gradient

Vector gradient

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

gradient(f, x)
gradient(f, x, ogCoord, <c>)

Description

gradient(f, x) computes the vector gradient of the scalar function with respect to in Cartesian coordinates. This is the vector .

ogCoord can be the name of a three-dimensional orthogonal coordinate system predefined in the table linalg::ogCoordTab. See Example 2.

Alternatively, ogCoord can be a list of vector of algebraic expressions representing the scale factors of the coordinate system. See example Example 3. For details, see the description of the Scales option on the linalg::ogCoordTab page.

Examples

Example 1

Compute the vector gradient of the scalar function f(x, y) = x2 + y in Cartesian coordinates:

delete x, y:
gradient(x^2 + y, [x, y])

Example 2

Compute the gradient of the function f(r, ϕ, z) = rcos(ϕ) z (0 ≤ ϕ < 2π) in cylindrical coordinates:

delete r, z, phi:
gradient(r*cos(phi)*z, [r, phi, z], Cylindrical)

Example 3

Compute the gradient of the function f(r, ϕ, θ) = rsin(ϕ) cos(θ) in spherical coordinates given by

with 0 ≤ θ ≤ π, 0 ≤ ϕ < 2 π.

The vectors

form an orthogonal system in spherical coordinates.

The scaling factors of the corresponding coordinate transformation (see linalg::ogCoordTab) are: , which we use in the following example to compute the gradient of the function f in spherical coordinates:

delete r, Theta, phi:
gradient(r*sin(phi)*cos(Theta), [r, Theta, phi],
                           [1, r, r*sin(Theta)])

The spherical coordinates are already defined in linalg::ogCoordTab. The last result can also be achieved with the input gradient(r*sin(phi)*cos(Theta), [r, Theta, phi], Spherical[RightHanded]):

gradient(r*sin(phi)*cos(Theta), [r, Theta, phi],
                         Spherical[RightHanded])

Parameters

f

An arithmetical expression in the variables given in x

x

A list of (indexed) identifiers

ogCoord

The name of a 3 dimensional orthogonal coordinate system predefined in the table linalg::ogCoordTab, or a list of algebraic expressions representing the scale factors of an orthogonal coordinate system.

c

The parameter of the coordinate systems EllipticCylindrical and Torus, respectively: an arithmetical expression. The default value is c = 1.

Return Values

Column vector of the domain Dom::Matrix().

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