$$ L = u_1 (x_1,y_1) + \lambda \left[ u_2(x_2, y_2) - u_0^2 \right] + \mu \left[0 - T(x_1,x_2,y_1,y_2) \right] $$

$\lambda$ and $\mu$ are the multipliers.

There's a number of variables to partially differentiate, but just focusing on $x_1$

$$ \frac{\partial L}{\partial x_1} = \underbrace{\frac{\partial u_1}{\partial x_1}}_{1)} - \mu \underbrace{\frac{\partial T}{\partial x}\cdot\frac{\partial x}{\partial x_1}}_{2)} =0 $$

My questions are

  • in 1) it has $\frac{\partial u_1}{\partial x_1}$ after partially differentiating ${u_1(x_1,y_1)}$, but in 2) it simply has $x$ underneath $\frac{\partial T}{\partial x}$, not $x_1$. How can it simply become $x$ and drop the $_1$?

  • I understand the chain rule is being used to get $\frac {\partial x}{\partial x_1}$ on the end in 2), why does it come out like this and not $\frac {\partial u_2}{\partial x_1}$, with ${u_2(x_1,y_1)}$ being differentiated in a similar way to ${u_1(x_1,y_1)}$?

Hope I haven't made any errors, quite new to writing math like this.


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