Earlier quoted context omitted.
I'm not a big Riemannian geometry buff, but I took a look at the definition in Do Carmo's book and it appears that "grad f" actually lies in TM, consistent with what I said above. Would love to learn more if I've got this mixed up. This would be nice, because it would generalize the "gradient" from vector calculus, which is clearly and unambiguously a vector.
It's probably just a notation/definition issue. I'm not sure if "grad f" is 100% consistently defined I'm a simple-minded physicist. I just know if you apply the same coordinate transformation to the gradient and to the displacement vector, you get the wrong answer. My usual reference is Schutz's Geometrical Methods of Mathematical Physics, and he defines the gradient as df, but other sources call that the "different…
Re: Matrix Calculus (For Machine Learning and Beyond)
#31And I'm just a simple applied mathematician. For me, the gradient is the vector that points in the direction of steepest increase of a scalar field, and the Jacobian (or indeed, "differential") is the linear map in the Taylor expansion. I'll be curious to take a look at your reference: looks like a good one, and I'm definitely interested in seeing what the physicist's perspective is. Thanks!