Question

Let F = (5xy,7y^2) be a vector field in the plane, and C the path y = 4x^2 , joining (0,0) to (1,4) in the plane.

A. Evaluatestudent submitted image, transcription available belowB. Does the integral in part (A) depend on the path joining (0,0) to (1,4)? (y/n)

Please include steps and answers.

PLease include steps and answers.

Answer 1

To evaluate the work done along the path C, we need to compute the line integral of the vector field F after parametrizing the path using x. Then integrate F along C. To determine path dependence, the curl of F should be checked; if zero, the field is conservative and the work done is path-independent.

Step-by-step explanation:

The question asks to evaluate the work done by a variable force field F along a given path C and to determine whether this work is path-dependent. To evaluate the work done, we need to calculate the line integral of the force field along the path C. This requires parametrizing the path with a convenient parameter, in this case, x, and then substituting the obtained expressions into the integral.

We can parametrize the path C using y = 4x^2, which gives us dy = 8x dx. Substituting y and dy into F, and integrating, we will get the work done over the path C. Furthermore, to check if the line integral is path-independent, one needs to examine whether the vector field F is conservative or not, which could be done by checking if the curl of F is zero.

Steps for Evaluating the Line Integral

1. Parametrize the path y = 4x^2 using x as the parameter.
2. Substitute y and dy in terms of x into the vector field F.
3. Perform the line integral of F along C using the dot product of F with dx and dy.
4. Evaluate the integral from the given starting point (x = 0) to the endpoint (x = 1).

Checking for Path Independence

To determine if the integral depends on the path, check the curl of the vector field. If the curl is zero, then the force field is conservative and the work done is path-independent. Otherwise, the work done is path-dependent.