1 / 12

Exploring the Klein Bottle: Spherical Coordinates and More

Dive into the world of the Klein bottle, a 4-dimensional object in 3D, and learn about triple integrals using spherical coordinates. Explore conversions between spherical and other coordinate systems with detailed examples and solutions.

yoshio-austin
Télécharger la présentation

Exploring the Klein Bottle: Spherical Coordinates and More

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. This is a Klein bottle, It is a 4 dimensional objected depicted here in 3 dimensions This object has only 1 side. 14.7 Day 2 Triple IntegralsUsing Spherical Coordinatesand more applications of cylindrical coordinates More information about the Klein bottle can be found at http://www-maths.mcs.standrews.ac.uk/images/klein.html

  2. Conversions between Spherical and other Coordinate systems

  3. Converting the differential(finding the Jacobian) 2 dxdydz=ρ sinφ dρdφdθ Why? To find volume of the box at the left, use V=lwh V = dρ * ρdφ * rdθ (the r is from cylindrical coordinates) From chapter 11 r = ρsin φ Hence dxdydz=ρ sinφ dρdφdθ 2

  4. Example 4

  5. Example 4 Solution

  6. Example 4 explanation

  7. Problem 22

  8. Problem 22 Solution

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

  10. Problem 14 (spherical coordinates only) Convert the integral from rectangular to spherical coordinates

  11. Problem 14 (spherical coordinates only)

  12. Problem 14 Solution (cylindrical)(from yesterday)

More Related