# Why does the Maxwell curl equation after incident on the boundary applies either to the instantaneous values of the field or to the amplitudes?

$$\mathbf{H}=\frac{1}{\mu\omega} \mathbf{k\times E}\;\;\;\;(incident)\;\;(2.49)$$

$$\mathbf{H^{'}}=\frac{1}{\mu\omega} \mathbf{k^{'}\times E^{'}}\;\;\;\;(reflected)\;\;(2.50)$$

$$\mathbf{H^{''}}=\frac{1}{\mu\omega} \mathbf{k^{''}\times E^{''}}\;\;\;\;(transmitted)\;\;(2.51)$$

It should be noted that the above equations apply either to the instantanous values of the fields or to the amplitudes, since the exponential factors $$\exp i(\mathbf{k\cdot r} -\omega t)$$, and so forth, are common to both the electric and associated magnetic fields.

I am trying to understand the paragraph written in the picture.

I understand that when the light incident at a boundary, there will be part of the light that gets transmitted and some of it gets reflected and We get these equations.

I don't understand the reason for the statement that "above equations apply either to the instantaneous values of the fields or to the amplitudes".

The instantaneous electric and magnetic fields are \begin{align} \mathbf{E}(\mathbf{r},t) &= \mathbf{E}_0 e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}\\ \mathbf{H}(\mathbf{r},t) &= \mathbf{H}_0 e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)} \end{align} and $$\mathbf{E}_0$$, $$\mathbf{H}_0$$ are the amplitudes.
If we have \begin{align} \mathbf{H} = \frac{1}{\mu \omega}\mathbf{k}\times \mathbf{E}, \end{align} for the fields, then this means \begin{align} \mathbf{H}_0 e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)} = \frac{1}{\mu \omega}\mathbf{k}\times\mathbf{E}_0 e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}. \end{align} Since the same (scalar, non-zero) factor $$e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}$$ appears on both sides, we can cancel it to get \begin{align} \mathbf{H}_0 = \frac{1}{\mu \omega} \mathbf{k}\times \mathbf{E}_0 \end{align} So the relation $$\mathbf{H} = \frac{1}{\mu \omega}\mathbf{k}\times \mathbf{E},$$ applies to both the instantaneous fields $$\mathbf{E}(\mathbf{r},t)$$, $$\mathbf{H}(\mathbf{r},t)$$ and the amplitudes $$\mathbf{E}_0$$, $$\mathbf{H}_0$$.