gradient

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gradient

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gradient

Definition/Summary

The gradient is a vector operator denoted by the symbol

Click to see the LaTeX code for this image

or grad. The gradient of a differentiable scalar function

of a vector

is a vector field whose components are the partial derivatives of

with respect to the variables

Explicitly,

Equations

For a function of three variables in Cartesian Coordinates,

In Cylindrical Polar Coordinates,

Where

and

are unit vectors in the radial and angular directions respectively.

In spherical coordinates,

where

is the angle from the +z-axis to the point

. Also

, etc., denote unit vectors.

NOTE: this definition of

is the one commonly used in math and engineering textbooks. PHYSICS TEXTBOOKS USUALLY HAVE

DEFINED THE OTHER WAY ROUND.

Scientists

Recent forum threads on gradient

Breakdown

Mathematics
> Calculus/Analysis
>> Calculus

See Also

tangent to curve
directional derivative

Images

Extended explanation

The main property of the gradient of

, is that it lies in the domain of the function

, and points in the direction in which

is increasing fastest. In particular the gradient at a point

is perpendicular to the "level set" of

through

, where

is constantly equal to

Commentary

Redbelly98 @ 11:06 PM Jan15-09

When listing the independent variables in an n-dimensional space, isn't it customary for the subscripts on the x's to run from 1 to n, and not 0 to n?

EDITS Jan16-09:

I've gone ahead and made the subscripts 1-to-n in the Definition/Summary section. (I am aware that in computer programming, array subscripts are usually indexed from 0 to n-1.)
Added spherical coordinates form of gradient to the Equations section.
Added cautionary note about φ and θ.
Added line spacing in Equations section.

robphy @ 05:12 PM May12-08

\imath and \jmath look nice with \hat

G01 @ 10:29 PM May10-08

Caught a latex error for you...

Hootenanny @ 11:49 AM May2-08

I agree Ouabache, bold font does look better and less 'cluttered' than underlined text for vectors. Overall, I think it would be best of we could standardise our use of notation across the library. I'll bring it up later.

Ouabache @ 04:11 AM May2-08

Hoot, my suggested editions are in representation of vector notation. My thoughts are that the 'bold' notation is easier to read, especially when you write unit vectors. I saw your suggestion (which I like), to compare notations in the topic where it may be first introduced. In this case it would be in a topic 'vector' which doesn't exist yet.