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}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D++a_i%28x%29%26%3D%5Cboldsymbol%7B%5Comega%7D%28%5Ctext%7BX%7D_i%29%28%5Cchi%5E%7B-1%7D%28x%29%29+%5C%5C++%26%3D+%28a%27+%5Ccirc+%5Cchi%27%29+%5Ctilde%7B%5Ctext%7BX%7D%27%7D%28%5Ctext%7BX%7D_i%29+%28%5Cchi%5E%7B-1%7D%28x%29%29+%5C%5C+%5C%5C++++%5Ctilde%7B%5Ctext%7BX%7D%7D%27%28%5Ctext%7BX%7D_i%29+%26%3D+%5Ctilde%7B%5Ctext%7BX%7D%7D%28%5Ctext%7BX%7D_i%29+%5Cleft%5B+%28D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%5Ccirc+%5Cchi%27+%5Cright%5D%5E%7B-1%7D+%5C%5C+%5C%5C+++++a_i%28x%29++%26%3D+%28a%27+%5Ccirc+%5Cchi%27%29++%5Ctilde%7B%5Ctext%7BX%7D%7D%28%5Ctext%7BX%7D_i%29+%5Cleft%5B+D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%5Ccirc+%5Cchi%27+%5Cright%5D%5E%7B-1%7D++%28%5Cchi%5E%7B-1%7D%28x%29%29+%5C%5C+++%26%3D+%5Csum_j+%28a%27_j+%5Ccirc+%5Cchi%27%29+%5Ctilde%7B%5Ctext%7BX%7D%5Ej%7D%28%5Ctext%7BX%7D_i%29+%5Cleft%5B+D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%5Ccirc+%5Cchi%27+%5Cright%5D%5E%7B-1%7D++%28%5Cchi%5E%7B-1%7D%28x%29%29+%5C%5C+++%26%3D+%28a%27_i+%5Ccirc+%5Cchi%27%29++%5Cleft%5B+D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%5Ccirc+%5Cchi%27+%5Cright%5D%5E%7B-1%7D++%28%5Cchi%5E%7B-1%7D%28x%29%29+%5C%5C+++++%5Cend%7Baligned%7D&bg=ffffff&fg=333333&s=1&c=20201002)
![\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}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D++a_i%28%5Cchi%28%5Cmathbf%7Bm%7D%29%29++%26%3D+%28a%27_i+%5Ccirc+%5Cchi%27%29++%5Cleft%5B+D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%5Ccirc+%5Cchi%27+%5Cright%5D%5E%7B-1%7D++%28%5Cmathbf%7Bm%7D%29+%5C%5C+++++%26%3D+%28a%27_i+%28%5Cchi%27%28%5Cmathbf%7Bm%7D%29%29%29++%5Cleft%5B+D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%28%5Cchi%27%28%5Cmathbf%7Bm%7D%29%29%29+%5Cright%5D%5E%7B-1%7D+++%5C%5C+++++%5Cend%7Baligned%7D&bg=ffffff&fg=333333&s=1&c=20201002)
— Me@2024-10-25 06:31:16 AM
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2024.11.10 Sunday (c) All rights reserved by ACHK
Daily Archives: November 10, 2024
Mounting a virtual disk
# In the host OS, if you want to read the contents of a virtual disk file:
mkdir ~/vdi_test_mount
sudo guestmount -a my_virtual_drive.vdi -m /dev/sda1 -o allow_other ~/vdi_test_mount
# After that, unmount the virtual disk:
sudo guestunmount ~/vdi_test_mount
rmdir ~/vdi_test_mount
— Me@2023-11-25 10:03:37 PM
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2024.11.10 Sunday (c) All rights reserved by ACHK
Physics Town 