I have wind vectors with 2 components $(u, v)$ that are in the Lambert Conformal Conical Projection. I want to compare the vectors with observations of the wind $(u_{obs}, v_{obs})$ that are created in a regular latitude and longitude grid (a locally Cartesian frame). Components in Lambert and latitude/longitude grid can't be compared directly as they are in different projections.
In the case of points I know how to go from Lambert to latitude/longitude and viceversa: the formulas are long but solvable. Here we have the equations: http://mathworld.wolfram.com/LambertConformalConicProjection.html These formulas transform points, but I don't know how to do it with vectors.
Any help will be very much appreciated. Thanks.
Ok, I self-anwser my question, in case it can helps someone. What we need is the Jacobian. Wikipedia provides details: https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
With the equations of transformations from latitude and longitude to Lambert space: $x=x(\phi, \lambda)$ and $y=y(\phi, \lambda)$ we compute the Jacobian:
$$J = \begin{pmatrix} \partial_{\phi}x & \partial_{\lambda}x \\ \partial_{\phi}y & \partial_{\lambda}y \end{pmatrix} $$ Then, for vectors like velocity:
$$ \begin{pmatrix} dx \\ dy \end{pmatrix} = J \begin{pmatrix}d\phi \\ d\lambda\end{pmatrix}$$
In case you want geographical coordinates $(X, Y)$, $dX = \cos\phi \;R d\lambda$, $dY = R d\phi$ you can apply a second Jacobian to go from latitude and longitude to geographical coordinates:
$$ J_1 = \frac{1}{R}\begin{pmatrix} 0 & 1 \\ \frac{1}{\cos\phi} & 0 \end{pmatrix} ; \begin{pmatrix} d\phi \\ d\lambda \end{pmatrix} = J_1 \begin{pmatrix}dX \\ dY\end{pmatrix}$$
So, in the end:
$$ \begin{pmatrix} dx \\ dy \end{pmatrix}_{Lambert} = JJ_1 \begin{pmatrix}dX \\ dY\end{pmatrix}_{Geographical}$$