Summation conventions:

A repeated (dummy) index i denotes a summation with respect to that index.

Kronecker delta and permutation symbol

Vector Transformations:

A vector u can be defined in any coordinate system. The vector remains the same regardless of the coordinate system. Matrices can be used to transform a vector from one coordinate system to another coordinate system. A coordinate system in three dimensions with axes x,y,z can be rotated to a new coordinate system with axes x',y',z'. A vector can be expressed in terms of its' components in the two orthogonal coordinate systems with unit base vectors ê:

The direction cosines form a transformation matrix A. For a counter clockwise rotation around the Z axis the transformation matrix A is:

The process can also be reversed. The transformation matrix below is just the transpose of the one above.

A very nice tool for matrix manipulation is MATLAB© by Mathworks©.

Some useful references:

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