Thank you again for another fine lesson. I'd like to make a comment on the slide at 14:15, however. I believe the manipulations starting with dz=... need more explanation because there is a lot going on conceptually under the notation. On the one hand, dz is a basis vector of the 1 forms, a covariant vector in the cotangent space. On the other hand, dz is the differential of a scalar function whose value can be set equal to the coordinate z. SO it is a function of the coordinates x, y, and z on the manifold. It turns out that the differential of this function 'z ' can be shown to equal 0 dx + 0 dy + 1 dz in the basis for 1 forms associated with these coordinates, and it follows from the derivation for general f on previous slides. Regardless, it is important to understand what is really going on and not to just naively say 'z is z'... because there are two meanings for z, whose equivalence required a nontrivial proof. Please correct me if I'm wrong. Thanks.
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Thank you again for another fine lesson. I'd like to make a comment on the slide at 14:15, however.
I believe the manipulations starting with dz=... need more explanation because there is a lot going on conceptually under the notation.
On the one hand, dz is a basis vector of the 1 forms, a covariant vector in the cotangent space. On the other hand, dz is the differential of a scalar function whose value can be set equal to the coordinate z. SO it is a function of the coordinates x, y, and z on the manifold. It turns out that the differential of this function 'z ' can be shown to equal 0 dx + 0 dy + 1 dz in the basis for 1 forms associated with these coordinates, and it follows from the derivation for general f on previous slides. Regardless, it is important to understand what is really going on and not to just naively say 'z is z'... because there are two meanings for z, whose equivalence required a nontrivial proof.
Please correct me if I'm wrong. Thanks.