   Chapter 16.5, Problem 21E

Chapter
Section
Textbook Problem

Show that any vector field of the formF(x, y, z) = f(x) i + g(y) j + h(z) kwhere f, g, h are differentiable functions, is irrotational.

To determine

To show: The vector field of the form F(x,y,z)=f(x)i+g(y)j+h(z)k is irrotational.

Explanation

Given data:

F(x,y,z)=f(x)i+g(y)j+h(z)k

Formula used:

Consider the standard equation of an curl F for F=Pi+Qj+Rk .

curlF=|ijkxyzPQR| (1)

Substitute f(x) for P , g(y) for Q and h(z) for R in equation (1),

curlF=|ijkxyzf(x)g(y)h(z)|={[y[h(z)]z[g(y)]]i[z[f(x)]x[h(z)]]j+[x[g(y)]y[f(x)]]k}={[[h(z)]

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