Question

Check the fundamental theorem for gradients using T = x2 4xy-2yz3, the points a- (0,0,0), b (1,1,1), and the following three paths: c. The parabolic path z x2,y-x.

Answer 1

The student's question concerns validating the fundamental theorem for gradients on a given scalar field between two points across different paths. The theorem will hold if the line integral results are the same for each path.

Step-by-step explanation:

The question concerns using the fundamental theorem for gradients to check a given scalar field T, defined as T = x2 - 4xy - 2yz3, between points a = (0,0,0) and b = (1,1,1) and across different paths including a parabolic path defined as z=x2, y=x. The fundamental theorem for gradients states that the line integral of a scalar field between two points is independent of path and can be calculated using the gradient of the field. In this case, we would need to find the gradient vector of T, which would be ∇T = (2x - 4y, -4x - 2z3, -6yz2), and evaluate the line integral over the specified paths. Checking against each path would demonstrate the theorem if the results are identical. However, the question seems to have irrelevant parts that are not connected to the main task, such as additional points and trajectory information which are not required for solving this particular problem of proving the theorem