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    Differential Geometry : Chain Rule and Differentiable Mappings

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    Chain Rule. Let M, N and Q be differentiable manifolds, and let φ : M ?> N and N ?> Q be differentiable mappings. Prove that
    or simply written
    [Comment: To familiarize yourself with notations in Differential Geometry try to check the form that the above equality takes when you express the (differentials of the) maps and in terms of the (Jacobian) matrices obtained by choosing coordinate systems.]

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    Solution Summary

    The chain rule and differentiable mappings are investigated. The solution is detailed and well presented.