compute the curl of a vector field in R^3

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Student[VectorCalculus][Curl] - compute the curl of a vector field in R^3

	Calling Sequence
	Curl(F) Curl(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 Curl(F) calling sequence computes the curl of the vector field F in R^3. This is equivalent to and CrossProduct(Del, F).

•	If F is a Vector-valued procedure, the default coordinate system is used, and it must be indexed by the coordinate names. Otherwise, F must be a vector field.

•	If F is a procedure, the result is a procedure. Otherwise, the result is a vector field.

•	The Curl(c) calling sequence returns the differential form of the curl operator in the coordinate system specified by c, 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 Curl() calling sequence returns the differential form of the curl operator in the current coordinate system. For more information, see SetCoordinates.

Examples

(1)

(2)

To display the differential form of the curl operator:

Vector[column]([[Diff(`VF `[3](x, y, z), y)-(Diff(`VF `[2](x, y, z), z))], [Diff(`VF `[1](x, y, z), z)-(Diff(`VF `[3](x, y, z), x))], [Diff(`VF `[2](x, y, z), x)-(Diff(`VF `[1](x, y, z), y))]], ["x", "y", "z"], "field")

(3)

Vector[column]([[(Diff(`VF `[3](r, theta, z), theta)-(Diff(r*`VF `[2](r, theta, z), z)))/r], [Diff(`VF `[1](r, theta, z), z)-(Diff(`VF `[3](r, theta, z), r))], [(Diff(r*`VF `[2](r, theta, z), r)-(Diff(`VF `[1](r, theta, z), theta)))/r]], ["r", "theta", "z"], "field")

(4)

Vector[column]([[(Diff(`VF `[3](s, phi, w), phi)-(Diff(s*`VF `[2](s, phi, w), w)))/s], [Diff(`VF `[1](s, phi, w), w)-(Diff(`VF `[3](s, phi, w), s))], [(Diff(s*`VF `[2](s, phi, w), s)-(Diff(`VF `[1](s, phi, w), phi)))/s]], ["s", "phi", "w"], "field")

(5)

Vector[column]([[(Diff(r*sin(phi)*`VF `[3](r, phi, theta), phi)-(Diff(r*`VF `[2](r, phi, theta), theta)))/(r^2*sin(phi))], [(Diff(`VF `[1](r, phi, theta), theta)-(Diff(r*sin(phi)*`VF `[3](r, phi, theta), r)))/(r*sin(phi))], [(Diff(r*`VF `[2](r, phi, theta), r)-(Diff(`VF `[1](r, phi, theta), phi)))/r]], ["r", "phi", "theta"], "field")

(6)

Vector[column]([[(Diff(alpha*sin(psi)*`VF `[3](alpha, psi, gamma), psi)-(Diff(alpha*`VF `[2](alpha, psi, gamma), gamma)))/(alpha^2*sin(psi))], [(Diff(`VF `[1](alpha, psi, gamma), gamma)-(Diff(alpha*sin(psi)*`VF `[3](alpha, psi, gamma), alpha)))/(alpha*sin(psi))], [(Diff(alpha*`VF `[2](alpha, psi, gamma), alpha)-(Diff(`VF `[1](alpha, psi, gamma), psi)))/alpha]], ["alpha", "psi", "gamma"], "field")

(7)

Nabla is a synonym for Del.

(8)

(9)

(10)

(11)

(12)

(13)

proc (r, theta, z) options operator, arrow; VectorCalculus:-Vector([(diff(h(r, theta, z), theta)-r*(diff(g(r, theta, z), z)))/r, diff(f(r, theta, z), z)-(diff(h(r, theta, z), r)), (g(r, theta, z)+r*(diff(g(r, theta, z), r))-(diff(f(r, theta, z), theta)))/r], attributes = [vectorfield, coords = cylindrical[r, theta, z]]) end proc

(14)

	See Also
	Student[VectorCalculus], Student[VectorCalculus][CrossProduct], Student[VectorCalculus][Del], Student[VectorCalculus][Divergence], Student[VectorCalculus][Laplacian], Student[VectorCalculus][Nabla], Student[VectorCalculus][SetCoordinates], Student[VectorCalculus][Vector], Student[VectorCalculus][VectorField]