I have difficulty in this chain rule. Can anyone explain this to me in simple words??

Question

I have difficulty in understanding the chain rule given in the picture. The derivative with $x$ do not have $\lambda$, while $y$ term have. Why is it so???

Answer 1

By the chain rule, $$\begin{align} \frac{\partial}{\partial x} &= \frac{\partial \zeta}{\partial x} \frac{\partial}{\partial \zeta} + \frac{\partial \eta}{\partial x} \frac{\partial}{\partial \eta} \\&= \frac{\partial}{\partial \zeta} + \frac{\partial}{\partial \eta}, \end{align} $$ since $\frac{\partial \zeta}{\partial x} = \frac{\partial \eta}{\partial x} = 1$. The same reasoning works for $\frac{\partial}{\partial y}$.

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Edit: for the other chain rule, $$\begin{align} \frac{\partial}{\partial y} & = \frac{\partial \zeta}{\partial y} \frac{\partial}{\partial \zeta} + \frac{\partial \eta}{\partial y} \frac{\partial}{\partial \eta} \\&= \lambda_1 \frac{\partial}{\partial \zeta} + \lambda_2\frac{\partial}{\partial \eta}. \end{align}$$