compute the divergence of a vector field

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Student[VectorCalculus][Divergence] - compute the divergence of a vector field

	Calling Sequence
	Divergence(F) Divergence(c)

Parameters

F	-	(optional) vector field or Vector-valued procedure; specify the components of the vector field
c	-	(optional) specify the coordinate system

Description

•	The Divergence(F) calling sequence computes the divergence of the vector field . This calling sequence is equivalent to and DotProduct(Del, F).

•	If is a Vector-valued procedure, the default coordinate system is used. The default coordinate system must be indexed by the coordinate names.

Otherwise, must be a Vector with the vectorfield attribute set, and it must have a coordinate system attribute that is indexed by the coordinate names.

•	If is a procedure, the returned object is a procedure. Otherwise, the returned object is an expression.

•	The Divergence(c) calling sequence returns the differential form of the divergence operator in the coordinate system specified by , which can be given as:

* an indexed name, e.g.,

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

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

•	The Divergence() calling sequence returns the differential form of the divergence operator in the current coordinate system. For more information, see SetCoordinates.

Examples

To create a vector field, use the Student[VectorCalculus][VectorField] command.

(1)

(2)

(3)

(4)

(5)

(6)

To display the differential form of the divergence operator:

(7)

(Diff(r*`VF `[1](r, theta, z), r)+Diff(`VF `[2](r, theta, z), theta)+Diff(r*`VF `[3](r, theta, z), z))/r

(8)

(Diff(s*`VF `[1](s, phi, w), s)+Diff(`VF `[2](s, phi, w), phi)+Diff(s*`VF `[3](s, phi, w), w))/s

(9)

(Diff(r^2*sin(phi)*`VF `[1](r, phi, theta), r)+Diff(r*sin(phi)*`VF `[2](r, phi, theta), phi)+Diff(r*`VF `[3](r, phi, theta), theta))/(r^2*sin(phi))

(10)

(Diff(alpha^2*sin(psi)*`VF `[1](alpha, psi, gamma), alpha)+Diff(alpha*sin(psi)*`VF `[2](alpha, psi, gamma), psi)+Diff(alpha*`VF `[3](alpha, psi, gamma), gamma))/(alpha^2*sin(psi))

(11)

To display the divergence of an arbitrary vector-valued function (r,theta) -> <f(r,theta),g(r,theta)> in the polar coordinate system:

(12)

proc (r, theta) options operator, arrow; (f(r, theta)+r*(diff(f(r, theta), r))+diff(g(r, theta), theta))/r end proc

(13)

	See Also
	Student[VectorCalculus], Student[VectorCalculus][Curl], Student[VectorCalculus][Del], Student[VectorCalculus][DotProduct], Student[VectorCalculus][Laplacian], Student[VectorCalculus][Nabla], Student[VectorCalculus][SetCoordinates], Student[VectorCalculus][Vector], Student[VectorCalculus][VectorField]