Q:

calculus of vatiations; finding the minimum value

Find the minimum value of the quadratic form:  
$$x^{2}+2y^{2}+3z^{2}+2xy+2xz$$ 
subject to the condition:  
$$x^{2}+y^{2}+z^{2}=1$$

A:

You could try to use the method of Lagrange multipliers.
Introduce the function
$$\Lambda(x,y,z,\lambda)=x^{2}+2y^{2}+3z^{2}+2xy+2xz +\lambda (x^2+y^2+z^2-1) $$
And try to solve the system $\nabla_{x,y,z,\lambda}\Lambda=0$:
$$
\frac{\partial \Lambda}{\partial x}=(1+\lambda)x+y+z=0\\
\frac{\partial \Lambda}{\partial y}=x+(2+\lambda)y=0\\
\frac{\partial \Lambda}{\partial z}=x+(3+\lambda) z=0\\
\frac{\partial \Lambda}{\partial \lambda}=x^2+y^2+z^2-1=0$$
The last equation is just the original constraint (the points $(x,y,z)$ have to lie on the unit sphere).

