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