By F. H. C. Marriott (Auth.)
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Extra info for Basic Mathematics for the Biological and Social Sciences
Three-dimensional polar coordinates There are two ways of extending the idea of polar coordinates to three dimensions. In the first, cylindrical polars, the coordinates x and y are replaced by r and Θ, defined as for ordinary polar coordinates, and the third coordinate z has the same meaning as for Cartesian coordinates in three dimensions. Spherical polars describe the position of a point in terms of its distance from the origin and two angles. In Fig. t Thus x = r sin Θ cos φ, y = r sin Θ sin ψ, z — r cos Θ.
Arcsec x and arccosec x are defined only when \x\ ^ 1, but the functions are hardly ever used in any case. Of course, arccosec x = arcsin (l/x) and arcsec x = arccos(l/x). Arctan x is defined for all, positive and negative, values of x. These inverse functions are not uniquely defined when the value of x is given; in fact, they are infinitely many valued. If x = siny, then x = sin(y + 2π), sin(y + 4π) and so on, and equally TRIGONOMETRICAL OR CIRCULAR FUNCTIONS 51 x — sin(7r — y), sin(37r — y).
Many elliptical shapes appear in photomicrographs; they may represent sections through circular cylinders, or through egg-shaped objects, or even through spheres if there has been distortion in one direction in the preparation of the slide or the photographic process. When x and y are large, the first three terms in equation (2) f The form given here is conventional. Writing, for example, 2h rather than h for the coefficient of xy makes some of the algebra a little easier. The choice of these particular letters seems more reasonable when the expression is compared with the homogeneous form ax2 + by2 + cz2 + 2fyz + 2gxz + 2hxy.
Basic Mathematics for the Biological and Social Sciences by F. H. C. Marriott (Auth.)