What is a gradient

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Definition/Summary

The gradient is a vector operator denoted by the symbol [itex]\mathbf\nabla[/itex] or grad. The gradient of a differentiable scalar function [itex]f\left({\mathbf x}\right)[/itex] of a vector [itex]\mathbf{x}=\left(x_1,x_2,\ldots,x_n\right)[/itex] is a vector field whose components are the partial derivatives of [itex]f\left({\mathbf x}\right)[/itex] with respect to the variables [itex]x_1,x_2,\ldots,x_n\,.[/itex] Explicitly,

[tex]\mathbf\nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n}\right)[/tex]

Equations

For a function of three variables in Cartesian Coordinates,

[tex]\nabla f\left(x,y,z\right)
= \frac{\partial f}{\partial x}\hat{\mathbf{i}}
+ \frac{\partial f}{\partial y}\hat{\mathbf{j}}
+ \frac{\partial f}{\partial z}\hat{\mathbf{k}}[/tex]


In Cylindrical Polar Coordinates,

[tex]\nabla f\left(r,\theta,z\right)
= \frac{\partial f}{\partial r}\hat{\mathbf{e_r}}
+ \frac{1}{r}\frac{\partial f}{\partial \theta}\hat{\mathbf{e_\theta}}
+ \frac{\partial f}{\partial z}\hat{\mathbf{k}}[/tex]

Where [itex]\hat{\mathbf{e_r}}[/itex] and [itex]\hat{\mathbf{e_\theta}}[/itex] are unit vectors in the radial and angular directions respectively.


In spherical coordinates,

[tex]\nabla f\left(r,\phi,\theta\right)
= \frac{\partial f}{\partial r}\hat{\mathbf{e_r}}
+ \frac{1}{r} \ \frac{\partial f}{\partial \phi} \hat{\mathbf{e_\phi}}
+ \frac{1}{r \ \sin \phi} \ \frac{\partial f}{\partial \theta} \hat{\mathbf{e_\theta}} [/tex]

where [itex]\phi[/itex] is the angle from the +z-axis to the point [itex](r, \phi, \theta ) [/itex]. Also [itex]\hat{\mathbf{e_r}}[/itex], etc., denote unit vectors.

NOTE: this definition of [itex]\phi, \theta[/itex] is the one commonly used in math and engineering textbooks. PHYSICS TEXTBOOKS USUALLY HAVE [itex]\phi, \theta[/itex] DEFINED THE OTHER WAY ROUND.

Extended explanation

The main property of the gradient of [itex]f[/itex], is that it lies in the domain of the function [itex]f[/itex], and points in the direction in which [itex]f[/itex] is increasing fastest. In particular the gradient at a point [itex]\mathbf{p}[/itex] is perpendicular to the "level set" of [itex]f[/itex] through [itex]\mathbf{p}[/itex], where [itex]f[/itex] is constantly equal to [itex]f(\mathbf{p})[/itex].

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
 

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