Rotation of axes
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Invariance with respect to a rotation of the axes
Let [Equation goes here - download the original pdf to see it.] and [Equation goes here - download the original pdf to see it.] be two sets of rectangular Cartesian axes with the same origin such that [Equation goes here - download the original pdf to see it.] is obtained from [Equation goes here - download the original pdf to see it.] by a rotation of the axes. Let [Equation goes here - download the original pdf to see it.] be a function and let [Equation goes here - download the original pdf to see it.] be the value of the same function where [Equation goes here - download the original pdf to see it.] are the elements corresponding to x, y, z under the rotation of axes from [Equation goes here - download the original pdf to see it.] to [Equation goes here - download the original pdf to see it.] . Then, if [Equation goes here - download the original pdf to see it.] then f is said to be invariant with respect to a rotation of axes. Example (9) Show that the distance of a point from the origin is invariant [Example goes here - download the original pdf to see it.] The above following from the orthonormality relations. So d is invariant. Example (10) Show that the cosine of the angle between two lines through the origin is invariant with respect to a rotation of the axes. [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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