91Ó°ÊÓ

A Parabolic.

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 parabolic is $\stackrel{Ë™}{\mathrm{s}}=\stackrel{Ë™}{\mathrm{u}}\sqrt{{\mathrm{u}}^2+{\mathrm{v}}^2}{\hat{\mathrm{e}}}_{\mathrm{u}}+\stackrel{Ë™}{\mathrm{v}}\sqrt{{\mathrm{u}}^2+{\mathrm{v}}^2}{\hat{\mathrm{e}}}_{\mathrm{v}}+\stackrel{Ë™}{\mathrm{Ï•³Ü\pm ¹}}{\hat{\mathrm{e}}}_{\mathrm{Ï•}}$.

Find the acceleration.

$$\begin{gathered} \stackrel{¨}{\mathrm{s}}=\left(\stackrel{¨}{\mathrm{u}}+{\mathrm{Γ}}_{\mathrm{ij}}^{\mathrm{u}}{\stackrel{˙}{\mathrm{q}}}_{\mathrm{i}}{\stackrel{˙}{\mathrm{q}}}_{\mathrm{j}}\right){\mathrm{h}}_{\mathrm{u}}{\hat{\mathrm{e}}}_{\mathrm{u}}+\left(\stackrel{¨}{\mathrm{v}}+{\mathrm{Γ}}_{\mathrm{ij}}^{\mathrm{v}}{\stackrel{˙}{\mathrm{q}}}_{\mathrm{i}}{\stackrel{˙}{\mathrm{q}}}_{\mathrm{j}}\right){\mathrm{h}}_{\mathrm{u}}{\hat{\mathrm{e}}}_{\mathrm{u}}+\left(\stackrel{¨}{\mathrm{ϕ}}+{\mathrm{Γ}}_{\mathrm{ij}}^{\mathrm{ϕ}}{\stackrel{˙}{\mathrm{q}}}_{\mathrm{i}}{\stackrel{˙}{\mathrm{q}}}_{\mathrm{j}}\right){\mathrm{h}}_{\mathrm{ϕ}}{\hat{\mathrm{e}}}_{\mathrm{ϕ}}. \\ \frac{{∂}^2\mathrm{s}}{∂{\mathrm{q}}_{\mathrm{i}}∂{\mathrm{q}}_{\mathrm{j}}}={\mathrm{Γ}}_{\mathrm{ij}}^{\mathrm{k}}\frac{∂\mathrm{s}}{∂{\mathrm{q}}_{\mathrm{k}}} \end{gathered}$$

Solve further.

$$\begin{gathered} \frac{{∂}^{2}s}{∂u^2}={\hat{e}}_z\frac{u}{u^2+v^2}\frac{∂s}{∂u}-\frac{v}{u^2+v^2}\frac{∂s}{∂v} \\ \frac{{∂}^{2}s}{∂u∂v}=\frac{v}{u^2+v^2}\frac{∂s}{∂u}+\frac{u}{u^2+v^2}\frac{∂s}{∂v} \\ \frac{{∂}^{2}s}{∂v^2}=-\frac{v}{u^2+v^2}\frac{∂s}{∂u}+\frac{u}{u^2+v^2}\frac{∂s}{∂v} \\ \frac{{∂}^{2}s}{∂u∂\mathrm{ϕ}}=\frac{1}{s}\frac{∂s}{∂\mathrm{ϕ}} \end{gathered}$$

Solve further.

role="math" localid="1659335844957" $$\begin{gathered} \frac{{∂}^{2}s}{∂v∂\mathrm{ϕ}}=\frac{1}{v}\frac{∂s}{∂\mathrm{ϕ}} \\ \frac{{∂}^{2}s}{∂{\mathrm{ϕ}}^2}=-\frac{u{v}^2}{v^2+u^2}\frac{∂s}{∂u}-\frac{u^{2}v}{v^2+u^2}\frac{∂s}{∂v} \end{gathered}$$

Find the components of acceleration.

$$\begin{gathered} {\stackrel{¨}{s}}_u=\left(\stackrel{¨}{u}+\frac{u}{u^2+v^2}{\stackrel{˙}{u}}^2+\frac{2v}{u^2+v^2}\stackrel{˙}{u}\stackrel{˙}{v}+\frac{u}{u^2+v^2}{\stackrel{˙}{v}}^2-\frac{u{v}^2}{u^2+v^2}{\stackrel{˙}{\mathrm{ϕ}}}^2\right)\sqrt{u^2+v^2} \\ {\stackrel{¨}{s}}_v=\left(\stackrel{¨}{u}+\frac{u}{u^2+v^2}{\stackrel{˙}{u}}^2+\frac{2u}{u^2+v^2}\stackrel{˙}{u}\stackrel{˙}{v}+\frac{v}{u^2+v^2}{\stackrel{˙}{v}}^2-\frac{u^{2}v}{u^2+v^2}{\stackrel{˙}{\mathrm{ϕ}}}^2\right)\sqrt{u^2+v^2}.6 \\ {\stackrel{¨}{s}}_{\mathrm{ϕ}}=\left(\stackrel{¨}{\mathrm{ϕ}}+\frac{2}{u}\stackrel{˙}{u}\stackrel{˙}{\mathrm{ϕ}}+\frac{2}{u}\stackrel{˙}{v}\stackrel{˙}{\mathrm{ϕ}}\right)uv \end{gathered}$$