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## Volume integrals of vector quantities

A tensor divergence analogy of the above trick can easily be found. Using Einstein summation convention, one can write tensors on the LHS whose divergence gives vectors on the RHS, (H.15)

Notice that no dependence on has entered. The relation can be written (H.16)

Remember that Roman indices are spatial, whereas Greek indices label coefficients. Note that the vector of vectors does contain some different functions than from Section H.2.1. The matrix determinant is det , the same as for . The tensor Gauss' theorem is (H.17)

Applying this to the domain integral of (H.16) gives (H.18)

which are the desired closed-form expressions for volume integrals of (the component of) the vector functions . The symbolic inverse is found to be, (H.19)

Again, singular solutions can still be found when . The full consequences of these types of matrix relations have not been explored.   Next: Other matrix elements Up: Matrix trick for pushing Previous: Volume integrals of scalar
Alex Barnett 2001-10-03