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Show that any vector field of the form F(x, y, z) = f(x)i + g(y)j + h(z)k where f, g, h are differentiable functions, is irrotational. By the definition of irrotational, a vector field F is irrotational if and only if curl(F) curl(F) i a j Ә k Ә ax ay az f(x) g(y) h(z)| = 0. Let F(x, y, z) = f(x)i + g(y)j + h(z)k, where f, g, and h are differentiable functions. Then

Show that any vector field of the form F(x, y, z) = f(x)i + g(y)j + h(z)k where f, g, h are differentiable functions, is irrotational. By the definition of irrotational, a vector field F is irrotational if and only if curl(F) curl(F) i a j Ә k Ә ax ay az f(x) g(y) h(z)| = 0. Let F(x, y, z) = f(x)i + g(y)j + h(z)k, where f, g, and h are differentiable functions. Then

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Show that any vector field of the form F(x, y, z) = f(x)i + g(y)j + h(z) k where f, g, h are differentiable functions, is irrotational. By the definition of irrotational, a vector field F is irrotational if and only if curl(F) + curl(F) ♦ curl(F) = → = = i j k Ә Ə Thus, F is irrotational. € 80 ду дz f(x) g(y) h(z) Ə ду i - 0 € h(z) = ·)- Ә Əx + Ә Əx h(z) = Since f(x), g(y), and h(z) are each a function of one variable, x, y, and z, respectively, we get j k Ə Ə g(y) = Əx ду дz 8 f(x) g(y) h(z) Ә əz )- D)₁-( g(y) = Ә дz ə ду f(x) = f(x) = ])i + ( k = 0. = 0. Let F(x, y, z) = f(x)i + g(y)j + h(z)k, where f, g, and h are differentiable functions. Then
Transcribed Image Text:Show that any vector field of the form F(x, y, z) = f(x)i + g(y)j + h(z) k where f, g, h are differentiable functions, is irrotational. By the definition of irrotational, a vector field F is irrotational if and only if curl(F) + curl(F) ♦ curl(F) = → = = i j k Ә Ə Thus, F is irrotational. € 80 ду дz f(x) g(y) h(z) Ə ду i - 0 € h(z) = ·)- Ә Əx + Ә Əx h(z) = Since f(x), g(y), and h(z) are each a function of one variable, x, y, and z, respectively, we get j k Ə Ə g(y) = Əx ду дz 8 f(x) g(y) h(z) Ә əz )- D)₁-( g(y) = Ә дz ə ду f(x) = f(x) = ])i + ( k = 0. = 0. Let F(x, y, z) = f(x)i + g(y)j + h(z)k, where f, g, and h are differentiable functions. Then
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