# Volume of Hypersphere ## General Threads * [Volume of n balls](https://math.stackexchange.com/questions/15656/volumes-of-n-balls-what-is-so-special-about-n-5/15798#15798) * [Volume of unit sphere goes to zero](https://math.stackexchange.com/questions/67039/why-does-the-volume-of-the-unit-sphere-go-to-zero) * [N-ball - An adventure in the Nth Dimension](https://www.americanscientist.org/sites/americanscientist.org/files/201110101628308738-2011-11CompSciHayes.pdf) * [Concentration of volume in shell](https://photos.google.com/search/_eAF1QipOQyc3orxK6jfimrO9nd~uN7nyUdy~uuIN8A_whiteboards/photo/AF1QipOEjnJb8krEdgTgSv1tZ6y_QoxgvmReTbD0hy6F) ## Derivations #### 1. Spherical Proof (Dr Quark) * [Derivation of volume of n-ball](https://www.youtube.com/watch?v=eEJtMFdkzzU) #### 2. Dr Peyam proof * [Video](https://www.youtube.com/watch?t=975&v=ngu6fMYJB8E&feature=youtu.be) * [WB 1](https://photos.google.com/photo/AF1QipNf43kvGsqV-C-4Jc-1RYbimH8SgJzPtfy-Gk3t) * [WB 2](https://photos.google.com/photo/AF1QipO6zZ-TY5AsMmJceXb_xaDq-hmLUpV71d9h47ki) * When we change vars, look the equation we are integrating over [here](https://www.desmos.com/calculator/elh13lgbv6) * [Beta function visual](https://www.desmos.com/calculator/wyeag34pzc) * [WB 3](https://photos.google.com/photo/AF1QipPWMwZbP_TErCLE8bSRViDjQp7lFd-zdIPP9H-R) * [WB 4](https://photos.google.com/photo/AF1QipPa_h2PpPZ9c33wu_-Km27dBFKHfJhIvRVGqtZP) #### 2.1 Special Functions used * [Gamma functions](https://www.youtube.com/watch?v=7y-XTrfNvCs) * [Gamma function history](https://drive.google.com/file/d/0B4W-gdhbNpsDaTNvbF9VcGgyR1E/view) * [Gamma function history 2](http://eulerarchive.maa.org/hedi/HEDI-2007-09.pdf) * [Gamma function resources, math exchange](https://math.stackexchange.com/questions/119020/how-to-come-up-with-the-gamma-function) #### 3. Calculus Textbook Proof * [WB 1](https://photos.google.com/photo/AF1QipO2XvjkPeDm_eGFLAgcolqdxzF_AY5ePV_Ya_Gr) * Vector calculus, linear algebra and differential forms, pg 392 #### 4. Recursive process "proof" * Have built out a process that calculates the volume of a hypersphere via recursive sweeping in cartesian coordinates. * https://photos.google.com/photo/AF1QipP9Nb3hPWAQk2BC4Y5UuUhY27FCsE42QA5Xpg_q * https://photos.google.com/photo/AF1QipP9WcP1a_s3h_i3v1VyCbfRjZ6SFBZg46HCRNhF * https://photos.google.com/photo/AF1QipO-j5NgX_ToT_BySvn7ReZ0yWyKtQMETw50yamA * https://math.stackexchange.com/questions/15656/volumes-of-n-balls-what-is-so-special-about-n-5/15798#15798 * https://math.stackexchange.com/questions/67039/why-does-the-volume-of-the-unit-sphere-go-to-zero * [Notebook with recursive code](http://localhost:8888/notebooks/notebooks/Math-appendix/higher%20dimensions/Higher%20Dimensions.ipynb) * [From integration and calc text, on whiteboard](https://photos.google.com/photo/AF1QipO2XvjkPeDm_eGFLAgcolqdxzF_AY5ePV_Ya_Gr)