Gradients of the Vector-Valued Functions

Vector-valued functions and their gradients

A vector-valued function in mathematics is a function that takes one or more input variables and produces one or more output variables, where the output variables are components of a vector. For example, the function $f(\theta) = \begin{bmatrix} \cos(\theta) \\ \sin(\theta) \end{bmatrix}$ is a vector-valued function that describes a circle in a 2D plane. The function takes input $\theta \in [0, 2\pi]$ (the angle with $x$-axis) and returns a 2D vector representing the coordinates on a unit circle $x = \cos(\theta)$ and $y = \sin(\theta)$.

So far, we have discussed the partial derivatives and gradients of functions that take one or more input variables and output a scalar value, i.e., $f: \R^m \rightarrow \R$. Now, we will generalize the notion of gradients for vector-valued functions $f: \R^m \rightarrow \R^n$, where $m \geq 1$ and $n > 1$.

Given an $m$-dimensional input vector $x = [x_1, x_2, ..., x_m]^T \in \R^m$, a vector-valued function $f: \R^m \rightarrow \R^n$ can be written as follows: