Gradient
Overview
The gradient is an operator that takes a function of multiple variables
{% f: (x_1,...,x_n) \rightarrow \mathbb{R} %}
and returns a vector.
{% \nabla f(\vec{x}) = \begin{bmatrix}
\frac{\partial{f(x)}}{\partial{x_1}} \\
\frac{\partial{f(x)}}{\partial{x_2}} \\
... \\
\frac{\partial{f(x)}}{\partial{x_n}} \\
\end{bmatrix} %}
Function Approximation
The gradient can be used to approximate a multe-variable function near a given point.
{% \Delta f \approx \nabla f(\vec{x}) \cdot \Delta \vec{x} = \sum_i \frac{\partial{f}}{\partial{x_i}} \Delta x_i %}
This can be seen as an application of
Taylor's Theorem
for multi-variable functions.
