Euler-Lagrange equation in vector notation

[WackStr]

I read in hand and finch (analytical mechanics) that if you assume you have a lagrangian:

$$L=(\phi,\nabla\phi,x,y,z)$$

Then what does the euler lagrange equation look like in vector notation. I know that if you have a function with more than 1 independent variable then the euler-lagrange equation looks like:

$$\frac{\delta L}{\delta\phi}=\frac{\partial L}{\partial \phi}-\frac{\partial}{\partial x}\left(\frac{\partial L}{\partial\left(\frac{\partial\phi}{\partial x}\right)}\right)-\frac{\partial}{\partial y}\left(\frac{\partial L}{\partial\left(\frac{\partial\phi}{\partial y}\right)}\right)-\frac{\partial}{\partial z}\left(\frac{\partial L}{\partial\left(\frac{\partial\phi}{\partial z}\right)}\right)=0$$

How do I convert this into vector notation. The hint in the question says to use the following equations:

$$\nabla(\vec{F}.G)=\vec{F}.\nabla G+G\nabla.\vec{F}$$
$$\int\int\int_V\nabla.\vec{F}\,dx\,dy\,dz=\int\int_S\vec{F}.\,d\vec{S}=0$$

This equation does work but I'm not sure if this is the form of the euler lagrange equation in vector notation.

I don't know how to use them. I did get this though:

$$\frac{\partial L}{\partial \phi}-\nabla.(\nabla_{\frac{\partial \phi}{x_i}}L)$$

 

[member 553083]

= 0Where the gradient is in vector notation and the subscripts denote the components of the gradient. The equation above is the form of the Euler-Lagrange equation in vector notation. You can also express it as:\nabla.(\nabla_{\frac{\partial \phi}{x_i}}L) - \frac{\partial L}{\partial \phi} = 0