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Student[VectorCalculus]

  

Gradient

  

compute the gradient of a function

  

Del

  

Vector differential operator

  

Nabla

  

Vector differential operator

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Gradient(f,c)

Del(f,c)

Nabla(f,c)

Parameters

f

-

(optional) algebraic expression

c

-

(optional) specify the coordinate system

Description

• 

The Gradient(f) calling sequence computes the gradient of the expression f in the current coordinate system.  If no coordinate system has been explicitly specified, the command will assume a cartesian system with coordinates the variables which appear in the expression f.

• 

The Gradient(f,c) calling sequence computes the gradient of the expression f in the coordinate system specified by the parameter c, which can be given as:

  

* an indexed name, e.g., sphericalr,φ,θ

  

* a name, e.g., spherical; default coordinate names will be used

  

* a list of names, e.g., r,φ,θ; the current coordinate system will be used, with these as the coordinate names

  

For more information on coordinate systems, see SetCoordinates.

  

The Gradient(c) calling sequence returns the differential form of the gradient operator in the coordinate system specified by the parameter c.

  

The Gradient() calling sequence returns the differential form of the gradient operator in the current coordinate system.  If no coordinate system has been set (by a call to SetCoordinates), cartesian coordinates are assumed.

• 

In all cases, the result is a vector field.

• 

Nabla and Del are both synonyms for Gradient.

• 

However, you can also use the Del or Nabla commands (as the Vector differential operator) with . (DotProduct) and &x (CrossProduct) as synonyms for the Curl, Divergence, and Laplacian commands.

Command

Equivalent Command using Del

Equivalent Command Using Nabla

Curl

Del &x

Nabla &x

Divergence

Del .

Nabla .

Laplacian

Del . Del

Nabla . Nabla

Examples

withStudent[VectorCalculus]:

The Gradient, Del, and Nabla commands compute the gradient.

Gradientx2+y2

2xe_x+2ye_y

(1)

Delu2+av2,u,v

2ue_u+2ave_v

(2)

Nablax2+y2

2xe_x+2ye_y

(3)

Gradientr2φ,sphericalr,φ,θ

2rφe_r+re_φ

(4)

SetCoordinatessphericalr,φ,θ

sphericalr,φ,θ

(5)

Delr2φ

2rφe_r+re_φ

(6)

To display the differential form of the gradient operator, use the Gradient() or Gradient(c) calling sequence.

Gradient

rSF r,φ,θe_r+φSF r,φ,θre_φ+θSF r,φ,θrsinφe_θ

(7)

Gradientpolarr,θ

rSF r,θe_r+θSF r,θre_θ

(8)

You can compute the divergence, curl, and gradient using the corresponding commands, or the Del or Nabla command and the . or &x operator.

SetCoordinatescartesianx,y,z

cartesianx,y,z

(9)

DivergenceVectorFieldy,x,0

0

(10)

Del.VectorFieldy,x,0

0

(11)

CurlVectorFieldy,x,0

2e_z

(12)

Nabla &x VectorFieldy,x,0

2e_z

(13)

SetCoordinatessphericalr,φ,θ

sphericalr,φ,θ

(14)

Laplacianrsinθ

2rsinφsinθrsinθsinφr2sinφ

(15)

Del.Delrsinθ

2rsinφsinθrsinθsinφr2sinφ

(16)

See Also

attributes

plots[gradplot3d]

plots[gradplot]

Student[MultivariateCalculus][Gradient]

Student[VectorCalculus]

Student[VectorCalculus][CrossProduct]

Student[VectorCalculus][Curl]

Student[VectorCalculus][Divergence]

Student[VectorCalculus][DotProduct]

Student[VectorCalculus][Laplacian]

Student[VectorCalculus][SetCoordinates]

Student[VectorCalculus][Vector]

Student[VectorCalculus][VectorField]

VectorCalculus[Gradient]

 


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