4.9-4.10 (HO L15) 3D polar coordinates, 3D sketching PDF

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4.9 Three-dimensional Polar Coordinate Systems Rotationally-symmetric regions are easier to describe mathematically if we use polar rather than Cartesian coordinates. In three dimensions there are two different forms of polar coordinates.

(a)Cylindrical polar coordinates In cylindrical polar coordinates, we retain the z axis but replace the (x,y)-position by one angle and one distance. Thus, we transform from (x, y, z) to (R, , z) where

(b) Spherical polar coordinates If we now use polar coordinates (r, ) to describe the position in the (z , R ) plane of cylindrical polar coordinates, we obtain spherical polar coordinates (r , ,  ), where

4.10 Sketching Graphs in 3D Although volume integrals are no more difficult than integrals over twodimensional planes, sketching the domain of integration is considerably harder. A rough sketch of the bounding surface is usually very helpful, and can be done by considering the following properties of a function. (a) Symmetry (i) a function that remains the same under the transformation x →  x has mirror symmetry in the plane x = 0. (ii) a function that depends uponx and y only in the combination x2+y2 is axisymmetric about the z axis. (b) Interesting features: Maxima, minima, asymptotes, etc. (c) Sections: Examine the intersections of surfaces with the plane x = 0 (and similarly y = 0 & z = 0 )....