Gradient

Gradient
Tag	`0ABK9V9Y64`
URI	`m2://0ABK9V9Y64`
Aliases	`gradient-vector`, `grad`
Kind	concept
Writing	W1 stated
MSC	26B12
Uses	Partial derivative, Differentiable function of several variables
Same as	Wikidata Q173582 Wikipedia Gradient

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Summary

Let $f$ be a differentiable real-valued function on an open subset of $\mathbb{R}^n$ . Its gradient at a point $a$ is the vector $\nabla f(a)$ of its partial derivatives at $a$ . It points in the direction in which $f$ increases fastest, and its length is that fastest rate of increase. The derivative of $f$ at $a$ is the dot product with the gradient: $Df(a)(v) = \nabla f(a) \cdot v$ .

What this node needs for W2

edge equivalent in: a statement for each pair of definitions that a source proves equivalent (0 of 1)
edge instance_of in: at least two examples (0 of 2)
edge not_instance_of in: at least one non-example (0 of 1)
edge specializes both: the more general notions and the special cases (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What depends on this

28 nodes use this node, directly or through other nodes.

Exterior derivative 004FRCXH6M
Mean value theorem in several variables 03HGK7EW6R
Lagrange multiplier with one constraint 04A2XEX4D4
Poincaré lemma for one-forms on convex sets 05MC3HND79
Area of a region as a line integral 09CBQB6V67
Chain rule for a composition with a curve 0A8TJ0VHC7
Stokes' theorem 0AKRKVHW94
Path independence of line integrals 0BFN21TF5F
Harmonic function 0DEB0JMN0W
Laplacian 0E0B6RA8K0
Closed and exact forms 0F1AVGX26N
Curl-free field that is not conservative 0GRM3BS0DD
Divergence theorem on a rectangle in the plane 0GZ3D8F921
Del operator 0H6DP8CHND
Scalar potential 0H7C3T1GJY
Generalised Stokes theorem 0HPHKCGEQ0
Green's theorem 0HR3N02317
Vector calculus identities 0JC5X4HWR2
Saddle point 0JW6MDHAZD
Divergence theorem 0JX86PSSA9
One-form 0MTENFBW3Q
Conservative vector field 0PKG97PX9Y
Differential form 0QHTX7WQEY
Wedge product of forms 0S166D363M
Gradient theorem 0SD237FEJD
Gradient is orthogonal to level sets 0VXYV3NQQ1
The exterior derivative squares to zero 0X308R2DBE
Homotopy invariance of line integrals of closed forms 0YVVN6R715

Sources

Web https://en.wikipedia.org/wiki/Gradient, lead section

What links here

Used by

Chain rule for a composition with a curve 0A8TJ0VHC7
Laplacian 0E0B6RA8K0
Del operator 0H6DP8CHND
Scalar potential 0H7C3T1GJY
Saddle point 0JW6MDHAZD
Conservative vector field 0PKG97PX9Y

Definitions

Having a gradient 0BXE0S1XER
Gradient 0NS2QYVC4Y