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Suppose that you are in a lab and you study a vector field F on Rº. From theoretical considerations, you know that the vector field is irrotational, and that it takes the form F(x, y, z) = (x + y², 2xy, a(x, y, z)), for some smooth function a : R³ → R. In your lab, you measure that: • The vector field vanishes at the origin; • Its divergence is V .F = 2x. Find the function a(x, y, z).

Suppose that you are in a lab and you study a vector field F on Rº. From theoretical considerations, you know that the vector field is irrotational, and that it takes the form F(x, y, z) = (x + y², 2xy, a(x, y, z)), for some smooth function a : R³ → R. In your lab, you measure that: • The vector field vanishes at the origin; • Its divergence is V .F = 2x. Find the function a(x, y, z).

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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Suppose that you are in a lab and you study a vector field F on R. From theoretical considerations, you know that the vector field is irrotational, and that it takes the form F(x, y, z) = (x + y², 2æy, a(x, y, z)), for some smooth function a : R´ → R. In your lab, you measure that: • The vector field vanishes at the origin; • Its divergence is V · F = 2x. Find the function a(x, y, z).
Transcribed Image Text:Suppose that you are in a lab and you study a vector field F on R. From theoretical considerations, you know that the vector field is irrotational, and that it takes the form F(x, y, z) = (x + y², 2æy, a(x, y, z)), for some smooth function a : R´ → R. In your lab, you measure that: • The vector field vanishes at the origin; • Its divergence is V · F = 2x. Find the function a(x, y, z).
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