Gradient

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This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.

This article deals with the advanced concept of gradient in a vector field

For the two point gradient used in early high school See:Gradient (two point)

In mathematics, the gradient of a real-valued differentiable function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_1,...,x_n)} at a point $p$ is a vector in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^n} which points in the direction in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} increases most rapidly at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} . The magnitude of the gradient at $p$ is equal to the maximum directional derivative of $f$ at $p$.

More precisely, we define the gradient, $\nabla f$ of $f$ to be the vector-field:

$\nabla f=({\frac {\partial f}{\partial x_{1}}},...,{\frac {\partial f}{\partial x_{n}}})$

If $u$ is a unit vector in $R^{n}$, then, by the chain rule, the directional derivative of $f$ in the direction of $u$ is simply the dot product:

$\nabla f\cdot u$

Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction $u$ is maximal in the direction of the gradient, and equal to $||\nabla f||$ for $u$ a unit vector in the direction of the gradient.

Properties of the Gradient

If $f$ is a differentiable function with smooth level sets $f^{-1}(c)$, then the gradient vector field $\nabla f$ is perpendicular to the level sets of $f$. For fix a level set $S=f^{-1}(c)$, and let $v$ be a vector tangent to $S$ at $p$. Then we can find a curve $\gamma (t)$ on $S$ with $\gamma '(0)=v$. Now

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since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a level set. Taking derivatives of both sides and applying the chain rule, we get that

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Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f} is perpendicular to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , i.e., the gradient of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is perpendicular to the level sets of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} .