I am following the solution of the following problem on the topic of the energy integral of a surface.
For a real-valued continuously differentiable function $u(x,y)$ on a closed domain $D$, the energy integral is defined as follows. $$E(u)=\iint_D \left( \left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2\right) dxdy.$$ If on $B$, the boundary of $D$, the function $u$ achieves the specific values of some function $f$, how can $E(u)$ be minimized within $D$ over all functions $u$, given the described boundary-constraint?
Here is how the author goes about finding out how to minimize the energy integral.
Suppose that the function $u_0$ produces this minimum: $$E(u_0)\leq E(u),\,\,\,\textrm{ for all functions }u\textrm{ with }u=f \textrm{ on } B$$ Let $v$ be any continuously differentiable function on $D$ that is zero on $B$. Then $u_0+\epsilon v$ equals $f$ on $B$ for any number $\epsilon$, and because $u_0$ has the minimum energy integral, $$\begin{align*} E(u_0)&\leq E(u_0+\epsilon v)\\ &= E(u_0) + 2\epsilon \iint_D \left(\frac{\partial u_0}{\partial x}\frac{\partial v}{\partial x}+\frac{\partial u_0}{\partial y}\frac{\partial v}{\partial y}\right)dxdy + \epsilon^2 E(v) \end{align*}$$
The last part is what I don't understand. Where did that expansion of $E(u_0+\epsilon v)$ come from? Could anybody explain what each term in that expansion is there for?
Using the simpler ($\LaTeX$-wise) notation
$u_{,x} = \dfrac{\partial u}{\partial x}, \tag{1}$
we have
$(u_0 + \epsilon v)_{,x} = u_{0,x} + \epsilon v_{,x}, \tag{2}$
whence
$((u_0 + \epsilon v)_{,x})^2 = (u_{0,x})^2 + 2\epsilon u_{0,x} v_{,x} + \epsilon^2 (v_{,x})^2; \tag{3}$
similarly
$((u_0 + \epsilon v)_{,y})^2 = (u_{0,y})^2 + 2 \epsilon u_{0,y} v_{,y} + \epsilon^2 (v_{,y})^2; \tag{4}$
when (3) and(4) are used in the expression for $E(u_0 + \epsilon v)$, the stated formula easily follows.
Hope this helps. Good Yule Season,
and as ever,
Fiat Lux!!!