Volume of Hyperspheres
1. Introductiona. Abstractb. Simple examplec. Motivationd. Concrete example
2. Definitiona. Definition
i. A hypersphere is a generalization of the circle/sphere to dimensions n > 3, where n is the number of dimensions. The formula for any hypersphere is given
by x12+x2
2+x32+…xn
2=r2.b. Mathematical concept
i. Using triple integrals to find volumesii. Spherical coordinates in 3-dimensions and 4-dimensions
3. Mathematical Theorya. Theorems
i. Jacobiansii. Gamma function
iii. Find generalized formulas for the volume of n-dimensional hyperspheres. b. Properties
i. Properties of spheres in general
1. For a sphere, we can note that the volume of a sphere, V= 43π r3 , is
the integral of the surface area, S=4 π r2; this property will be used to calculate the Volume of a hypersphere in n-dimensions.
ii. Hyperspheres of odd n dimensionsiii. Hyperspheres of even n dimensions
c. Further examplesi. Find the volume of the hypersphere x2+ y2+z2+w2=4
4. Application
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