By Kwong-Tin Tang

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**Extra info for Mathematical Methods - For Students of Physics and Related Fields2009**

**Example text**

To ﬁnd the third coordinate, we construct the plane through O and perpendicular to the polar axis, drop a projection from P to the plane meeting the latter at H, draw an arbitrary ﬁducial line through O in this plane, and measure the angle between this line and OH. This angle is ϕ(P ). Cartesian and cylindrical coordinate systems can be described similarly. 8. As indicated in the ﬁgure, the polar axis is usually taken to be the z-axis, and the ﬁducial line from which ϕ(P ) is measured is chosen to be the x-axis.

This meant inventing ways of adding, subtracting, multiplying, and dividing objects such as (x, y, z). The invention of a spatial analogue of the planar complex numbers is due to William R. Hamilton. Next to Newton, Hamilton is the greatest of all English mathematicians, and like Newton he was even greater as a physicist than as a mathematician. At the age of ﬁve Hamilton could read Latin, Greek, and Hebrew. At eight he added Italian and French; at ten he could read Arabic and Sanskrit, and at fourteen, Persian.

This is because polar coordinates are not deﬁned in terms of any ﬁxed axes. 11. 13) where the coordinates are subscripted to emphasize their dependence on the points at which the unit vectors are erected. In the case of Cartesian coordinates, this, of course, is not necessary because the unit vectors happen to be independent of the point. 11: (a) The vector a has the same components along unit vectors at P and Q in Cartesian coordinates. (b) The vector a has diﬀerent components along unit vectors at diﬀerent points for a polar coordinate system.