Introduction to Electrodynamics · Problem 1.25
Vector Analysis — David J. Griffiths
Physics · vector-calculus-product-rules
Problem
(a) Check product rule (iv) (by calculating each term separately) for the functions (b) Do the same for product rule (ii). (c) Do the same for rule (vi).Intuitive explanation
The key here is understanding that vector calculus product rules—like those for divergence, curl, and gradient of products—have specific forms that mirror familiar calculus rules but with extra terms. You'll verify these by computing the left side and right side of each rule separately using component-by-component differentiation, then confirming they match. This builds confidence that the rules work and shows you how to handle mixed derivatives of vector fields in practice.
Intuitive explanation
The key here is understanding that vector calculus product rules—like those for divergence, curl, and gradient of products—have specific forms that mirror familiar calculus rules but with extra terms. You'll verify these by computing the left side and right side of each rule separately using component-by-component differentiation, then confirming they match. This builds confidence that the rules work and shows you how to handle mixed derivatives of vector fields in practice.