Consider the vector field F = xe³i + (arctan z — e³)j + (9z - sin x cos y)k. (a) Determine whether F is incompressible, i.e. whether its divergence is zero, and whether it is conservative. If conservative, find a scalar potential. (b) Compute fFdS, S is the piecewise smooth surface that is the union of the hemisphere x² + y² + ² = 1 for z ≥ 0 and the disk x² + y² ≤ 1 in the xy-plane oriented outward.
Consider the vector field F = xe³i + (arctan z — e³)j + (9z - sin x cos y)k. (a) Determine whether F is incompressible, i.e. whether its divergence is zero, and whether it is conservative. If conservative, find a scalar potential. (b) Compute fFdS, S is the piecewise smooth surface that is the union of the hemisphere x² + y² + ² = 1 for z ≥ 0 and the disk x² + y² ≤ 1 in the xy-plane oriented outward.
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:Consider the vector field F = xei + (arctan z -e³)j + (9z - sin x cos y)k.
(a) Determine whether F is incompressible, i.e. whether its divergence is zero, and
whether it is conservative. If conservative, find a scalar potential.
(b) Compute fFdS, S is the piecewise smooth surface that is the union of the
hemisphere x² + y² + z² = 1 for z ≥ 0 and the disk x² + y² ≤ 1 in the xy-plane
oriented outward.
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