Gradient

${\frac {d}{dx}}\sin x=?\,$

This article/section deals with mathematical concepts appropriate for late high school or early college.

In mathematics, the gradient is a vector associated to a point 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} of a 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)} which takes real values. Specifically, the gradient at $p$ is a vector in $R^{n}$ which points in the direction in which $f$ increases most rapidly at $p$. The magnitude of the gradient at $p$ is equal to the maximum directional derivative of $f$ at $p$. The gradient is an extension of the idea of derivative to functions with more than one variable.

Stated another way, a gradient is a vector that has coordinate components that consist of the partial derivatives of a function with respect to each of its variables. For example, if $f(x,y)=x^{2}+y^{2}$, then

$\nabla f(x,y)=\left({\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}}\right)=(2x,2y)$. Observe that in this case, the gradient vector $(2x,2y)$ is orthogonal to the "level curve" defined by $x^{2}+y^{2}=r^{2}$, which here is a circle: the gradient points outward from the origin, which is the direction of steepest increase of $f$, and vectors outward from the origin are perpendicular to circles centered at the origin. We'll see later that this is a case of a more general property of the gradient.

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

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

consisting of the various partial derivatives of $f$. 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 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 = f^{-1}(c)} , and let 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} be a vector tangent 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 S} 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} . Then we can find a curve 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 \gamma(t)} on 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} with 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 \gamma'(0) = v} . Now

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\circ\gamma(t) = c }

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

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\cdot \gamma'(0) = \nabla f\cdot v = 0 }

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} .