Question

Let $\mathit{x}\mathbf{=}\mathit{u}\mathbf{+}\mathit{v}\mathbf{,}\mathit{y}\mathbf{=}\mathit{v}$. Find $\mathit{d}\mathit{s}$, the a vectors, and $\mathit{d}{\mathit{S}}^{\mathbf{2}}$for the u, v coordinate system and show that it is not an orthogonal system. Hint: Show that the vectors are not orthogonal, and that $\mathit{d}{\mathit{S}}^{\mathbf{2}}$contains du dv terms. Write the ${\mathit{g}}_{\mathbf{i}\mathbf{j}}$ matrix and observe that it is symmetric but not diagonal. Sketch the lines$\mathit{u}\mathbf{,}\mathit{v}\mathbf{=}\mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{t}$ and observe that they are not perpendicular to each other.

Short answer

The statement has been proven and the value of matrix is $g=\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]$.

Step-by-step solution

Given Information

The given values are mentioned below.

$$\begin{gathered} x=u+v \\ y=v \end{gathered}$$

Definition of cylindrical coordinates.

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.

Prove the statement and find the matrices.

The given values are mentioned below.

$$\begin{gathered} x=u+v \\ y=v \end{gathered}$$

The position vector is given by $s=(u+v){\stackrel{⏜}{e}}_x+v{\stackrel{⏜}{e}}_y$.

Find the value of $a_i$.

$\begin{array}{l}{a}_u={\stackrel{⏜}{e}}_x\\ a_v={\stackrel{⏜}{e}}_x+{\stackrel{⏜}{e}}_y\end{array}$

Find the value of metric tensor

$$\begin{gathered} g_{ij}=a_i.a_j \\ g=\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right] \end{gathered}$$

Find the value of $d{s}^2$

$$\begin{gathered} d{s}^2=g_{ij}d{q}_{i}d{q}_j \\ =d{u}^u+2dudv+2d{v}^2 \end{gathered}$$

The statement has been proven and the value of matrix is $g=\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]$.