Vector Calculus - II

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

We'll cover the following

Jacobian

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

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

A Jacobian matrix is defined as:

J=[∂f∂x1⋯∂f∂xn]=[∇Tf1⋮∇Tfm]=[∂f1∂x1⋯∂f1∂xn⋮⋱⋮∂fm∂x1⋯∂fm∂xn] 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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