I am trying to solve a problem currently and I have the setup needed that lets me find the Euler-Lagrange equation for a vector valued problem.
The Lagrangian for the equation relates to the energy of a surface that is parameterized in terms of variables e1 and e2 (where normally people may use variables u and v instead). The vector x is the general position vector describing the surface and the subsequent subscripts denote the derivatives of position vector in those parametrized coordinates.
$$ \mathcal L = \iint F(e_1,e_2,\vec x_1(e_1,e_2),\vec x_2(e_1,e_2),\vec x_{12}(e_1,e_2)) $$
The corresponding Euler-Lagrange equation we arrive at is.
$$ -\frac{\partial}{\partial e_1}(\frac{\partial F}{\partial \vec x_1}) - \frac{\partial}{\partial e_2}(\frac{\partial F}{\partial \vec x_2}) + \frac{\partial}{\partial e_1 \partial e_2}(\frac{\partial F}{\partial \vec x_{12}}) $$
With boundary conditions that I am not including. My question is, what is the proper approach to taking the derivative of a function F with respect to a vector? There is some uncertainty on my part on what exactly the notation is supposed to mean when a vector is involved in the derivative operator.
I will give a specific example of a function. How would I evaluate this?
$$\frac{\partial}{\partial \vec x_1}(\langle \vec x_1,\vec x_2 \rangle)$$
I guess more specifically, I am not even sure of how to go about evaluating the first part of this which would be
$$\frac{\partial}{\partial \vec x_1}(\vec x_1)$$
Am I just supposed to interpret this as taking the covariant derivative?