91Ó°ÊÓ

Spherical coordinates are given below.

$$\begin{gathered} x=r\cos \mathrm{ϕ}\sin \mathrm{θ} \\ y=r\sin \mathrm{ϕ}\sin \mathrm{θ} \\ z=r\cos \mathrm{θ} \end{gathered}$$

$$\begin{gathered} x=r\cos \mathrm{ϕ}\sin \mathrm{θ} \\ y=r\sin \mathrm{ϕ}\sin \mathrm{θ} \\ z=r\cos \mathrm{θ} \end{gathered}$$

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

Spherical coordinates are given below.

$$\begin{gathered} x=r\cos \mathrm{ϕ}\sin \mathrm{θ} \\ y=r\sin \mathrm{ϕ}\sin \mathrm{θ} \\ z=r\cos \mathrm{θ} \end{gathered}$$

The formula states that $ds=idx×jdy+kdz$.

$$\begin{gathered} dx=\cos \mathrm{ϕ}\sin \mathrm{θ}dr+r\cos \mathrm{ϕ}\cos \mathrm{θ}d\mathrm{θ}-r\sin \mathrm{ϕ}\sin \mathrm{θ}d\mathrm{ϕ} \\ dx=\sin \mathrm{ϕ}\sin \mathrm{θ}dr+r\sin \mathrm{ϕ}\cos \mathrm{θ}d\mathrm{θ}+r\cos \mathrm{ϕ}\sin \mathrm{θ}d\mathrm{ϕ} \\ dz=\cos \mathrm{θ}dr-r\sin \mathrm{θ}d\mathrm{θ} \end{gathered}$$

Substitute the above value in the formula.

$$\begin{gathered} ds=idx+jdy+kdz \\ ds=\left(i\cos \mathrm{ϕ}\sin \mathrm{θ}+j\sin \mathrm{ϕ}+k\cos \mathrm{θ}\right)dr+\left(ir\cos \mathrm{ϕ}\cos \mathrm{θ}+jr\sin \mathrm{ϕ}\cos \mathrm{θ}-kr\sin \mathrm{θ}\right)d\mathrm{θ} \\ +\left(-ir\sin \mathrm{ϕ}\sin \mathrm{θ}+jr\sin \mathrm{ϕ}\sin \mathrm{θ}d\mathrm{θ}\right)d\mathrm{ϕ} \end{gathered}$$

Find the other values.

localid="1659347919099" $$\begin{gathered} h_r{e}_r=i\cos \mathrm{ϕ}\sin \mathrm{θ}+j\sin \mathrm{ϕ}\sin \mathrm{θ}+k\cos \mathrm{θ} \\ {h}_{\mathrm{θ}}{e}_{\mathrm{θ}}=ir\cos \mathrm{ϕ}\cos \mathrm{θ}+jr\sin \mathrm{ϕ}\cos \mathrm{θ}-kr\sin \mathrm{θ} \\ {h}_{\mathrm{ϕ}}{e}_{\mathrm{ϕ}}=-ir\sin \mathrm{ϕ}\sin \mathrm{θ}+jr\cos \mathrm{ϕ}\sin \mathrm{θ} \end{gathered}$$

Find the other values.

localid="1659348180974" $$\begin{gathered} e_r=i\cos \mathrm{ϕ}\sin \mathrm{θ}+j\sin \mathrm{ϕ}\sin \mathrm{θ}+k\cos \mathrm{θ} \\ {e}_{\mathrm{θ}}=i\cos \mathrm{ϕ}\cos \mathrm{θ}+j\sin \mathrm{ϕ}\cos \mathrm{θ}-k\sin \mathrm{θ} \\ {e}_{\mathrm{ϕ}}=-i\sin \mathrm{ϕ}+j\cos \mathrm{ϕ} \end{gathered}$$

$$\begin{gathered} e_r=i\cos \mathrm{ϕ}\sin \mathrm{θ}+j\sin \mathrm{ϕ}\sin \mathrm{θ}+k\cos \mathrm{θ} \\ {e}_{\mathrm{θ}}=i\cos \mathrm{ϕ}\cos \mathrm{θ}+j\sin \mathrm{ϕ}\cos \mathrm{θ}-kr\sin \mathrm{θ} \\ {e}_{\mathrm{ϕ}}=-i\sin \mathrm{ϕ}+j\cos \mathrm{ϕ} \end{gathered}$$

Derivate the values mentioned above.

$$\begin{gathered} \stackrel{\mathbf{.}}{e_r}=\stackrel{}{\stackrel{\mathbf{.}}{\mathrm{ϕ}}\sin \mathrm{θ}{e}_{\mathrm{ϕ}}}+\stackrel{\mathbf{.}}{\mathrm{θ}}{e}_{\mathrm{θ}} \\ \stackrel{\mathbf{.}}{e_{\mathrm{θ}}}=\stackrel{}{\stackrel{\mathbf{.}}{\mathrm{ϕ}}\cos \mathrm{θ}{e}_{\mathrm{ϕ}}}+\stackrel{\mathbf{.}}{\mathrm{θ}}{e}_r \\ \stackrel{\mathbf{.}}{e_{\mathrm{ϕ}}}=\stackrel{\mathbf{.}}{\mathrm{ϕ}}\left(\stackrel{}{\sin \mathrm{θ}{e}_{\mathrm{ϕ}}}+\stackrel{\mathbf{.}}{\cos \mathrm{θ}}{e}_{\mathrm{θ}}\right) \end{gathered}$$

$$\begin{gathered} \stackrel{\mathbf{.}}{e_r}=\stackrel{}{\stackrel{\mathbf{.}}{\mathrm{ϕ}}\sin \mathrm{θ}{e}_{\mathrm{ϕ}}}+\stackrel{\mathbf{.}}{\mathrm{θ}}{e}_{\mathrm{θ}} \\ \stackrel{\mathbf{.}}{e_{\mathrm{θ}}}=\stackrel{}{\stackrel{\mathbf{.}}{\mathrm{ϕ}}\cos \mathrm{θ}{e}_{\mathrm{ϕ}}}+\stackrel{\mathbf{.}}{\mathrm{θ}}{e}_r \\ \stackrel{\mathbf{.}}{e_{\mathrm{ϕ}}}=\stackrel{\mathbf{.}}{\mathrm{ϕ}}\left(\stackrel{}{\sin \mathrm{θ}{e}_{\mathrm{ϕ}}}+\stackrel{\mathbf{.}}{\cos \mathrm{θ}}{e}_{\mathrm{θ}}\right) \end{gathered}$$

Let the velocity be$\frac{ds}{dt}$.

The value of ds is mentioned below.

$$\begin{gathered} ds=e_r{h}_{r}dr+e_{\mathrm{θ}}{h}_{\mathrm{θ}}d\mathrm{θ}+e_{\mathrm{ϕ}}{h}_{\mathrm{ϕ}}d\mathrm{ϕ} \\ ds=e^{r}dr+e_{\mathrm{θ}}rd\mathrm{θ}+e_{\mathrm{ϕ}}r\sin \mathrm{θ}d\mathrm{ϕ} \end{gathered}$$

The value of velocity becomes as follows.

$$\begin{gathered} \frac{ds}{dt}=\frac{d}{dt}\left(i\cos \mathrm{ϕ}\sin \mathrm{θ}+j\sin \mathrm{ϕ}\sin \mathrm{θ}+kr\sin \mathrm{θ}\right) \\ \frac{ds}{dt}=\frac{d}{dt}\left(r{e}_r\right) \\ \frac{ds}{dt}=e_r\stackrel{\mathbf{.}}{r+e_{\mathrm{θ}}}+r\stackrel{\mathbf{.}}{\mathrm{θ}}+e_{\mathrm{ϕ}}r\sin \mathrm{θ}\stackrel{\mathbf{.}}{\mathrm{ϕ}} \end{gathered}$$

Let the acceleration be $\frac{d^{2}s}{d{t}^2}$.

$$\begin{gathered} \frac{d^{2}s}{d{t}^2}=e_r\stackrel{••}{r}+\stackrel{•}{e_r+}\stackrel{•}{r}\stackrel{•}{e_{\mathrm{θ}}}\stackrel{••}{\mathrm{θ}}+r\stackrel{•}{e_{\mathrm{θ}}}\stackrel{•}{\mathrm{θ}}+\stackrel{•}{r}\stackrel{•}{\mathrm{ϕ}}{e}_{\mathrm{ϕ}}{\sin }_{\mathrm{θ}}+\stackrel{•}{e_{\mathrm{ϕ}}}r\stackrel{•}{\mathrm{ϕ}\sin \mathrm{θ}} \\ \frac{d^{2}s}{d{t}^2}=\left(\stackrel{••}{r}-r{\stackrel{•}{\mathrm{θ}}}^2-r{\stackrel{•}{\mathrm{ϕ}}}^2{\sin }^2\mathrm{θ}\right)e_2+\left(\stackrel{••}{r\mathrm{θ}}-2\stackrel{•}{r}{\stackrel{•}{\mathrm{θ}}}^2-r{\stackrel{•}{\mathrm{ϕ}}}^2\sin \mathrm{θ}\cos \mathrm{θ}\right)+\left(\stackrel{••}{2r\mathrm{ϕ}}\sin \mathrm{θ}+2\stackrel{•}{r}\stackrel{•}{\stackrel{•}{\mathrm{ϕ}}\cos \mathrm{θ}}+r{\stackrel{•}{\mathrm{ϕ}}}^2\sin \mathrm{θ}\right)e_{\mathrm{ϕ}} \end{gathered}$$

The velocity is$\frac{ds}{dt}=e_r\stackrel{•}{r}+e_{\mathrm{θ}}r\stackrel{•}{\mathrm{θ}+e_{\mathrm{ϕ}}}r\sin \mathrm{θ}\stackrel{•}{\mathrm{ϕ}}$ $\frac{ds}{dt}=e_r\stackrel{•}{r}+e_{\mathrm{θ}}r\stackrel{•}{\mathrm{θ}+e_{\mathrm{ϕ}}}r\sin \mathrm{θ}\stackrel{•}{\mathrm{ϕ}}$.

The acceleration is mentioned belowlocalid="1659350430667" $$\begin{gathered} (\stackrel{••}{r}-\stackrel{•}{r}{\mathrm{θ}}^2-r{\stackrel{•}{\mathrm{ϕ}}}^{2}si{n}^2\mathrm{θ})e_r+(r\stackrel{••}{\mathrm{θ}}-2\stackrel{•}{r}\stackrel{•}{\mathrm{θ}}-r{\stackrel{•}{\mathrm{ϕ}}}^{2}sin\mathrm{θ}cos\mathrm{θ}) \\ {e}_{\mathrm{θ}}+(2\stackrel{•}{r}\mathrm{ϕ}sin\mathrm{θ}+2r\stackrel{•}{\mathrm{θ}}\stackrel{•}{\mathrm{ϕ}}┴•cos\mathrm{θ}+r\stackrel{••}{\mathrm{ϕ}}sin\mathrm{θ})e_{\mathrm{ϕ}}. \end{gathered}$$