В цилиндрических координатах:

\[\frac{\partial}{\partial x} = \cos \alpha \frac{\partial}{\partial r} - \frac{\sin \alpha}{r} \frac{\partial}{\partial \alpha}\] \[\frac{\partial}{\partial y} = \sin\alpha \frac{\partial}{\partial r} + \frac{\cos \alpha}{r} \frac{\partial}{\partial \alpha}\] \[\vec{A} = A_x \vec{e}_x + A_y \vec{e}_y = A_r \vec{e}_r + A_\alpha \vec{e}_\alpha\] \[\begin{cases} \vec{e}_r = \vec{e}_x \cos \alpha + \vec{e}_y \sin \alpha, \\ \vec{e}_\alpha = - \vec{e}_x \sin \alpha + \vec{e}_y \cos \alpha. \end{cases}\]

Откуда:

\[\begin{cases} A_x = A_r \cos \alpha - A_\alpha \sin \alpha, \\ A_y = A_r \sin \alpha + A_\alpha \cos \alpha. \end{cases}\] \[\begin{aligned} \frac{\partial u_x}{\partial x} - \frac{\partial u_y}{\partial y} = & \left(\cos \alpha \frac{\partial}{\partial r} - \frac{\sin \alpha}{r} \frac{\partial}{\partial \alpha}\right)\left(u_r \cos \alpha - u_\alpha \sin \alpha\right) - \\ & - \left(\sin\alpha \frac{\partial}{\partial r} + \frac{\cos \alpha}{r} \frac{\partial}{\partial \alpha}\right) \left(u_r \sin \alpha + u_\alpha \cos \alpha\right) = \\ & = \cos^2 \alpha \frac{\partial u_r}{\partial r} - \sin \alpha \cos \alpha \frac{\partial u_\alpha}{\partial r} - \\ & - \frac{\sin \alpha \cos \alpha}{r} \frac{\partial u_r}{\partial \alpha} + \frac{\sin^2 \alpha }{r} u_r + \frac{\sin^2 \alpha }{r} \frac{\partial u_\alpha}{\partial \alpha} + \frac{\sin \alpha \cos \alpha}{r} u_\alpha \\ & - \sin^2 \alpha \frac{\partial u_r}{\partial r} - \sin \alpha \cos \alpha \frac{\partial u_\alpha}{\partial r} - \\ & - \frac{\sin \alpha \cos \alpha}{r} \frac{\partial u_r}{\partial \alpha} - \frac{\cos^2 \alpha }{r} u_r - \frac{\cos^2 \alpha }{r} \frac{\partial u_\alpha}{\partial \alpha} + \frac{\sin \alpha \cos \alpha}{r} u_\alpha \\ & = \cos 2\alpha \left(\frac{\partial u_r}{\partial r} - \frac{1}{r} \frac{\partial u_\alpha}{\partial \alpha} - \frac{u_r}{r} \right) - \sin 2\alpha \left(\frac{\partial u_\alpha}{\partial r} + \frac{1}{r} \frac{\partial u_r}{\partial \alpha} - \frac{u_\alpha}{r}\right) \end{aligned}\] \[\begin{aligned} \frac{\partial u_y}{\partial x} + \frac{\partial u_x}{\partial y} = & \left(\cos \alpha \frac{\partial}{\partial r} - \frac{\sin \alpha}{r} \frac{\partial}{\partial \alpha}\right) \left(u_r \sin \alpha + u_\alpha \cos \alpha\right) + \\ & + \left(\sin\alpha \frac{\partial}{\partial r} + \frac{\cos \alpha}{r} \frac{\partial}{\partial \alpha}\right) \left(u_r \cos \alpha - u_\alpha \sin \alpha\right) = \\ & = \cos \alpha \sin \alpha \frac{\partial u_r}{\partial r} + \cos^2 \alpha \frac{\partial u_\alpha}{\partial r} - \\ & - \frac{\sin^2 \alpha}{r} \frac{\partial u_r}{\partial \alpha} - \frac{\sin \alpha \cos \alpha}{r} u_r - \\ & - \frac{\sin \alpha \cos \alpha}{r} \frac{\partial u_\alpha}{\partial \alpha} + \frac{\sin^2 \alpha}{r} u_\alpha + \\ & + \sin\alpha \cos \alpha \frac{\partial u_r}{\partial r} - \sin^2\alpha \frac{\partial u_\alpha}{\partial r} + \\ & + \frac{\cos^2 \alpha}{r} \frac{\partial u_r}{\partial \alpha} - \frac{\sin\alpha \cos \alpha}{r} u_r - \\ & - \frac{\sin\alpha \cos \alpha}{r} \frac{\partial u_\alpha}{\partial \alpha} - \frac{\cos^2 \alpha}{r} u_\alpha = \\ & = \sin 2\alpha \left(\frac{\partial u_r}{\partial r} - \frac{1}{r} \frac{\partial u_\alpha}{\partial \alpha} - \frac{u_r}{r} \right) + \cos 2\alpha \left(\frac{\partial u_\alpha}{\partial r} + \frac{1}{r} \frac{\partial u_r}{\partial \alpha} - \frac{u_\alpha}{r}\right) \end{aligned}\]

Введём обозначения:

\[\begin{aligned} & U_a = \frac{\partial u_r}{\partial r} - \frac{1}{r} \frac{\partial u_\alpha}{\partial \alpha} - \frac{u_r}{r} \\ & U_b = \frac{\partial u_\alpha}{\partial r} + \frac{1}{r} \frac{\partial u_r}{\partial \alpha} - \frac{u_\alpha}{r} \\ & V_a = \frac{\partial v_r}{\partial r} - \frac{1}{r} \frac{\partial v_\alpha}{\partial \alpha} - \frac{v_r}{r} \\ & V_b = \frac{\partial v_\alpha}{\partial r} + \frac{1}{r} \frac{\partial v_r}{\partial \alpha} - \frac{v_\alpha}{r} \end{aligned}\]

Тогда

\[\begin{aligned} (6) + (1) - (2) = & \left( U_a \sin 2 \alpha + U_b \cos 2\alpha \right) \left( V_a \sin 2 \alpha + V_b \cos 2\alpha \right) + \\ & + \left( U_a \cos 2 \alpha - U_b \sin 2\alpha \right) \left( V_a \cos 2 \alpha - V_b \sin 2\alpha \right) = \\ & = U_a V_a + U_b V_b \end{aligned}\]

Завершим теорию с вариационным принципом. Найдём в цилиндрических координатах:

\[e_{k,ij} \frac{\partial \varphi}{\partial x_k} \frac{\partial u_j}{\partial x_i} = (**)\]

Как и выше здесь будет несколько слагаемых с различными коэффициентами:

\[(**) = e_{31} \frac{\partial \varphi}{\partial z} \left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y}\right) +\] \[+ e_{33} \frac{\partial \varphi}{\partial z} \frac{\partial u_z}{\partial z} + e_{15} \left(\frac{\partial \varphi}{\partial x} \left[\frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z}\right] + \frac{\partial \varphi}{\partial y} \left[\frac{\partial u_z}{\partial y} + \frac{\partial u_y}{\partial z}\right]\right) =\] \[= e_{31} \nabla_\perp \cdot \vec{u}_\perp + e_{33} \frac{\partial \varphi}{\partial z} + e_{15} \nabla_\perp \varphi \cdot \left(\nabla_\perp u_z + \frac{\partial \vec{u}_\perp}{\partial z}\right)\]