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Tensor[Laplacian] - find the Laplacian of a differential form with respect to a metric.
Calling Sequences
Laplacian(g, omega)
Parameters
g - a covariant metric tensor on an n-dimensional manifold M
omega - a differential form on M
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Description
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The Laplace-Beltrami operator Delta is the second order linear differential operator which acts on p-forms omega by Delta(omega) = (d o delta + delta o d) (omega).
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The delta differential operator is the first-order linear differential operator defined in terms of the exterior derivative operator d and the Hodge star operator * by
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delta(omega) = (-1)^(k)(* d *)(omega),
where omega is a p-form and k = n*p +n +1. The form delta(omega) has degree p-1. Since a metric tensor is needed to define the Hodge star operator * , a metric is also needed to define delta.
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The Laplacian(g, omega) computes the Laplacian Delta(omega) of the differential form with respect to the metric tensor g.
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The command Laplacian:-ExteriorDerivativeStar(g, omega) computes delta(omega).
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This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form Laplacian(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-Laplacian.
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Examples
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Define a manifold M with coordinates [x, y, z] and a metric g on M.
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![DGsetup([x, y, z], M)](/support/helpjp/helpview.aspx?si=5601/file05888/math105.png)
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M >
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| (2.1) |
Example 1.
Define a differential 1-form alpha and suppress the printing of the arguments of its coefficients with the PDEtools[declare] command.
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M >
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![alpha := DGzip(([a, b, c])(x, y, z), [dx, dy, dz], "plus")](/support/helpjp/helpview.aspx?si=5601/file05888/math127.png)
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| (2.2) |
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M >
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(x, y, z))](/support/helpjp/helpview.aspx?si=5601/file05888/math134.png)
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Compute the Laplacian of alpha.
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M >
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![(-a/(x^3*z)+(3/2)*(a[x]/(x^2*z))-a[x, x]/(x*z)-c[x]/z-a[y, y]/z^2-(1/2)*(a[z]/z)-a[z, z])*dx+(-b[y, y]/z^2-2*c[y]/z+(1/2)*(b[x]/(x^2*z))-b[x, x]/(x*z)+(1/2)*(b[z]/z)-b[z, z])*dy+(-a/(2*x^2*z^2)+a[x]/(x*z^2)+2*b[y]/z^3+3*c/(2*z^2)-(3/2)*(c[z]/z)-c[z, z]+(1/2)*(c[x]/(x^2*z))-c[x, x]/(x*z)-c[y, y]/z^2)*dz](/support/helpjp/helpview.aspx?si=5601/file05888/math154.png)
| (2.4) |
Example 2.
Define a 2 form beta and compute its Laplacian.
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![beta := DGzip(([a, b, c])(x, y, z), [`&wedge`(dx, dy), `&wedge`(dx, dz), `&wedge`(dy, dz)], "plus")](/support/helpjp/helpview.aspx?si=5601/file05888/math166.png)
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| (2.5) |
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M >
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![(-a/(x^3*z)+(3/2)*(a[x]/(x^2*z))-a[x, x]/(x*z)+c[x]/z-a[y, y]/z^2-2*b[y]/z+(3/2)*(a[z]/z)-a[z, z])*dx*`^`*dy+(-b/(z*x^3)+(3/2)*(b[x]/(x^2*z))-b[x, x]/(x*z)+2*a[y]/z^3+b/(2*z^2)-(1/2)*(b[z]/z)-b[z, z]-b[y, y]/z^2)*dx*`^`*dz+(-c[y, y]/z^2+a/(2*x^2*z^2)-a[x]/(x*z^2)-c/(2*z^2)+(1/2)*(c[z]/z)-c[z, z]+(1/2)*(c[x]/(x^2*z))-c[x, x]/(x*z))*dy*`^`*dz](/support/helpjp/helpview.aspx?si=5601/file05888/math176.png)
| (2.6) |
Example 3.
Compute the delta derivative of the 2 form beta.
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![(1/2)*((2*a[y]+z*b+2*z^2*b[z])*dx/z^2)+(1/2)*((a-2*a[x]*x-x^2*c+2*x^2*z*c[z])*dy/(z*x^2))-(1/2)*((-z*b+2*z*b[x]*x+2*x^2*c[y])*dz/(x^2*z^2))](/support/helpjp/helpview.aspx?si=5601/file05888/math191.png)
| (2.7) |
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