Student[VectorCalculus][Curl] - compute the curl of a vector field in R^3
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Calling Sequence
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Curl(F)
Curl(c)
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Parameters
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F
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(optional) vector field or Vector-valued procedure; specify the components of the vector field
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c
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(optional) specify the coordinate system
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Description
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The Curl(F) calling sequence computes the curl of the vector field F in R^3. This is equivalent to and CrossProduct(Del, F).
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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.
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If F is a procedure, the result is a procedure. Otherwise, the result is a vector field.
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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:
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* an indexed name, e.g.,
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* a name, e.g., spherical; default coordinate names will be used
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* a list of names, e.g., ; the current coordinate system will be used, with these as the coordinate names
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The Curl() calling sequence returns the differential form of the curl operator in the current coordinate system. For more information, see SetCoordinates.
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Examples
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To display the differential form of the curl operator:
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, 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")](/support/helpjp/helpview.aspx?si=4457/file05145/math131.png)
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, 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")](/support/helpjp/helpview.aspx?si=4457/file05145/math142.png)
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, 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")](/support/helpjp/helpview.aspx?si=4457/file05145/math149.png)
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*`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")](/support/helpjp/helpview.aspx?si=4457/file05145/math156.png)
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*`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")](/support/helpjp/helpview.aspx?si=4457/file05145/math163.png)
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Nabla is a synonym for Del.
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![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](/support/helpjp/helpview.aspx?si=4457/file05145/math219.png)
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