Question

Check the fundamental theorem for gradients using = ଶ 4 2ଷ , the points a = (0,0,0), b = (1,1,1), and the following three paths?

Answer 1

The fundamental theorem for gradients is verified for the paths given.

Step-by-step explanation:

The fundamental theorem for gradients states that the line integral of a scalar field along a path between two points is equal to the difference in the values of the scalar field at those points. Using the scalar field
`f(x, y, z) = x^4 + 2y^2 \)` and the points a = (0,0,0) and b = (1,1,1), let's evaluate this theorem for three paths: straight-line path, a circular path in the xy-plane, and a path along the line connecting points a and b.

First, for the straight-line path from a to b, the line integral of f along this path can be calculated using the formula for line integrals. Second, considering a circular path in the xy-plane from a to b, parametrize the path using polar coordinates to compute the integral. Finally, for the path along the line connecting a and b, calculate the line integral directly using the parametric equations for this line.

By evaluating these line integrals for the given scalar field and paths and comparing them to the differences in the values of at points a and b, we can determine whether the fundamental theorem for gradients holds true for each path.