Hello StackExchange community,
I've been trying to prove Marsden and Tromba's Vector Calculus 5/E Exercise 7.3.17 part (c), which reads:
Prove that the tangent plane of surface S is well-defined independently of the regular (one-to-one) parametrization, namely Φ(u,v) and Ψ(s(u,v),t(u,v)), using the inverse function theorem.
The two normal vectors Tu x Tv and Ts x Tt are supposed to be scalar multiples of each other, by a factor of the Jacobian determinant between the two sets of variables (u,v) and (s,t) but I can't seem to find a way to match the Jacobian.
$$ \dfrac{\partial \Phi}{\partial u} = \dfrac{\partial \Psi}{\partial s}\dfrac{\partial s}{\partial u} +\dfrac{\partial \Psi}{\partial t}\dfrac{\partial t}{\partial u} $$
and
$$ \dfrac{\partial \Phi}{\partial v} = \dfrac{\partial \Psi}{\partial s}\dfrac{\partial s}{\partial v} +\dfrac{\partial \Psi}{\partial t}\dfrac{\partial t}{\partial v} $$
So that
$$ \mathbf{T_u^\Phi} \times \mathbf{T_v^\Phi} = \dfrac{\partial \Phi}{\partial u} \mathbf i \times \dfrac{\partial \Phi}{\partial v} \mathbf j = \left( \dfrac{\partial \Psi}{\partial s}\dfrac{\partial s}{\partial u} +\dfrac{\partial \Psi}{\partial t}\dfrac{\partial t}{\partial u} \right) \cdot \left( \dfrac{\partial \Psi}{\partial s}\dfrac{\partial s}{\partial v} +\dfrac{\partial \Psi}{\partial t}\dfrac{\partial t}{\partial v} \right) \mathbf k $$
but I'm supposed to show that
$$ \mathbf{T_u^\Phi} \times \mathbf{T_v^\Phi} = \left( \dfrac{\partial{(s,t)}}{\partial{(u,v)}} \right) (\mathbf{T_s^\Psi} \times \mathbf{T_t^\Psi}) $$
Thank you in advance!