JetCalculus[HorizontalExteriorDerivative] - calculate the horizontal exterior derivative of a bi-form on a jet space
Calling Sequences
HorizontalExteriorDerivative(omega)
Parameters
omega - a differential bi-form on the jet space of a fiber bundle E -> M
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Description
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Every differential form on a jet space can be expressed as a sum of wedge products of 1-forms on M and contact 1-forms. A differential form omega is called a bi-form of degree (r, s) if it is a sum of wedge products of r 1-forms on M and s contact 1-forms. Alternatively, omega is of type (r, s) if omega(X_1, X_2, ... X_(r + s)) = 0 whenever more than r of the vector fields X_i are total vector fields or more than s of the vector fields X_i are vertical vector fields on J^k(E) -> M. The non-negative integer r is called the horizontal degree of omega. The non-negative integer s is called the vertical degree of omega.
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If omega is called a bi-form of degree (r, s), then its exterior derivative d(omega) is a sum of two bi-forms, one of type (r + 1, s), the other of type (r, s + 1). The type (r + 1, s) part is called the horizontal exterior derivative of omega and is denoted by dH(omega). The type (r, s + 1) part is called the vertical exterior derivative of omega and is denoted by dV(omega). Thus d(omega) = dH(omega) + dV(omega) from which it follows that dH^2 = 0, dH dV = - dV dH and dH^2 = 0.
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If (x^i) are local coordinates on M and D_i denotes total differentiation with respect to x^i, then dH(omega) = D_i(omega) Dx^i.
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The command HorizontalExteriorDerivative is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form HorizontalExteriorDerivative(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-HorizontalExteriorDerivative(...).
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Examples
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Example 1.
Create the jet space J^2(E) for the bundle E = R^2 x R^2 with coordinates (x, y, u, v) -> (x, y).
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Calculate the horizontal exterior derivative of a function.
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Calculate the horizontal exterior derivative of a type (1, 0) bi-form.
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Calculate the horizontal exterior derivative of a type (0, 2) bi-form.
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