• Introduction
• Proof of Jacobi's Identity
• [1] S. Grossman, Calculus, 4th edition, Harcourt Brace Jovanovich, 1988.

Proof of Jacobi's Identity

by Laurie L. Lacey, Ph.D., Schenectady County Community College, Schenectady NY, USA, laceyll@gw.sunysccc.edu, 2002 Laurie L. Lacey

This worksheet demonstrates how certain algebra-heavy proofs may be done using Maple.

Introduction

Calculus III is one of the first courses where students are introduced to proofs.   Often the algebraic proofs concerning vectors can be stressful for the students.  Moreover, students tend to substitute numbers in for the vector components and "prove" the theorem by example rather than in generality.

Such a proof can be the proof of Jacobi's Identiy:

Jacobi's Identity [1, p. 588].  Given vectors u , v , and w  in R3, u  x ( v  x w ) + v  x ( w  x u ) + w  x ( u  x v ) = 0 .

>

restart;

Proof of Jacobi's Identity

>	with( VectorCalculus );

Warning, the assigned names <,> and <|> now have a global binding

Warning, these protected names have been redefined and unprotected: *, +, ., Vector, diff, int, limit, series

>	u := Vector([u1, u2, u3]); v := Vector([v1, v2, v3]); w := Vector([w1, w2, w3]);

Maple was instructed to find the cross product of v  and w  and cross the result with u .  

Note that in the VectorCalculus  package, &x  denotes the cross product operator.

>	a :=  v &x w;

>	b := u &x a;

Likewise, Maple was asked to obtain the cross product of w  and u  and cross that result with v .

>	c := w &x u;

>	d := v &x c;

Next, Maple was also asked to determine the cross product of u  and v  and cross that result with w.

>	e := u &x v;

>	f := w &x e;

Finally, Maple was asked to add the cross products b , d , and f determined above.

>

b+d+f;

>	simplify(%);

>

Most students are able to follow the proof and even check the algebra once they believe the proof "works."  

The reader is challenged to construct other similar proofs using Maple.

[1] S. Grossman, Calculus, 4th edition, Harcourt Brace Jovanovich, 1988.  

Disclaimer:  While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.