# Definitions > [!NOTE] Definition > Let $f: U \subset \mathbb{R}^{n}\to \mathbb{R}$ be a [[Real-Valued Function on Real n-Space (Multivariable Function)|real valued function of several real variables]]. The *gradient* of $f$, denoted $\nabla f,$ is defined as the [[List|n-tuple]] $\nabla f = \left( \frac{ \partial f }{ \partial x_{1} }, \dots, \frac{ \partial f }{ \partial x_{n} } \right)$where $\frac{ \partial f }{ \partial x_{1} }$ denotes the $i$-th [[Partial Derivatives (of Real-Valued Function on Real n-Space)|partial derivative]] of $f.$ # Applications - [[Directional Derivative of Real-Valued Function of Several Real Variables]]. - [[Linear Approximation of Real-Valued Function on Real n-Space near a Point]]. - [[Normal Vector of Surface]] shows gradient vector is normal to [[Level Sets of Real-Valued Function of Several Real Variables]]. - [[Critical Point of Real-Valued Function on Real 2-Space]]. > [!Example] > Consider $f:\mathbb{R}^{3} \to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ defined by $g(t)=f(\underline{r}(t))$, where $\underline{r}:\mathbb{R}\to \mathbb{R}^{3}$ is parametrised curve in $\mathbb{R}^{3}$. Show that $g'(t) = \nabla f \cdot \underline{r}'(t)$ > >*Solution*. Using [[Chain Rule for Differentiability|chain rule]], $g'(t) = \frac{ \partial f }{ \partial x } \frac{dx}{dt} + \frac{ \partial f }{ \partial y } \frac{dy}{dt} +\frac{ \partial f }{ \partial z } \frac{dz}{dt} = \nabla \cdot \underline{r}'(t) $