Check the fundamental theorem for gradients, using T = x2 + 4xy + 2yz3, the points a = (...

Check the fundamental theorem for gradients, using T = x² + 4xy + 2yz³, the points a = (0, 0, 0), b = (1, 1, 1), and the three paths in Fig. 1.28:

(a) (0, 0, 0) → (1, 0, 0) → (1, 1, 0) → (1, 1, 1);

(b) (0, 0, 0) → (0, 0, 1) → (0, 1, 1) → (1, 1, 1);

(c) the parabolic path z = x²; y = x.

Fig. 1.28