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gts87
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Hello, I'm trying (somewhat haphazardly) to teach myself about differential forms. A question I have which is confusing me at the moment is about the tangent and cotangent spaces.
In https://www.physicsforums.com/showthread.php?t=2953" the basis for the tangent space was described in terms of the directional derivatives [tex]\partial/\partial x_{i}[/tex], with the basis for the cotangent space being the differentials [tex]dx_{i}[/tex]. This is consistent with other materials I've found (such as the articles on Wikipedia regarding the tangent space). However, in Saunders Mac Lane's Mathematics: Form and Function he calls the cotangent space the set of all directional derivatives, with the tangent space being the space of tangent vectors [tex]<dx/dt, dy/dt>[/tex] to the points of the parameterized curve given by x = g(t), y = h(t). (He began by describing the chain rule [tex]dz/dt = (\partial z/\partial x)(dx/dt) + (\partial z/\partial y)(dy/dt)[/tex] on terms of the inner product [tex]<\partial z/\partial x, \partial z/\partial y>\bullet<dx/dt, dy/dt>[/tex], with the first vector (grad(z)) being an element of the cotangent space, and the second an element of the tangent space.) (All this was in chapter VI.9, if anyone has the book.)
Also, in David Bachman's book A Geometric Approach to Differential Forms, he describes the differentials dx, dy, etc. as coordinate functions of the tangent space. I'm wondering why these books would mix up these two. The problem is that both explanations make intuitive sense to me, (at least for the tangent space basis) so are they just two different formulations of the tangent and cotangent spaces, or is someone wrong? Or am I just misunderstanding something?
If anyone can shed some light on this I'd be grateful!
In https://www.physicsforums.com/showthread.php?t=2953" the basis for the tangent space was described in terms of the directional derivatives [tex]\partial/\partial x_{i}[/tex], with the basis for the cotangent space being the differentials [tex]dx_{i}[/tex]. This is consistent with other materials I've found (such as the articles on Wikipedia regarding the tangent space). However, in Saunders Mac Lane's Mathematics: Form and Function he calls the cotangent space the set of all directional derivatives, with the tangent space being the space of tangent vectors [tex]<dx/dt, dy/dt>[/tex] to the points of the parameterized curve given by x = g(t), y = h(t). (He began by describing the chain rule [tex]dz/dt = (\partial z/\partial x)(dx/dt) + (\partial z/\partial y)(dy/dt)[/tex] on terms of the inner product [tex]<\partial z/\partial x, \partial z/\partial y>\bullet<dx/dt, dy/dt>[/tex], with the first vector (grad(z)) being an element of the cotangent space, and the second an element of the tangent space.) (All this was in chapter VI.9, if anyone has the book.)
Also, in David Bachman's book A Geometric Approach to Differential Forms, he describes the differentials dx, dy, etc. as coordinate functions of the tangent space. I'm wondering why these books would mix up these two. The problem is that both explanations make intuitive sense to me, (at least for the tangent space basis) so are they just two different formulations of the tangent and cotangent spaces, or is someone wrong? Or am I just misunderstanding something?
If anyone can shed some light on this I'd be grateful!
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