Difference between revisions of "Gradient (mathematics)"
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Similarly to the usual derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. | Similarly to the usual derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. | ||
| − | The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. This characterizing property of the gradient allows it to be defined independently of a choice of coordinate system, as a vector field whose components in a coordinate system will transform when going from one coordinate system to another. | + | The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. |
| + | |||
| + | This characterizing property of the gradient allows it to be defined independently of a choice of [[coordinate system]], as a vector field whose components in a coordinate system will transform when going from one coordinate system to another. | ||
The Jacobian is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative. | The Jacobian is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative. | ||
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== See also == | == See also == | ||
| + | * [[Coordinate system]] | ||
| + | * [[Curl (mathematics)]] | ||
| + | * [[Del]] | ||
| + | * [[Divergence]] | ||
| + | * [[Gradient theorem]] | ||
| + | * [[Graph of a function]] | ||
| + | * [[Hessian matrix]] | ||
* [[Mathematics]] | * [[Mathematics]] | ||
| + | * [[Skew gradient]] | ||
== External links == | == External links == | ||
Latest revision as of 12:51, 16 May 2016
In mathematics, the gradient is a generalization of the usual concept of derivative of a function in one dimension to a function in several dimensions.
Description
If f(x1, ..., xn) is a differentiable, scalar-valued function of standard Cartesian coordinates in Euclidean space, its gradient is the vector whose components are the n partial derivatives of f. It is thus a vector-valued function.
Similarly to the usual derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction.
The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph.
This characterizing property of the gradient allows it to be defined independently of a choice of coordinate system, as a vector field whose components in a coordinate system will transform when going from one coordinate system to another.
The Jacobian is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative.
See also
- Coordinate system
- Curl (mathematics)
- Del
- Divergence
- Gradient theorem
- Graph of a function
- Hessian matrix
- Mathematics
- Skew gradient
External links
- Gradient @ Wikipedia
- Grad, Div and Curl @ YouTube
