This is a Klein bottle, It is a 4 dimensional objected depicted here in 3 dimensions
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Integrals Using 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. ore information about the Klein bottle can be found ttp://www- maths.mcs.standrews.ac.uk/images/klein.ht
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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 Integrals Using Spherical Coordinates and more applications of cylindrical coordinates. More information about the Klein bottle can be found at - PowerPoint PPT Presentation
Transcript of 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
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
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Problem 14 (spherical coordinates only)
Convert the integral from rectangular to spherical coordinates