Section 9.7 Polar, Cylindrical, and Spherical Coordinates Math 21a February 15, 2008 Announcements No class Monday 2/18. The bipolar cylindrical coordinates are produced by projecting in the "z"-direction.

Illustration of cylindrical coordinates with interactive graphics. As with two dimensional space the standard $$\left( {x,y,z} \right)$$ coordinate system is called the Cartesian coordinate system. Section 6-12 : Cylindrical Coordinates. Bipolar coordinates form the basis for several sets of three-dimensional orthogonal coordinates. "Cylindrical" polar-cylindrical coordinates "Spherical" spherical coordinates with poles along the axis and coordinates in the order radius, polar angle, azimuthal angle {"BipolarCylindrical", {a}} bipolar-cylindrical coordinates with focal length 2 a in the order focal angle, logarithmic radius, {"Bispherical", {a}}
We will look at polar coordinates for points in the xy-plane, using the origin (0;0) and the positive x-axis for reference. Recall that the position of a point in the plane can be described using polar coordinates $(r,\theta)$. A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. There are several notational conventions, and whereas is used in this work, Arfken (1970) prefers . The bipolar cylindrical coordinates are produced by projecting in the z-direction. A set of curvilinear coordinates defined by x = (asinhv)/(coshv-cosu) (1) y = (asinu)/(coshv-cosu) (2) z = z, (3) where u in [0,2pi), v in (-infty,infty), and z in (-infty,infty). A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. In the last two sections of this chapter we’ll be looking at some alternate coordinate systems for three dimensional space.

Cylindrical coordinates are a simple extension of the two-dimensional polar coordinates to three dimensions.

There are several notational conventions, and whereas (u,v,z) is used in this work, Arfken (1970) prefers (eta,xi,z). Cylindrical Polar Coordinates With the axis of the circular cylinder taken as the z-axis, the perpendicular distance from the cylinder axis is designated by r and the azimuthal angle taken to be Φ.

(4) (5) Lesson 6: Polar, Cylindrical, and Spherical coordinates 1. Cylindrical coordinates are a simple extension of the two-dimensional polar coordinates to three dimensions. Illustration of cylindrical coordinates with interactive graphics. Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand.

The following identities show that curves of constant and are Circles in -space. 2. Convert the cylindrical coordinates defined by corresponding entries in the matrices theta, rho, and z to three-dimensional Cartesian coordinates x, y, and z. theta …

Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in theperpendicular z-direction.The two lines of foci F_{1} and F_{2} of the projected Apollonian circles are generally taken to be defined by x=-a and x=+a, respectively, (and by y=0) in the Cartesian coordinate system. Applications Recall that the position of a point in the plane can be described using polar coordinates $(r,\theta)$. A point P in the plane, has polar coordinates (r; ), where r … The following identities show that curves of constant u and v are circles in xy-space. Bipolar Cylindrical Coordinates. No oﬃce hours Tuesday 2/19.

Yes oﬃce hours Wednesday 2/20 2–4pm SC 323. A set of Curvilinear Coordinates defined by (1) (2) (3) where , , and . Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in the perpendicular -direction.The two lines of foci and of the projected Apollonian circles are generally taken to be defined by and , respectively, (and by ) in the Cartesian coordinate system. Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular -direction.Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae.The two foci and are generally taken to be fixed at − and +, respectively, on the -axis of the Cartesian coordinate system