An exercise on the gradient of a vector field Consider a potential function of the form • U(x, y) = Ax² + Bxy + Cy² + Dx + Ey+F Compute the gradient vector VU (x, y). Answer: U(x, y) = (2Ax+By+D,Bx+2C y +E) ⚫ Pick some values for A, B, C, D, E, F out of a hat (keep it simple!) • Ask yourself: does there exist a point (x, y) at which the gradient vector VU(x, y) is the zero vector? If so, is that point unique? • Repeat as necessary. • What conditions on A, B, C, D, E, F are necessary and sufficient for the existence of a point (x, y) at which VU (x, y) is the zero vector? If that point exists, is it unique?
An exercise on the gradient of a vector field Consider a potential function of the form • U(x, y) = Ax² + Bxy + Cy² + Dx + Ey+F Compute the gradient vector VU (x, y). Answer: U(x, y) = (2Ax+By+D,Bx+2C y +E) ⚫ Pick some values for A, B, C, D, E, F out of a hat (keep it simple!) • Ask yourself: does there exist a point (x, y) at which the gradient vector VU(x, y) is the zero vector? If so, is that point unique? • Repeat as necessary. • What conditions on A, B, C, D, E, F are necessary and sufficient for the existence of a point (x, y) at which VU (x, y) is the zero vector? If that point exists, is it unique?
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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Transcribed Image Text:An exercise on the gradient of a vector field
Consider a potential function of the form
•
U(x, y) = Ax² + Bxy + Cy² + Dx + Ey+F
Compute the gradient vector VU (x, y).
Answer: U(x, y) = (2Ax+By+D,Bx+2C y +E)
⚫ Pick some values for A, B, C, D, E, F out of a hat (keep it simple!)
• Ask yourself: does there exist a point (x, y) at which the gradient
vector VU(x, y) is the zero vector?
If so, is that point unique?
•
Repeat as necessary.
• What conditions on A, B, C, D, E, F are necessary and sufficient for
the existence of a point (x, y) at which VU (x, y) is the zero vector? If
that point exists, is it unique?
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