Chapter 16.5, Problem 24E

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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*)

**24.** curl(**F** + **G**) = curl **F** + curl **G**

To determine

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

Explanation

**Formula used:**

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

Consider

Find

Substitute

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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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