91Ó°ÊÓ

Spherical coordinates are given below.

$$\begin{gathered} x=r\cos \mathrm{ϕ}\sin \mathrm{θ} \\ y=r\sin \mathrm{ϕ}\sin 0 \\ z=r\cos \mathrm{θ} \end{gathered}$$

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.

Assume the following quantity.

$$\begin{gathered} C=\left(i,j,k\right) \\ S=\left(e_r,e_0,e_{\mathrm{ϕ}}\right) \\ {e}_r=i\cos \mathrm{ϕ}\sin \mathrm{θ}+j\sin \mathrm{ϕ}\sin \mathrm{θ}+k\cos \mathrm{θ} \\ {e}_{\mathrm{θ}}=i\cos \mathrm{ϕ}\sin \mathrm{θ}+j\sin \mathrm{ϕ}\sin \mathrm{θ}k\cos \mathrm{θ} \\ {e}_{\mathrm{ϕ}}=-i\cos \mathrm{ϕ}+j\sin \mathrm{ϕ} \end{gathered}$$

T is the transformation operator defined by $T{c}_j=s_j$.

$$\begin{gathered} T\left(C\right)=\left(\begin{array}{ccc}\cos \mathrm{ϕ}\sin \mathrm{θ}& \cos \mathrm{ϕ}\sin \mathrm{θ}& -\sin \mathrm{ϕ}\\ \cos \mathrm{ϕ}\sin \mathrm{θ}& \cos \mathrm{ϕ}\sin \mathrm{θ}& \cos \mathrm{ϕ}\\ \cos \mathrm{θ}& -\sin \mathrm{θ}& 0\end{array}\right) \\ T{\left(C\right)}^{-1}=\left(\begin{array}{ccc}\cos \mathrm{ϕ}\sin \mathrm{θ}& \cos \mathrm{ϕ}\sin \mathrm{θ}& -\sin \mathrm{ϕ}\\ \cos \mathrm{ϕ}\sin \mathrm{θ}& \cos \mathrm{ϕ}\sin \mathrm{θ}& \cos \mathrm{ϕ}\\ -\sin \mathrm{ϕ}& \cos \mathrm{ϕ}& 0\end{array}\right) \end{gathered}$$.

Formula states that $V\left(S\right)=T{\left(C\right)}^{-1}V\left(C\right)$

$$\begin{gathered} T\left(C\right)=\left(\begin{array}{ccc}\cos \mathrm{ϕ}\sin \mathrm{θ}& \sin \mathrm{ϕ}\sin \mathrm{θ}& \cos \mathrm{θ}\\ \cos \mathrm{ϕ}\cos \mathrm{θ}& \cos \mathrm{ϕ}\sin \mathrm{θ}& \sin \mathrm{θ}\\ -\sin \mathrm{ϕ}& \cos \mathrm{ϕ}& 0\end{array}\right) \\ T{\left(C\right)}^{-1}=\left(\begin{array}{c}\cos \mathrm{θ}\\ -\sin \mathrm{θ}\\ -r\sin \mathrm{θ}\end{array}\right) \\ V\left(S\right)=r_r\cos \mathrm{θ}-e_{\mathrm{θ}}-e_{\mathrm{ϕ}}r\sin \mathrm{θ} \end{gathered}$$

Hence, the vector is $V\left(S\right)=e_r\cos \mathrm{θ}-e_{\mathrm{θ}}\sin \mathrm{θ}-e_{\mathrm{ϕ}}r\sin \mathrm{θ}$.