如何理解切空间里的向量可以看成微分算子？
定义倒是清楚，包括各种定义的等价性，但是有点不太能理解的是，如何将微分算子和微分流形看成是“相切的”？有没有比较直观的解释？谢谢。
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6 个回答
我们之所以要研究切向量、余切向量，正是因为流形上的曲线和函数在局部上有很好的性质，比如说线性性质。而我们考察线性空间的性质时，当然两个线性空间如果同构就几乎可以看做一回事。所以相切的概念本身就是在讲，局部上如果可以用线性空间的行为去刻画流形的性质，就说这个空间在某种意义上与流形相切。
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I think this is because T (maps manifold to its tangent bundle) is a covariant End-functor of category Manifold. usually we get the feeling of tangent via Euclidian space, while in Manifold there are enough arrows Rn -> M (curves etc.) and M -> Rm (embedding thms), and the functority of T extends that feeling to general manifolds, ie, we wanna this hold: tangent vectors should be preserved under morphisms.
as for me this definition more natural: first we have a manifold, with naturally an algebra C(M), then consider the derivations of C(M), denoted by Der(C(M)), and recognise it as TM. (this process should be adjusted when considering more rigid structures like complex)
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i mean morphisms between manifolds naturally induce morphisms between the function algebras and thus induce morphisms between Der() and we get the functor T. this is the link of the two paragraphs above.
as for me this definition more natural: first we have a manifold, with naturally an algebra C(M), then consider the derivations of C(M), denoted by Der(C(M)), and recognise it as TM. (this process should be adjusted when considering more rigid structures like complex)
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i mean morphisms between manifolds naturally induce morphisms between the function algebras and thus induce morphisms between Der() and we get the functor T. this is the link of the two paragraphs above.
谢邀。
我觉得最直观的理解就是在
里，一个vector 
对应一个linear differential operator 
，给出函数的directional derivative。而一个n dimensional manifold M上面的函数局部来看都是定义在上，所以切向量可以看成是linear differential operator。（不过我猜你大概都知道了。。。。。）
======刚才这里写了一段被@Liph 指出其实是cotangent space，可耻地删掉了T_T======
我记得warner的书就是用这种方法定义的切空间，讲得挺清楚的可以看看。
我觉得最直观的理解就是在
======刚才这里写了一段被@Liph 指出其实是cotangent space，可耻地删掉了T_T======
我记得warner的书就是用这种方法定义的切空间，讲得挺清楚的可以看看。
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