Ex 3.2 Verification, 2.2

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Functional Differential Geometry

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Eq. (3.44):

\begin{aligned} a_i(x)&=\boldsymbol{\omega}(\text{X}_i)(\chi^{-1}(x)) \\ &= (a' \circ \chi') \tilde{\text{X}'}(\text{X}_i) (\chi^{-1}(x)) \\ \\ \tilde{\text{X}}'(\text{X}_i) &= \tilde{\text{X}}(\text{X}_i) \left[ (D(\chi \circ (\chi')^{-1}) \circ \chi' \right]^{-1} \\ \\ a_i(x) &= (a' \circ \chi') \tilde{\text{X}}(\text{X}_i) \left[ D(\chi \circ (\chi')^{-1}) \circ \chi' \right]^{-1} (\chi^{-1}(x)) \\ &= \sum_j (a'_j \circ \chi') \tilde{\text{X}^j}(\text{X}_i) \left[ D(\chi \circ (\chi')^{-1}) \circ \chi' \right]^{-1} (\chi^{-1}(x)) \\ &= (a'_i \circ \chi') \left[ D(\chi \circ (\chi')^{-1}) \circ \chi' \right]^{-1} (\chi^{-1}(x)) \\ \end{aligned}

\begin{aligned} a_i(\chi(\mathbf{m})) &= (a'_i \circ \chi') \left[ D(\chi \circ (\chi')^{-1}) \circ \chi' \right]^{-1} (\mathbf{m}) \\ &= (a'_i (\chi'(\mathbf{m}))) \left[ D(\chi \circ (\chi')^{-1}) (\chi'(\mathbf{m}))) \right]^{-1} \\ \end{aligned}

— Me@2024-10-25 06:31:16 AM

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