MATH 2110Q Lecture Notes - Lecture 9: Spherical Coordinate System

• School: UConn
• Department: Mathematics
• Course: MATH 2110Q
• Professor: Myron Minn-Thu-Aye

Document Summary

What are spherical coordinates (cid:4666),, (cid:4667): (cid:2025) (rho) to the distance from the origin to the point, still measures counterclockwise rotation about the z-axis, (phi) the angle downwards from the positive z-axis. =(cid:1871)(cid:1866) (cid:2025)(cid:1871)(cid:1866: (cid:1876)=(cid:1870)cos, (cid:1877)=(cid:1871)(cid:1866)(cid:1871)(cid:1866, (cid:1878)=(cid:2025)(cid:1867)(cid:1871, (cid:1876)(cid:2870)+(cid:1877)(cid:2870)+(cid:1878)(cid:2870)=(cid:2025)(cid:2870) In cartesian, dv = dx dy dz (in some order). In cylindrical, dv = dr dz (in some order). It"s possible to use diagram to find dv = (cid:2025)(cid:2870)(cid:1871)(cid:1866) : a triple integral in spherical coordinates looks like (cid:4666)(cid:1868),, (cid:4667) , spherical coordinates are helpful when dealing with. Example: cones, spheres, (cid:1868)= (cid:1876)(cid:2870)+(cid:1877)(cid:2870)+(cid:1878)(cid:2870)=(cid:2870) of x + y + z = 16, and outside (below) the cone z = (cid:885)(cid:1876)(cid:2870)+(cid:885)(cid:1877)(cid:2870). Set up a triple integral to find the volume of the solid region inside the upper (z (cid:3410) 0) hemisphere: 0 (cid:3409) p (cid:3409) 4, 0 (cid:3409) (cid:3409) 2(cid:2024, (cid:3409) (cid:3409) (cid:3095)(cid:2870, = (cid:3095)(cid:2870) is the xy-plane. Cone removed from the upper half of the sphere.