Vector Calculus - II

The appendix complements the vector calculus lesson from the linear algebra chapter with a bit more details.

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Jacobian

The gradient is used for real-valued functions, $f:R^n \to R$.

The concept of gradients can be extended to vector-valued functions, $f: R^n \to R^m$ by Jacobian matrices.

A Jacobian matrix is defined as:

$J = \begin{bmatrix} \dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n} \end{bmatrix} = \begin{bmatrix} \nabla^\mathsf{T} f_1 \\ \vdots \\ \nabla^\mathsf{T} f_m \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix}$

JAX provides both forward and reverse mode auto-differentiation functions to calculate the Jacobian.

1. jacfwd() - calculates the Jacobian column-by-column
2. jacrev() - calculates the Jacobian row-by-row

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