When do partial derivatives commute

The only difference between the two cases is the order in which we are taking limits. In the case, we take the limit with respect to on the inside, and then take the limit with respect to on the outside. Jump to: navigation , search. Template:Commuting operators Contents. Navigation menu Personal tools Log in. Namespaces Page Discussion. Views Read View source View history. Most viewed methods Integration by parts First derivative test Product rule Second derivative test Piecewise differentiation.

The book i use doesn't have very many proofs. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.

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Linked 1. Related 4. It only says that for scalar functions. Show 1 more comment. Active Oldest Votes. Improve this answer. Liviu Nicolaescu Liviu Nicolaescu Add a comment. Deane Yang Deane Yang This makes sense, although in a way it kind of cancels the entire point of differential geometry as I see it so far which is not a lot : I thought the motivation is to define calculus again in an intrinsic, coordinate-free way on general smooth manifolds; specifically, since we believe our universe is modeled well by one.

Reducing it all back to the Euclidian setting when we stumble along some complication does not seem to follow the same spirit. Perhaps this means we should think our universe is some complicated embedding in a larger Euclidian space, unreachable to us? Nor does it mean that the geometric structure is not co-ordinate-free. The study of Riemannian geometry does not depend on co-ordinates. Of course, even the study of submanifolds in Euclidean space does not, too.

Peter Michor Peter Michor 24k 1 1 gold badge 56 56 silver badges bronze badges. It is possible to explain this in simpler terms "soldering form" seems to be advanced from the basic place in diff. In the diagram, a small piece of the x,y plane is shown.

I've assumed f 0,0 is zero to keep the algebra simple. The proof is contained in the diagram, and can be understood just by looking at it.