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Solved: Alternative construction of potential functions
Chapter 13, Problem 59AE(choose chapter or problem)
Problem 59AE
Alternative construction of potential functions Use the procedure in Exercise to construct potential functions for the following fields.
Alternative construction of potential functions in ℝ2 Assume that the vector field F is conservative in ℝ2, so that the line integral ∫CF·dr is independent of path. Use the following procedure to construct a potential function φ for the vector field F = 〈f, g〉= 〈2x –y, −x+2y〉
a. Let A be (0, 0) and let B be an arbitrary point (x,y).Define φ(x, y)to be the work required to move an object from A to B, where φ(A) = 0. Let C1 be the path from A to (x, 0) to B and let C2be the path from A to (0, y)to B. Draw a picture.
b. Evaluate and conclude that φ(x,y) = x2 − xy + y2.
c. Verify that the same potential function is obtained by evaluating the line integral over C2.
F = 〈x,y〉
Questions & Answers
QUESTION:
Problem 59AE
Alternative construction of potential functions Use the procedure in Exercise to construct potential functions for the following fields.
Alternative construction of potential functions in ℝ2 Assume that the vector field F is conservative in ℝ2, so that the line integral ∫CF·dr is independent of path. Use the following procedure to construct a potential function φ for the vector field F = 〈f, g〉= 〈2x –y, −x+2y〉
a. Let A be (0, 0) and let B be an arbitrary point (x,y).Define φ(x, y)to be the work required to move an object from A to B, where φ(A) = 0. Let C1 be the path from A to (x, 0) to B and let C2be the path from A to (0, y)to B. Draw a picture.
b. Evaluate and conclude that φ(x,y) = x2 − xy + y2.
c. Verify that the same potential function is obtained by evaluating the line integral over C2.
F = 〈x,y〉
ANSWER:
Solution 59AE