2021-04-27t18-05-20z.md

date	title	id
2021-04-27 11:05:20 -0700	The Gradient in Vector Calculus	2021-04-27t18-05-20z

The gradient of a scalar-valued differentiable function $f$ is the vector field $\nabla f$ whose value at point $\mathbf{p}$ is a vector whose components are each of $f$'s partial derivatives at $p$.

$$ \nabla f(p)={ \begin{bmatrix} {\frac{\partial f}{\partial x_{1}}}(p) \\ \vdots \\ {\frac{\partial f}{\partial x_{n}}}(p) \end{bmatrix} } $$

The gradient of $f$ at $\mathbf{p}$ points in the direction of steepest ascent, i.e. the direction in the vector field where $f$ is closest to its maximum value.