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ps3 - Physics 523 Introduction to Relativity Homework 3 Due...

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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 Levi-Civita symbols 1 , and = d + A is the Levi-Civita connection of the metric g on M , given exterior derivative d and matrix-valued one-forms ( A i ) k j = Γ k ij that define this connection (more generally, for any connection, the one-form 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 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 ∂r , ∂θ , ∂φ (though this will lead us to expressions that are not immediately comparable to those found using the orthonormal basis ∂r , 1 r ∂θ , 1 r sin θ ∂φ , for example). Computing explicitly at a point p IR 3
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