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DifferentialGeometry:-Tools[&MatrixMinus, &MatrixMult, &MatrixPlus, &MatrixWedge]
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Calling Sequence
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A &MatrixMinus B - subtract two Matrices/Vectors of vectors, differential forms or tensors
A &MatrixMult C - multiply a Matrix/Vector A of vectors, differential forms or tensors by a scalar C or a Matrix/Vector C of scalars
C &MatrixMult A - multiply a Matrix A of vectors, differential forms or tensors by a scalar C or a Matrix/Vector C of scalars
A &MatrixPlus B - add two Matrices/Vectors of vectors, differential forms or tensors
E &MatrixWedge F - calculate the Matrix wedge product of two Matrices/Vectors of differential forms.
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Parameters
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A, B
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two Matrices/Vectors of vectors, differential forms or tensors
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C
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a scalar or a Matrix/Vector of scalars
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E, F
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two Matrices/Vectors of differential forms
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Description
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These commands provide, within the DifferentialGeometry environment, the basic arithmetical operations for Matrices or Vectors of: vectors, differential forms, or tensors. They are particularly useful for curvature calculations for connections on principle bundles of matrix groups.
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These commands are part of the DifferentialGeometry:-Tools package, and so can be used in the form described above only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.
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Examples
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Define a 3-dimensional manifold M with coordinates [x, y, z].
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![DGsetup([x, y, z], M)](/support/helpjp/helpview.aspx?si=6660/file05946/math112.png)
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Example 1
Define two column Vectors of 1 forms A, B; a 2x2 matrix C of scalars; a row Vector of 1 forms E and a 2x2 Matrix of 1 forms F.
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![A := Vector(evalDG([dx-dy, dy+dx]))](/support/helpjp/helpview.aspx?si=6660/file05946/math133.png)
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](/support/helpjp/helpview.aspx?si=6660/file05946/math136.png)
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![B := Vector(evalDG([dx+2*dy, dx+3*dy]))](/support/helpjp/helpview.aspx?si=6660/file05946/math140.png)
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](/support/helpjp/helpview.aspx?si=6660/file05946/math143.png)
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![C := Matrix([[1, 2], [3, 4]])](/support/helpjp/helpview.aspx?si=6660/file05946/math147.png)
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](/support/helpjp/helpview.aspx?si=6660/file05946/math157.png)
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![F := Matrix(evalDG([[dx-dz, dy], [dz, dx+dy+3*dz]]))](/support/helpjp/helpview.aspx?si=6660/file05946/math161.png)
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Perform various arithmetic operations with the quantities A, B, C, E, F.
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](/support/helpjp/helpview.aspx?si=6660/file05946/math183.png)
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](/support/helpjp/helpview.aspx?si=6660/file05946/math190.png)
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](/support/helpjp/helpview.aspx?si=6660/file05946/math197.png)
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](/support/helpjp/helpview.aspx?si=6660/file05946/math204.png)
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](/support/helpjp/helpview.aspx?si=6660/file05946/math218.png)
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](/support/helpjp/helpview.aspx?si=6660/file05946/math232.png)
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| (14) |
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M >
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