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{{math-h}}

{{math-h}}

In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[derivative]] to functions with more than one [[variable]].

In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[Derivative (calculus)|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 <math>f(x,y) = x^2+y^2</math>, then  

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 <math>f(x,y) = x^2+y^2</math>, then  

::<math>\nabla f(x,y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (2x,2y)</math>.  Observe that in this case, the gradient vector <math>(2x,2y)</math> is orthogonal to the "level curve" defined by <math>x^2+y^2=r^2</math>, which here is a circle: the gradient points outward from the origin, which is the direction of steepest increase of <math>f</math>, 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.

::<math>\nabla f(x,y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (2x,2y)</math>.   

Observe that in this case, the gradient vector <math>(2x,2y)</math> is orthogonal to the "level curve" defined by <math>x^2+y^2=r^2</math>, which here is a circle: the gradient points outward from the origin, which is the direction of steepest increase of <math>f</math>, 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, <math>\nabla f</math> of <math>f</math> to be the [[vector field]]:  

More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the [[vector field]]:  

Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.

Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.

[[Category:Calculus]]

[[Category:Calculus]]