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# Tensor Operators

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# Tensor Operators   FlexPDE supports limited use of TENSOR quantities, to parallel the results of GRAD(vector).

A TENSOR is a vector of vectors, potentially 3 x 3 components.

TENSOR( ( T11, T12, T13) , (T21, T22, T23) , (T31, T32, T33))

This operator returns a TENSOR object with the indicated components.  Each of the Tij may be any scalar expression.

DOT(vector, tensor)

This operator returns a VECTOR with components ( (V1*T11+V2*T21+V3*T31), (V1*T12+V2*T22+V3*T32), (V1*T13+V2*T23+V3*T33) ).

DOT(tensor, vector)

This operator returns a VECTOR with components ( (T11*V1+T12*V2+T13*V3), (T21*V1+T22*V2+T23*V3), (T31*V1+T32*V2+T33*V3) ).

DOT(tensor, tensor)

This operator returns a TENSOR representing the matrix product of the tensors.  The operator ** can be used to produce the same result (i.e. tensor**tensor).

DIV(tensor)

This operator returns a VECTOR value whose components depend on the metric coefficients of the selected problem geometry.  In Cartesian geometry, the result is a VECTOR made up of the divergences of the tensor columns.

TRANSPOSE(tensor)

This operator returns a TENSOR which is the transpose of the argument tensor.

vector * vector

produces a tensor of all combinations of component products.

XXCOMP ( tensor )
XYCOMP ( tensor )
XZCOMP ( tensor )
YXCOMP ( tensor )
YYCOMP ( tensor )
YZCOMP ( tensor )
ZXCOMP ( tensor )
ZYCOMP ( tensor )
ZZCOMP ( tensor )

These operators returns a scalar whose value is the indicated component of the tensor argument ( X indicates the first coordinate component, Y the second and Z the third, regardless of the actual assigned names of the coordinates).