This is a Klein bottle, It is a 4 dimensional objected depicted here in 3 dimensions

14.7 Day 2 Triple IntegralsUsing Spherical Coordinates

and more applications of cylindrical coordinates

This is a Klein bottle, It is a 4 dimensional objected depicted here in 3 dimensions

This object has only 1 side.

More information about the Klein bottle can be found at

http://www-maths.mcs.standrews.ac.uk/images/klein.html

Conversions between Spherical and other Coordinate systems

Converting the differential(finding the Jacobian)

dxdydz=ρ sinφ dρdφdθ 2

Why? To find volume of the box at the left, use V=lwhV = dρ * ρdφ * rdθ(the r is from cylindrical coordinates)

From chapter 11r = ρsin φ

Hence dxdydz=ρ sinφ dρdφdθ

2

Example 4

Example 4 Solution

Example 4 explanation

Problem 22

Problem 22 Solution

(the really sad part of this example is that the example provided by the teacher is also incorrect)

Problem 14 (spherical coordinates only)

Convert the integral from rectangular to spherical coordinates

Problem 14 (spherical coordinates only)

Problem 14 Solution (cylindrical)(from yesterday)