Good on 2D surface, but for when there are more two parameters, better to remember that can just integrate det(g)**0.5 to get area; where g is all dot products between all pairs of tangent vectors (dot product only allowed in orthonormal coordinates, which is important to treat correctly when both spaces are curved). If g diagonal, area is product of 1D integrals also. Determinant is product of finite eigenvalues that are in this cause always positive (due to dot products), which can be useful method when doing it numerically.
I've been watching this very closely and still I can not understand why the paralelogram is aproximated with delta u/delta v. What is the link? Further that, if we make the tangent vectors lenght equal to 1 (we normalize the tangent vectors) then ru*delta_u and rv*delta_v could be better aproximations to the patch? Edit: I have seen this approach in every single text book, but they used to make ru*delta_u and rv*delta_v as something obvious without deep explanation, and I just don't get it😅
Great stuff and good energy!! I think you are doing yourself a disservice though by using a blackboard and chalk. The chalk smudges are making the details hard to see. No matter what, very useful videos! Thank you!
Good on 2D surface, but for when there are more two parameters, better to remember that can just integrate det(g)**0.5 to get area; where g is all dot products between all pairs of tangent vectors (dot product only allowed in orthonormal coordinates, which is important to treat correctly when both spaces are curved). If g diagonal, area is product of 1D integrals also. Determinant is product of finite eigenvalues that are in this cause always positive (due to dot products), which can be useful method when doing it numerically.
I've been watching this very closely and still I can not understand why the paralelogram is aproximated with delta u/delta v. What is the link?
Further that, if we make the tangent vectors lenght equal to 1 (we normalize the tangent vectors) then ru*delta_u and rv*delta_v could be better aproximations to the patch?
Edit: I have seen this approach in every single text book, but they used to make ru*delta_u and rv*delta_v as something obvious without deep explanation, and I just don't get it😅
Great stuff and good energy!! I think you are doing yourself a disservice though by using a blackboard and chalk. The chalk smudges are making the details hard to see. No matter what, very useful videos! Thank you!
Disagreed black board and chalk is the best thing ever!