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Representation of Vectors in Parabolic Coordinate System

1. Vectors are represented in Parabolic Coordinate System using Unit Basis Vectors in the Directions of its \(U\) Axis and \(V\) Axis. These Directions are given by Vectors \(\vec{u}\) (for \(U\) Axis) and \(\vec{v}\) (for \(V\) Axis) in Standard Basis and their corresponding Unit Basis Vectors are denoted by \(\hat{u}\) and \(\hat{v}\) respectively.
2. The Vectors \(\vec{u}\) and \(\vec{v}\) which give the Directions of \(U\) Axis and \(V\) Axis respectively in Standard Basis for Parabolic Coordinate System are derived from the Partial Derivatives of the Standard Basis Position Vector Function formed by Cartesian Parameterization Functions of the Parabolic Coordinate System with respect to its Coordinate Variables \(u\) and \(v\) as follows
   
   \(x=u-v\hspace{7mm}y=\pm2\sqrt{uv}\)   (for Horizontally Aligned Parabolic Coordinate System)
   
   \(x=\pm2\sqrt{uv}\hspace{7mm}y=u-v\)   (for Vertically Aligned Parabolic Coordinate System)
   
   These Cartesian Parameterization Functions can be represented in form of a the Standard Basis Position Vector Function \(\vec{R}\) as
   
   \(\vec{R} = (u-v) \mathbf{\hat{i}} \pm \hspace{1mm}2\sqrt{uv} \mathbf{\hat{j}}\)   (for Horizontally Aligned Parabolic Coordinate System)
   
   \(\vec{R} = \pm2\sqrt{uv} \mathbf{\hat{i}} + (u-v) \mathbf{\hat{j}}\)   (for Vertically Aligned Parabolic Coordinate System)
   
   Now, for Horizontally Aligned Parabolic Coordinate System the Vectors \(\vec{u}\) and \(\vec{v}\) are calculated by taking the Partial Derivatives of the Position Vector Function \(\vec{R}\) with respect to Coordinate Variables \(u\) and \(v\) as
   
   \(\vec{u} = {\Large \frac{\partial \vec{R}}{\partial u}} ={\Large \frac{\partial (u-v)}{\partial u}} \mathbf{\hat{i}} \pm {\Large \frac{\partial (2\sqrt{uv})}{\partial u}} \mathbf{\hat{j}}= \mathbf{\hat{i}} \pm {\Large \sqrt{\frac{v}{u}}} \mathbf{\hat{j}}\)
   
   \(\vec{v} = {\Large \frac{\partial \vec{R}}{\partial v}} ={\Large \frac{\partial (u-v)}{\partial v}} \mathbf{\hat{i}} \pm {\Large \frac{\partial (2\sqrt{uv})}{\partial v}} \mathbf{\hat{j}}= -\mathbf{\hat{i}} \pm {\Large \sqrt{\frac{u}{v}}} \mathbf{\hat{j}}\)
   
   The Vectors \(\vec{u}\) and \(\vec{v}\) are called the Natural Basis Vectors or Covariant Basis Vectors of Parabolic Coordinate System.
   
   The Magnitude of the Vectors \(\vec{u}\) and \(\vec{v}\) (given by \(|\vec{u}|={\Large \sqrt{\frac{u+v}{u}}}\) and \(|\vec{v}|={\Large \sqrt{\frac{u+v}{v}}}\) respectively) are called the Scale Factors of the Parabolic Coordinate System.
   
   And, the Unit Basis Vectors \(\hat{u}\) and \(\hat{v}\) in the Directions of Vectors \(\vec{u}\) and \(\vec{v}\) are obtained as follows
   
   \(\hat{u} = {\Large \frac{\vec{u}}{|\vec{u}|}}={\Large \sqrt{\frac{u}{u+v}}} \mathbf{\hat{i}} \pm {\Large \sqrt{\frac{v}{u+v}}} \mathbf{\hat{j}},\hspace{6mm}\) \(\hat{v} = {\Large \frac{\vec{v}}{|\vec{v}|}}=-{\Large \sqrt{\frac{v}{u+v}}} \mathbf{\hat{i}} \pm {\Large \sqrt{\frac{u}{u+v}}} \mathbf{\hat{j}}\)
   
   Similarly, for Vertically Aligned Parabolic Coordinate System the Vectors \(\vec{u}\) and \(\vec{v}\) are calculated by taking the Partial Derivatives of the Position Vector Function \(\vec{R}\) with respect to Coordinate Variables \(u\) and \(v\) as
   
   \(\vec{u} = {\Large \frac{\partial \vec{R}}{\partial u}} =\pm {\Large \frac{\partial (2\sqrt{uv})}{\partial u}} \mathbf{\hat{i}} + {\Large \frac{\partial (u-v)}{\partial u}} \mathbf{\hat{j}}= \pm {\Large \sqrt{\frac{v}{u}}} \mathbf{\hat{i}} + \mathbf{\hat{j}}\)
   
   \(\vec{v} = {\Large \frac{\partial \vec{R}}{\partial v}} =\pm {\Large \frac{\partial (2\sqrt{uv})}{\partial v}} \mathbf{\hat{i}} + {\Large \frac{\partial (u-v)}{\partial v}} \mathbf{\hat{j}}= \pm {\Large \sqrt{\frac{u}{v}}} \mathbf{\hat{i}} - \mathbf{\hat{j}}\)
   
   The Vectors \(\vec{u}\) and \(\vec{v}\) are called the Natural Basis Vectors or Covariant Basis Vectors of Parabolic Coordinate System.
   
   The Magnitude of the Vectors \(\vec{u}\) and \(\vec{v}\) (given by \(|\vec{u}|={\Large \sqrt{\frac{u+v}{u}}}\) and \(|\vec{v}|={\Large \sqrt{\frac{u+v}{v}}}\) respectively) are called the Scale Factors of the Parabolic Coordinate System.
   
   And, the Unit Basis Vectors \(\hat{u}\) and \(\hat{v}\) in the Directions of Vectors \(\vec{u}\) and \(\vec{v}\) are obtained as follows
   
   \(\hat{u} = {\Large \frac{\vec{u}}{|\vec{u}|}}= \pm {\Large \sqrt{\frac{v}{u+v}}} \mathbf{\hat{i}} + {\Large \sqrt{\frac{u}{u+v}}} \mathbf{\hat{j}},\hspace{6mm}\) \(\hat{v} = {\Large \frac{\vec{v}}{|\vec{v}|}}=\pm {\Large \sqrt{\frac{u}{u+v}}} \mathbf{\hat{i}} - {\Large \sqrt{\frac{v}{u+v}}} \mathbf{\hat{j}}\)
3. The Covariant Basis Vectors of Parabolic Coordinate System \(\vec{r}\) and \(\vec{\theta}\) can be arranged as Columns of a Matrix. Such Matrix is called the Jacobian Matrix of the Parabolic Coordinate System. It is given as follows
   
   \(J = \begin{bmatrix}\vec{u} & \vec{v}\end{bmatrix} = \begin{bmatrix}{\Large \frac{\partial \vec{R}}{\partial u}} & {\Large \frac{\partial \vec{R}}{\partial v}}\end{bmatrix} = \begin{bmatrix}{\Large \frac{\partial (u-v)}{\partial u}} & {\Large \frac{\partial (u-v)}{\partial v}} \\ \pm {\Large \frac{\partial (2\sqrt{uv})}{\partial u}} & \pm {\Large \frac{\partial (2\sqrt{uv})}{\partial v}}\end{bmatrix} = \begin{bmatrix}1 & -1 \\ \pm {\Large \sqrt{\frac{v}{u}}} & \pm {\Large \sqrt{\frac{u}{v}}}\end{bmatrix}\)   (for Horizontally Aligned Parabolic Coordinate System)
   
   \(J = \begin{bmatrix}\vec{u} & \vec{v}\end{bmatrix} = \begin{bmatrix}{\Large \frac{\partial \vec{R}}{\partial u}} & {\Large \frac{\partial \vec{R}}{\partial v}}\end{bmatrix} = \begin{bmatrix}\pm {\Large \frac{\partial (2\sqrt{uv})}{\partial u}} & \pm {\Large \frac{\partial (2\sqrt{uv})}{\partial v}} \\ {\Large \frac{\partial (u-v)}{\partial u}} & {\Large \frac{\partial (u-v)}{\partial v}}\end{bmatrix} = \begin{bmatrix}\pm {\Large \sqrt{\frac{v}{u}}} & \pm {\Large \sqrt{\frac{u}{v}}} \\ 1 & -1\end{bmatrix}\)   (for Vertically Aligned Parabolic Coordinate System)