Unformatted text preview: ± ijk =± ikj . We shall adopt Einstein’s summation convention, that is repeated index implies summation. Let’s look at the ith 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.
 Fall '04
 POLLACK
 Physics, Magnetism

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