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I'm kind of rusty in my computations. I'm trying to compute the pullback of the form dθ on S^{1}by a differentiable map f: M→S^{1}, where f is differentiable and M is a 3-manifold; please tell me if this is correct:

0)Say we use the basis vectors {∂/∂x^{1},∂/∂x^{2}, ∂/∂x^{3}, }

for TxM ;

1)We compute the pushforwards of the three basis vectors, and get:

f_{*}(∂/∂x^{i})=∂f/∂x^{i}∂/∂θ , for i=1,2,3.

2)We evaluate dθ at each of the pushforwards of the basis vectors, to get:

dθ (∂f/∂x^{i}∂/∂θ)= (∂f/∂x^{i}); i=1,2,3.

3)We conclude :

f^{*}dθ = ∂f/∂x^{1}dx+ ∂f/∂x^{2}dy+ ∂f/∂x^{3}dz

Is this correct?

Thanks for your comments.

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# Computing Pullback of 1-Form

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