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Unformatted text preview: Physics 523, Introduction to Relativity Homework 3 Due Tuesday, 18 th October 2011 Hans Bantilan Gradient, Divergence, and Curl Let ∂ ∂x i be the coordinate basis of T p M associated with coordinate chart x : U → Ω, where U ⊆ M is a neighborhood containing p ∈ M , and Ω ⊂ IR d is open. For a scalar field f : M → IR , we may define its gradient as grad f := g ij ∂f ∂x i ∂ ∂x j . The divergence of a vector field Z = Z i ∂ ∂x i is defined as div Z := 1 √ g ∂ ∂x i ( √ gZ i ) and its curl is defined as curl Z := 1 √ g ijk ∇ j Z k ∂ ∂x i where ijk are the LeviCivita symbols 1 , and ∇ = d + A is the LeviCivita connection of the metric g on M , given exterior derivative d and matrixvalued oneforms ( A i ) k j = Γ k ij that define this connection (more generally, for any connection, the oneform A that defines it is sometimes called the gauge field). We are to consider these objects evaluated on IR 3 in spherical x i = { r, θ, φ } , with the flat metric then given by g = dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 . To emphasize the simplicity of this calculation, we display it in its most elementary form by using the coordinate basis n ∂ ∂r , ∂ ∂θ , ∂ ∂φ o (though this will lead us to expressions that are not immediately comparable to those found using the orthonormal basis n ∂ ∂r , 1 r ∂ ∂θ , 1 r sin θ ∂ ∂φ o , for example). Computing, for example)....
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This note was uploaded on 02/02/2012 for the course PHY 523 at Princeton.
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