Changes which would complicate something for this year but should be performed for next year:
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Explain that vector bundles are useful to extend meaning of vector valued functions, where vector spaces are moving with the point, and mention it is important to understand geometry of injections and topology of spaces.
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Add pictures (video?) on how to draw a vector bundle on a cylinder and on a Moebius strip. Link it with the topology justification above.
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Swap chapter 2.7 and 2.8
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Add inverse fn theorem on submanifolds with boundary in the submanifolds section.
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Add exercise: pullback of partition of unity is a partition of unity.
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Add divergence_\mu X = L_X \mu and use it to show divergence theorem
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Mention Hodge star operator and maybe Laplace-Beltrami in appendix after volume forms
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Add exercise on Liouville theorem for hamiltonians in symplectic geometry
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Add exercise on fixed-points for gradient flows and Hamiltonian flows
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Revise de Rham cohomology, maybe simplify proof to only use $K\omega = \int_0^1 \phi_t^(\iota_T\omega)$, then $dK\omega -K(d\omega)= \int_0^1 \phi_t L_T \omega t = \phi_1\omega - \omega$ and by defining the homotopy with $\tilde K = i_0^ K$ the theorem is proved. It is less explicit but maybe simpler to reason about
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More computational exercises