Equation error in calculus.md

[wsehjk]

In section 2.4.3 2.4.3. Partial Derivatives and Gradients, the equation seems to be wrong,

It should be

If so, I would like to create a pr to fix.
Thanks

[rdiaz02]

I find that line confusing or ambiguous, just simply for the very first expression:

Using what I think is a common definition of Jacobian (e.g., Scardapane, 2024, "Alice’s Adventures in a differentiable wonderland", p. 36 of version 2024-10-03, https://www.sscardapane.it/assets/alice/Alice_book_volume_1.pdf) where $J_{ij} = \frac{\partial y_i}{\partial x_j}$ we can do as follows:

If y is a vector valued function, $\frac{\partial \mathbf{y}}{\partial \mathbf{x}}$ is the Jacobian of y. Now write $\mathbf{y} = A\mathbf{x}$. Then ${\partial x} [A\mathbf{x}] = A $, where A is the Jacobian.

But other authors (e.g., Prince, 2023, "Understanding deep learning", p. 448 in version of 2025-03-25: https://github.com/udlbook/udlbook/releases/download/v5.0.1/UnderstandingDeepLearning_03_26_25_C.pdf) use the transpose for the Jacobian: $J_{ij} = \frac{\partial y_j}{\partial x_i}$ . In this case, then, D2L's first expression is correct: $\frac{\partial \mathbf{y}}{\partial \mathbf{x}} = A^{T}$

It would have been nice to clarify how exactly they lay out or stack $\nabla (A\mathbf{x})$. It seems in D2L they are using the same convention as Price, guessing from their eq. 2.4.8.