handout02 - ± ijk =-± ikj We shall adopt Einstein’s...

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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
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Unformatted text preview: ± ijk =-± ikj . We shall adopt Einstein’s summation convention, that is repeated index implies summation. Let’s look at the i-th component of ∇× ( ∇ ϕ ), namely, ± ∇× ( ∇ ϕ ) ² i = ± ijk ∂ j ( ∇ ϕ ) k = ± ijk ∂ j ∂ k ϕ = 1 2 ( ± ijk + ± ijk ) ∂ j ∂ k ϕ = 1 2 ( ± ijk ∂ j ∂ k ϕ + ± ikj ∂ k ∂ j ϕ ) = 1 2 ( ± ijk ∂ j ∂ k ϕ-± ijk ∂ j ∂ k ϕ ) = 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 ϕ = ∂ k ∂ j ϕ . 1...
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This note was uploaded on 09/28/2008 for the course PHYS 218 taught by Professor Pollack during the Fall '04 term at Cornell.

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