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Well, I was reading about plane waves in Principles of Optics by Born and Wolf.

Here, $\mathbf r$ is the position vector of a point $P$ and $\mathbf s$ is a unit vector perpendicular to the plane.

Now, they chose a new set of coordinate axes such that $\mathrm{O\zeta}$ in the direction of $\mathbf s\,.$

Thus, $$\mathbf r\cdot \mathbf s~=~ \zeta\,.$$

But then they wrote $$\frac{\partial}{\partial x} = s_x\frac{\partial}{\partial \zeta}, ~~ \frac{\partial}{\partial y} = s_y\frac{\partial}{\partial \zeta},~~\frac{\partial}{\partial z} = s_z\frac{\partial}{\partial \zeta}\,.$$

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How did they transform the coordinates? I'm not getting how they wrote the differential operator relations above.

Could anyone shed some light on how to derive the relations from $\mathbf r\cdot\mathbf s~=~\zeta$?

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Writing $u$ instead of $\zeta$, we first find an expression for the differential of $u$. $$u=\vec{r}\cdot\hat{s} = s_x x+ s_y y + s_z z \\ du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy + \frac{\partial u}{\partial z}dz \\ du = s_x dx + s_y dy + s_z dz $$ Now consider the differential of a function $f(u) = f(\vec{r}\cdot\hat{s})$. Considering $f$ to be a function of $u$ we can write its differential as $$df= \frac{\partial f}{\partial u}du = \frac{\partial f}{\partial u}\left( s_x dx + s_y dy + s_z dz \right) \\ df = \left(s_x\frac{\partial f}{\partial u}\right) dx + \left(s_y\frac{\partial f}{\partial u}\right) dy + \left(s_z\frac{\partial f}{\partial u}\right) dz \;.$$ Now consider $f$ to be a function of $x$, $y$, and $z$ and write its differential as $$df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz \;.$$ Now we compare the last two expressions for $df$ and we keep in mind that $x$, $y$ and $z$ can be varied independently. When we hold $y$ and $z$ constant and vary $x$ the last two expressions for $df$ give us $$\Delta f= \left(s_x\frac{\partial f}{\partial u}\right) \Delta x=\frac{\partial f}{\partial x} \Delta x\;.$$
Therefore we must have $$\frac{\partial f}{\partial x} = s_x\frac{\partial f}{\partial u}\;.$$ Since $f$ an arbitrary (differentiable) function, we conclude that the $$\frac{\partial}{\partial x} = s_x\frac{\partial}{\partial u} \;,$$ which is the first operator equation in Born & Wolf. The remaining equations follow by considering what happens when we hold $x$ and $z$ fixed and vary $y$, etc.

To continue beyond your question, the next steps that Born & Wolf do not show could be to write the second derivatives as $$\frac{\partial^2}{\partial x^2} = s_x^2\frac{\partial^2}{\partial u^2} \;, \\ \frac{\partial^2}{\partial y^2} = s_y^2\frac{\partial^2}{\partial u^2} \;, \\ \frac{\partial^2}{\partial z^2} = s_z^2\frac{\partial^2}{\partial u^2} \;,\\$$ which leads to $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2} = s_x^2\frac{\partial^2}{\partial u^2} + s_y^2\frac{\partial^2}{\partial u^2} + s_z^2\frac{\partial^2}{\partial u^2} \\ = \left( s_x^2 + s_y^2 + s_z^2 \right) \frac{\partial^2}{\partial u^2} = \frac{\partial^2}{\partial u^2} \;,$$ where the last step follows from the fact that $\hat{s}$ is a unit vector. Therefore, we have Born & Wolf's next equation $$\nabla^2 V = \frac{\partial^2 V}{\partial \zeta^2}\;.$$

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