Cylindrical and Spherical Coordinates

Cylindrical and Spherical Coordinates Representation and Conversions

 Representing 3D points in Cylindrical Coordinates. Recall polar representations in the plane r

 Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds a z-coordinate to the polar coordinates (r,). r

 r Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds a z-coordinate to the polar coordinates (r,).

 r  r Representing 3D points in Cylindrical Coordinates. (r,,z)

(r,,z)  r  r Converting between rectangular and Cylindrical Coordinates Cylindrical to rectangular No real surprises here! Rectangular to Cylindrical

Representing 3D points in Spherical Coordinates Spherical Coordinates are the 3D analog of polar representations in the plane. We divide 3-dimensional space into • a set of concentric spheres centered at the origin. • rays emanating outward from the origin

Representing 3D points in Spherical Coordinates (x,y,z) We start with a point (x,y,z) given in rectangular coordinates. Then, measuring its distance  from the origin, we locate it on a sphere of radius centered at the origin. Next, we have to find a way to describe its location on the sphere. 

Representing 3D points in Spherical Coordinates We use a method similar to the method used to measure latitude and longitude on the surface of the Earth. We find the great circle that goes through the “north pole,” the “south pole,” and the point.

Representing 3D points in Spherical Coordinates Note: all angles are measured in radians, as always. We measure the latitude or polar angle starting at the “north pole” in the plane given by the great circle. This angle is called . The range of this angle is 

Representing 3D points in Spherical Coordinates We use a method similar to the method used to measure latitude and longitude on the surface of the Earth. Next, we draw a horizontal circle on the sphere that passes through the point.

Representing 3D points in Spherical Coordinates And “drop it down” onto the xy-plane.

Representing 3D points in Spherical Coordinates We measure the latitude or azimuthal angle on the latitude circle, starting at the positive x-axis and rotating toward the positive y-axis. The range of the angle is Angle is called . Note that this is the same angle as the  in cylindrical coordinates!

Finally, a Point in Spherical Coordinates! ( , ,) Our designated point on the sphere is indicated by the three spherical coordinates ( ,  , ) ---(radial distance, azimuthal angle, polar angle). Please note that this notation is not at all standard and varies from author to author and discipline to discipline. (In particular, physicists often use  to refer to the azimuthal angle and  refer to the polar angle.) 

Converting Between Rectangular and Spherical Coordinates • First note that if r is the usual cylindrical coordinate for (x,y,z) • we have a right triangle with • acute angle , • hypotenuse , and • legs r and z. • It follows that (x,y,z) r  z  What happens if  is not acute?

Converting Between Rectangular and Spherical Coordinates (x,y,z) r Spherical to rectangular  z 

Converting from Spherical to Rectangular Coordinates Rectangular to Spherical (x,y,z) r  z 

Cylindrical and Spherical Coordinates Integration

Integration Elements: Rectangular Coordinates We know that in a Riemann Sum approximation for a triple integral, a summand This computes the function value at some point in the little “sub-cube” and multiplies it by the volume of the little cube of length , width , and height .

Integration Elements: Cylindrical Coordinates What happens when we consider small changes in the cylindrical coordinates r, q, and z? We no longer get a cube, and (similarly to the 2D case with polar coordinates) this affects integration.

 r  r Integration Elements: Cylindrical Coordinates What happens when we consider small changes in the cylindrical coordinates r, q, and z? Start with our previous picture of cylindrical coordinates:

Integration Elements: Cylindrical Coordinates What happens when we consider small changes in the cylindrical coordinates r, q, and z? Start with our previous picture of cylindrical coordinates: r Expand the radius by a small amount: r r+Dr

Integration Elements: Cylindrical Coordinates This leaves us with a thin cylindrical shell of inner radius r and outer radius r+D r. r r+Dr r r+Dr

Dq   Integration Elements: Cylindrical Coordinates Now we consider the angle q. We want to increase it by a small amount Dq.

 Integration Elements: Cylindrical Coordinates This give us a “wedge.” Combining this with the cylindrical shell created by the change in r, we get

Integration Elements: Cylindrical Coordinates This give us a “wedge.” Intersecting this wedge with the cylindrical shell created by the change in r, we get

Integration Elements: Cylindrical Coordinates Finally , we look at a small vertical change Dz .

Integration in Cylindrical Coordinates. We need to find the volume of this little solid. As in polar coordinates, we have the area of a horizontal cross section is. . .

So . . . Integration in Cylindrical Coordinates. We need to find the volume of this little solid. Since the volume is just the base times the height. . .

Cylindrical and Spherical Coordinates