Replace Heavier Math section with Geometeric section #147
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docs/src/index.md
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| an ``n``-dimensional vector ``(g_1, ..., g_m)`` of functions on ``M``. | ||
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| We define the _push-forward_ ``g_*:TM -> TN`` between tangent bundles by ``g_*(X)(h) = X(g\circ h)`` for any tangent vector ``X`` and function ``f``. | ||
| We have ``g_*(d/dx_i)(y_j) = dg_j/dx_i, so the push-forward corresponds to the Jacobian, given a chosen basis. |
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missing backticks after math expression, g_*(d/dx_i)(y_j) = dg_j/dx_i
docs/src/index.md
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| We have ``g_*(d/dx_i)(y_j) = dg_j/dx_i, so the push-forward corresponds to the Jacobian, given a chosen basis. | ||
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| Similarly, the pullback of the differential ``df`` is defined by | ||
| ``g^*(df) = d(g\circ f)``. So for a coordinate differential ``dy_j``, we have |
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Shouldn't the order of function application be opposite here. I.e g first maps M->N then, and f:N->R , g^*(df) = d(f\circ g)
docs/src/index.md
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| Let ``df`` denote the first order information of ``f`` at each point. This is called the differential of ``f``. | ||
| If the derivatives of ``f`` and ``g`` agree at ``p``, we say that ``df`` and ``dg`` represent the same cotangent at ``p``. | ||
| The covectors ``dx_1, ..., dx_m`` form the basis of the cotangent space T^*_pM at ``p``. Notice that this vector space is |
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Need to wrap T^*_pM in ticks and curly brackets
docs/src/index.md
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| We have ``g_*(d/dx_i)(y_j) = dg_j/dx_i, so the push-forward corresponds to the Jacobian, given a chosen basis. | ||
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| Similarly, the pullback of the differential ``df`` is defined by | ||
| ``g^*(df) = d(g\circ f)``. So for a coordinate differential ``dy_j``, we have |
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ah, the good old confusion about the order of function composition
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Yeah, its really bad.
a lot of my maths units at uni were taught by group theorists,
from a subsubfield that lits to write all functions in postfix application (x f g meaning g(f(x))),
which makes for extra fun of confusion.
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I'm pretty into it! You can actually read it all left to right! f \circ g (x) reads "take x, apply g then apply f"! 😭
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Is there a |
Yup, leftmost button at the top when commenting on commit changes :). Thanks for the corrections! |
Thanks. Happy to help with obvious typos. I'm a bit confused by the content, but my diff geometry knowledge is limited and I know this is difficult to explain succinctly. |
Co-Authored-By: Anton Isopoussu <[email protected]>
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@aisopous can I merge this? |
Co-Authored-By: Nick Robinson <[email protected]>
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Oh, yeah! It'll benefit from some better writing in the future, though! |
This takes @aisopous's PR from December that I never got round to merging,
and shifts it over here to where the docs are.
JuliaDiff/ChainRules.jl#135
I think @aisopous addressed all comments there, explaining why the points were correct.
It also deletes the heavier math section which has issues.
Since this section covers our need for rigerous math better than that section ever could.
This thus closes #139
A follow up PR to breakup this page in the docs is needed.
Preview of Docs:
http://www.juliadiff.org/ChainRulesCore.jl/preview-PR147/#Geometric-interpretation-of-reverse-and-forwards-mode-AD-1