Chapter 16.5, Problem 27E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If *f* is a scalar field and **F**, **G** are vector fields, then *f***F**, **F** · **G**, and **F** × **G** are defined by

(*f***F**)(*x*, *y*, *z*) = *f*(*x*, *y*, *z*) **F**(*x*, *y*, *z*)

(**F** · **G**)(*x*, *y*, *z*) = **F**(*x*, *y*, *z*) · **G**(*x*, *y*, *z*)

(**F** × **G**)(*x*, *y*, *z*) = **F**(*x*, *y*, *z*) × **G**(*x*, *y*, *z*)

**27.** div(**F** × **G**) = G · curl **F** − **F** · curl **G**

To determine

**To prove:** The vector field of the form

Explanation

**Formula used:**

Consider the standard equation of a divergence of vector field.

Consider the standard equation of a curl **F**.

Consider

Modify equation (1) to find

Simplify the equation.

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