quadratic term of the integrand: , then in the integral we substitute :

Make another substitution of :

Integrate by parts, taking

So, we have by the fundamental theorem of calculus that