91Ó°ÊÓ

An elliptical cylinder.

The coordinate system primarily utilized in three-dimensional systems is the cylindrical coordinates of the system. The cylindrical coordinate system is used to find the surface area in three-dimensional space.

Velocity in elliptical cylinder is$\stackrel{Ë™}{s}=\stackrel{Ë™}{u}a\sqrt{\sin h^{2}u+{\sin }^{2}v}{\hat{e}}_u+\stackrel{Ë™}{v}a\sqrt{\sin h^{2}u+{\sin }^{2}v}{\hat{e}}_v+\stackrel{Ë™}{z{\hat{e}}_z}$.

Find the acceleration.

localid="1659333210449" $$\begin{gathered} \stackrel{¨}{s}=\left(\stackrel{¨}{u}+{\mathrm{Γ}}_{ij}^u{\stackrel{˙}{q}}_i{\stackrel{˙}{q}}_j\right)h_u{\hat{e}}_u+\left(\stackrel{¨}{v}+{\mathrm{Γ}}_{ij}^v{\stackrel{˙}{q}}_i{\stackrel{˙}{q}}_j\right)h_v{\hat{e}}_u+\stackrel{¨}{z}{\hat{e}}_z \\ \frac{{∂}^{2}s}{∂q_i∂q_j}={\mathrm{Γ}}_{ij}^k\frac{∂s}{∂q_k} \end{gathered}$$

Solve further.

$$\begin{gathered} \frac{{∂}^2\mathrm{s}}{∂{\mathrm{u}}^2}=\mathrm{acoshucosv}{\hat{\mathrm{e}}}_{\mathrm{x}}+\mathrm{asinhusinv}{\hat{\mathrm{e}}}_{\mathrm{y}} \\ \frac{{∂}^2\mathrm{s}}{∂\mathrm{u}∂\mathrm{v}}=-\mathrm{asinhusinv}{\hat{\mathrm{e}}}_{\mathrm{x}}+\mathrm{acoshucosv}{\hat{\mathrm{e}}}_{\mathrm{y}} \\ \frac{{∂}^2\mathrm{s}}{∂{\mathrm{v}}^2}=\mathrm{acoshucosv}{\hat{\mathrm{e}}}_{\mathrm{x}}-\mathrm{asinhusinv}{\hat{\mathrm{e}}}_{\mathrm{y}} \end{gathered}$$

Find the value of ${\hat{e}}_x$and ${\hat{e}}_y$.

$$\begin{gathered} {\hat{e}}_x=\frac{\sin hu\cos v}{a\left(\sin h^{2}u+{\sin }^{2}v\right)}\frac{∂s}{∂u}-\frac{\cos hu\sin v}{a\left(\sin h^{2}u+{\sin }^{2}v\right)}\frac{∂s}{∂v} \\ {\hat{e}}_y=\frac{\cos hu\sin v}{a\left(\sin h^{2}u+{\sin }^{2}v\right)}\frac{∂s}{∂u}-\frac{\sin hu\cos v}{a\left(\sin h^{2}u+{\sin }^{2}v\right)}\frac{∂s}{∂v} \end{gathered}$$

Find the coefficients.

$$\begin{gathered} \frac{{∂}^2\mathrm{s}}{∂{\mathrm{u}}^2}=\frac{\mathrm{sinhucoshu}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}}\frac{∂\mathrm{s}}{∂\mathrm{u}}-\frac{\mathrm{sinvcosv}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}}\frac{∂\mathrm{s}}{∂\mathrm{v}} \\ {\mathrm{Γ}}_{\mathrm{uu}}^{\mathrm{u}}=\frac{\mathrm{sinhucoshu}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}} \\ {\mathrm{Γ}}_{\mathrm{uu}}^{\mathrm{u}}=-\frac{\mathrm{sinvcosv}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}} \\ \frac{{∂}^2\mathrm{s}}{∂{\mathrm{u}}^2}=\frac{\mathrm{sinvcosv}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}}\frac{∂\mathrm{s}}{∂\mathrm{u}}-\frac{\mathrm{sinhucoshu}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}}\frac{∂\mathrm{s}}{∂\mathrm{v}} \end{gathered}$$

Find the further coefficients.

$$\begin{gathered} {\mathrm{Γ}}_{uv}^u=\frac{\sin u\cos u}{\sin h^{2}u+{\sin }^{2}v};{\mathrm{Γ}}_{uv}^u=\frac{\sin hv\cos hv}{\sin h^{2}u+{\sin }^{2}v} \\ \frac{{∂}^{2}s}{∂v^2}=-\frac{\sin hu\cos hu}{\sin h^{2}u+{\sin }^{2}v}\frac{∂s}{∂u}+\frac{\sin v\cos v}{\sin h^{2}u+{\sin }^{2}v}\frac{∂s}{∂v} \\ {\mathrm{Γ}}_{vv}^u=-\frac{\sin hu\cos hu}{\sin h^{2}u+{\sin }^{2}v} \\ {\mathrm{Γ}}_{vv}^v=-\frac{\sin v\cos v}{\sin h^{2}u+{\sin }^{2}v} \end{gathered}$$

Find the components of acceleration.

$$\begin{gathered} {\stackrel{¨}{s}}_u=\left(\stackrel{¨}{u}+\frac{\sin hu\cos hu}{{\sin }^{2}u+{\sin }^{2}v}{\stackrel{˙}{u}}^2+\frac{2\sin u\cos u}{{\sin }^{2}u+{\sin }^{2}v}\stackrel{˙}{u}\stackrel{˙}{v}-\frac{\sin u\cos u}{{\sin }^{2}u+{\sin }^{2}v}{\stackrel{˙}{v}}^2\right)h_u \\ {\stackrel{¨}{s}}_v=\left(\stackrel{¨}{u}+\frac{\sin v\cos u}{{\sin }^{2}u+{\sin }^{2}v}{\stackrel{˙}{u}}^2+\frac{2\sin hv\cos hu}{{\sin }^{2}u+{\sin }^{2}v}\stackrel{˙}{u}\stackrel{˙}{v}-\frac{\sin v\cos u}{{\sin }^{2}u+{\sin }^{2}v}{\stackrel{˙}{v}}^2\right)h_u \\ {\stackrel{¨}{s}}_z=\stackrel{¨}{z} \end{gathered}$$.