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True/False: The vector field F = yzi + xzj + ryk is the gradient field of some differentiable function f(x, y, 2). (Justify your answer.)

True/False: The vector field F = yzi + xzj + ryk is the gradient field of some differentiable function f(x, y, 2). (Justify your answer.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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True/False: The vector field F = yzi + xzj + ayk is the gradient field of some differentiable function f(x, y, z). (Justify your answer.) True/False: The vector field G = ri + aj is not the gradient field of any function g(a, y) whose second order derivatives are continuous over R?. (Justify your answer.) Find the work of the vector field F(r, y, z) = yi along the curve which is obtained as the intersection of the surfaces z = x²+y² –6 and 6x +12y = z+6. Hint: you may find it useful to complete the squares and use the identity sin? t = }(1 – cos(2t)).
Transcribed Image Text:True/False: The vector field F = yzi + xzj + ayk is the gradient field of some differentiable function f(x, y, z). (Justify your answer.) True/False: The vector field G = ri + aj is not the gradient field of any function g(a, y) whose second order derivatives are continuous over R?. (Justify your answer.) Find the work of the vector field F(r, y, z) = yi along the curve which is obtained as the intersection of the surfaces z = x²+y² –6 and 6x +12y = z+6. Hint: you may find it useful to complete the squares and use the identity sin? t = }(1 – cos(2t)).
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