Rotation of axes
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The summation convention
The transformation matrix is [Equation goes here - download the original pdf to see it.] This can be summarised using the summation notation, thus [Equation goes here - download the original pdf to see it.] The transpose gives [Equation goes here - download the original pdf to see it.] A summation convention is used to simplify this even further. This is the convention that whenever a suffix appears twice in the same expression, that expression is to be summed over all possible values of the suffix. Thus [Equation goes here - download the original pdf to see it. is summarised by [Equation goes here - download the original pdf to see it.] The transpose becomes [Equation goes here - download the original pdf to see it.] The Kronecker delta is also employed in this context. This is defined by [Equation goes here - download the original pdf to see it.] Using the Kronecker delta the six orthonormal equations are summarised by the single equation [Equation goes here - download the original pdf to see it.] For example, if [Equation goes here - download the original pdf to see it.] this is [Equation goes here - download the original pdf to see it.] That is [Equation goes here - download the original pdf to see it.] In the expression [Equation goes here - download the original pdf to see it.], k is a repeated suffix, also known as a dummy suffix, and may be replaced by any other unused symbol, for example, [Equation goes here - download the original pdf to see it.] Example (8) [Example goes here - download the original pdf to see it.]
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Contents of
Rotation of axes
1 Rotation of Axes - Introduction
2 Right and left-hand coordinates
3 Direction cosines
4 Angles between lines through the origin
5 Rotation of axes
6 The summation convention
7 Invariance with respect to a rotation of the axes
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