handout02 - ijk =- ikj . We shall adopt Einsteins summation...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
PHYSICS 218 SOLUTION TO HANDOUT 2 Created: September 12, 2004 0:56am Last updated: September 12, 2004 8:48am Show that ∇× V = 0 when V is the gradient of a scalar function of position, V = ϕ . Solution: Those who have taken a course on vector calculus probably have seen this a few times, but we shall work this out again. We can work this out explicitly using Cartesian coordinates, but we will do it in a more general way using Levi-Civita symbols. Recall Levi-Civita symbols are defined as ± ijk = +1 if ( ijk ) = (123) , (231) , (312) , - 1 if ( ijk ) = (321) , (213) , (132) , 0 otherwise . The Levi-Civita has the anti-symmetric property
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ijk =- ikj . We shall adopt Einsteins summation convention, that is repeated index implies summation. Lets look at the i-th component of ( ), namely, ( ) i = ijk j ( ) k = ijk j k = 1 2 ( ijk + ijk ) j k = 1 2 ( ijk j k + ikj k j ) = 1 2 ( ijk j k - ijk j k ) = 0 . In the last but one step, we had made used of the fact that j and k are dummy variables, and the partial derivatives are interchangable, j k = k j . 1...
View Full Document

Ask a homework question - tutors are online