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Let F be a smooth vector field on a simply connected open subset U CR. Show that F is both irrotational and incompressible if and only if it can be written as F = Vf for a smooth function f: U →R satisfying V²ƒ = 0, where V² is the Laplacian operator. Note that a function f that is a solution of the equation V? f = 0 is called a harmonic function.

Let F be a smooth vector field on a simply connected open subset U CR. Show that F is both irrotational and incompressible if and only if it can be written as F = Vf for a smooth function f: U →R satisfying V²ƒ = 0, where V² is the Laplacian operator. Note that a function f that is a solution of the equation V? f = 0 is called a harmonic function.

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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Let F be a smooth vector field on a simply connected open subset U C Rº. Show that F is both irrotational and incompressible if and only if it can be written as F = Vf for a smooth function f : U → R satisfying V² f = 0, where V2 is the Laplacian operator. Note that a function f that is a solution of the equation V2 f = 0 is called a harmonic function.
Transcribed Image Text:Let F be a smooth vector field on a simply connected open subset U C Rº. Show that F is both irrotational and incompressible if and only if it can be written as F = Vf for a smooth function f : U → R satisfying V² f = 0, where V2 is the Laplacian operator. Note that a function f that is a solution of the equation V2 f = 0 is called a harmonic function.
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