Multivariate Calculus and Geometry by Seán Dineen

Author:Seán Dineen
Language: eng
Format: epub, pdf
Publisher: Springer London, London

Fig. 10.15.

Figure 10.15a and b gives the minimum and maximum values of , and , and there is no restriction on . Hence our parametrization has domain

From Fig. 10.14 we get the spherical polar coordinates in terms of the Cartesian coordinates :

The parametrization of the sphere can also be used to parametrize the inverted vertical cone of height and base radius (Fig. 10.16a).

Note that the angle between the -axis and the curved surface of the cone is . If we take the cross-section of the cone of height , , then we obtain a circle of colatitude on the sphere of radius (Fig. 10.16b). The spherical polar coordinates of this circle are

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