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Currently Playing: How to Describe an Entire Surface with Just Two Numbers

PDF link if you want a more detailed explanation: https://dibeos.net/how-to-describe-a-surface-with-2-numbers Attention! Notice how the parametrization that we use in this video (and also the whole algorithm itself) assumes that the surface is embedded in a 3-dimensional Euclidean space. This is being used here to find a property, namely the Gaussian curvature, which is actually an intrinsic property. It is intrinsic because of the famous Gauss's Theorema Egregium (which is Latin for "Remarkable Theorem"). It basically means that Gaussian curvature shouldn't depend on the embedding. Technically, you can compute Gaussian curvature without relying on an external ambient space (i.e., xyz-structure), as we are doing here. This could be done just with measurements within the surface itself (like angles, distances, and so on). However, in order to do so we would have to talk about tensors, and Christoffel symbols, which would be mathematically way more complex. So, there is a tradeoff here: it is easier to compute, and visualize, Gaussian curvature using an embedding because it avoids talking about tensors and Christoffel symbols, but unfortunately we sacrifice the core of Differential Geometry, i.e. the clear understanding of the fact that Gaussian curvature is an intrinsic property, and therefore can be calculated without an embedding -- it is just mathematically harder. Submit your research to us! https://dibeos.net/research/ We welcome any research you may have done and also welcome any peer review for the submitted material. The Mathematics of Minkowski Spacetime https://youtu.be/lebBoqx_b6s?si=IpReo3MjOIxrlGgJ Eigenvalues and Eigenvectors https://youtu.be/HFTbZ_rVhIA?si=qRIvHZwF66TmDi_- Visit our site to access all the PDF's: https://dibeos.net --- Our goal is to be the #1 math channel in the world. Please, give us your feedback, and help us achieve this ambitious dream. --- Some great books for learning math or physics https://www.amazon.com/hz/wishlist/ls/OUBVJVG21N5W?ref_=wl_share Need a VPN? https://go.nordvpn.net/aff_c?offer_id=15&aff_id=110880&url_id=858 🐦 Follow me on X: https://x.com/dibeoluca 📸 Follow me on Instagram: https://www.instagram.com/lucadibeo/ 🧵 Follow me on Threads: https://www.threads.net/@lucadibeo 😎 Become a member to have exclusive access: https://www.youtube.com/channel/UC3Z1rXCFFadHw69-PZpQRYQ/join 📈 Check out my Udemy courses (you may find something that interests you 😉): https://www.udemy.com/user/luca-di-beo/ 🥹 Consider supporting us on Patreon: https://www.patreon.com/user?u=86646021