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Covariant Metric Tensors for Coordinate Systems

1. The Gram Matrix / Gramian Matrix calculated for Jacobian Matrix of Any Coordinate System containing the Covariant Basis Vectors of the Coordinate System as its Columns is called Covariant Metric Tensor or simply Metric Tensor of the Coordinate System
   
   For example, if \(J\) is the Jacobian Matrix of an \(N\) Dimensional Coordinate System having Covariant Basis Vectors \(\mathbf{e_1}, \mathbf{e_2}, \cdots, \mathbf{e_N} \) as its Columns as follows
   
   \(J = \begin{bmatrix}\mathbf{e_1} & \mathbf{e_2} & \cdots & \mathbf{e_N} \end{bmatrix}\)   ...(1)
   
   then, the Covariant Metric Tensor \(g\) for the Coordinate System correnponding to the Jacobian Matrix \(J\) of the Coordinate System is calculated as
   
   \(g= J^TJ = \begin{bmatrix}\mathbf{e_1} \\ \mathbf{e_2} \\ \vdots \\ \mathbf{e_N} \end{bmatrix}\begin{bmatrix}\mathbf{e_1} & \mathbf{e_2} & \cdots & \mathbf{e_N} \end{bmatrix}= \begin{bmatrix}\mathbf{e_1} \cdot \mathbf{e_1} & \mathbf{e_1}\cdot \mathbf{e_2} & \cdots & \mathbf{e_1}\cdot\mathbf{e_N} \\ \mathbf{e_2} \cdot \mathbf{e_1} & \mathbf{e_2}\cdot \mathbf{e_2} & \cdots & \mathbf{e_2}\cdot\mathbf{e_N} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{e_N} \cdot \mathbf{e_1} & \mathbf{e_N}\cdot \mathbf{e_2} & \cdots & \mathbf{e_N}\cdot\mathbf{e_N} \end{bmatrix}\)   ...(2)
   
   \(g_{ij}\) denotes the Component/Element present at the \(i\)th Row and \(j\)th Column of the Metric Tensor \(g\).
2. The Inverse of the Metric Tensor Matrix of a Coordinate System (i.e. \({(J^TJ)}^{-1}\)) is called the Inverse Metric Tensor or Contravariant Metric Tensor of the Coordinate System. It is denoted by \(g^{-1}\). \(g^{ij}\) denotes the Component/Element present at the \(i\)th Row and \(j\)th Column of the Inverse Metric Tensor \(g^{-1}\).
3. Both Metric Tensor and Inverse Metric Tensor for any Coordinate System are Symmetric Matrices whose Order is Same as the Number of Basis Vectors in the Coordinate System and have as many Components/Elements as the Square of the Number of Basis Vectors in the Coordinate System.
4. For Cartesian Coordinate Systems, both Metric Tensor and Inverse Metric Tensor are Identity Matrices.
5. For Orthogonal Coordinate Systems, both Metric Tensor and Inverse Metric Tensor are Diagonal Matrices.
6. For 2D Coordinate Systems, the Metric Tensor is used for Representing the Differential Area (denoted by \(dA\)) as follows
   
   \(dA = \sqrt{|g|}dudv = |J|dudv\)   ...(3)
   
   where
   
   \(|g|= \) Determinant of the Metric Tensor
   
   \(|J|= \) Determinant of the Jacobian Matrix
   
   \(u, v= \) Coordinate Variables of the given 2D Coordinate System
7. For 3D Coordinate Systems, the Metric Tensor is used for Representing the Differential Volume (denoted by \(dV\)) as follows
   
   \(dV = \sqrt{|g|}dudvdw = |J|dudvdw\)   ...(4)
   
   where
   
   \(|g|= \) Determinant of the Metric Tensor
   
   \(|J|= \) Determinant of the Jacobian Matrix
   
   \(u, v, w= \) Coordinate Variables of the given 3D Coordinate System
8. For \(N\) Dimensional Coordinate Systems (where \(N \geq 2\)), the Metric Tensor is used for Representing the Differential Arc Length of a Curve (denoted by \(ds\)) as follows
   
   \(ds = \sqrt{g_{ij} {\Large \frac{du_i}{dt}}{\Large \frac{du_j}{dt}}}dt\)   ...(5)
   
   where
   
   \(g_{ij}= \) Components of Metric Tensor for the Coordinate System
   
   \(i,j = \) Indices whose Values can Range from \(1\) to \(N\)
   
   \(u_1,u_2,u_3...u_N= \) Coordinate Variables of the given \(N\) Dimensional Coordinate System
   
   \(t= \) Variable that Parameterises Coordinate Variables of the given Coordinate System
   
   For \(2\) Dimensional Coordinate Systems, the Differential Arc Length of a Curve \(ds\) as given in equation (5) above expands to
   
   \(ds = \sqrt{ g_{11} {\Large \frac{du_1}{dt}}{\Large \frac{du_1}{dt}} + g_{12} {\Large \frac{du_1}{dt}}{\Large \frac{du_2}{dt}} + g_{21} {\Large \frac{du_2}{dt}}{\Large \frac{du_1}{dt}} + g_{22} {\Large \frac{du_2}{dt}}{\Large \frac{du_2}{dt}}}dt\)
   
   \(\Rightarrow ds = \sqrt{ g_{11} {({\Large \frac{du_1}{dt}})}^2 + 2 g_{12} {\Large \frac{du_1}{dt}}{\Large \frac{du_2}{dt}} + g_{22} {({\Large \frac{du_2}{dt}})}^2 }dt\)   (since \(g\) is Symmetric \(g_{12}=g_{21}\)) ...(6)
   
   For \(3\) Dimensional Coordinate Systems, the Differential Arc Length of a Curve \(ds\) as given in equation (7) above expands to
   
   \(ds = \sqrt{ g_{11} {\Large \frac{du_1}{dt}}{\Large \frac{du_1}{dt}} + g_{12} {\Large \frac{du_1}{dt}}{\Large \frac{du_2}{dt}} + g_{13} {\Large \frac{du_1}{dt}}{\Large \frac{du_3}{dt}} + g_{21} {\Large \frac{du_2}{dt}}{\Large \frac{du_1}{dt}} + g_{22} {\Large \frac{du_2}{dt}}{\Large \frac{du_2}{dt}} + g_{23} {\Large \frac{du_2}{dt}}{\Large \frac{du_3}{dt}} + g_{31} {\Large \frac{du_3}{dt}}{\Large \frac{du_1}{dt}} + g_{32} {\Large \frac{du_3}{dt}}{\Large \frac{du_2}{dt}} + g_{33} {\Large \frac{du_3}{dt}}{\Large \frac{du_2}{dt}} }dt\)
   
   \(\Rightarrow ds = \sqrt{ g_{11} {({\Large \frac{du_1}{dt}})}^2 + g_{22} {({\Large \frac{du_2}{dt}})}^2 + g_{33} {({\Large \frac{du_3}{dt}})}^2 + 2 g_{12} {\Large \frac{du_1}{dt}}{\Large \frac{du_2}{dt}} + 2 g_{13} {\Large \frac{du_1}{dt}}{\Large \frac{du_3}{dt}} + 2 g_{23} {\Large \frac{du_2}{dt}}{\Large \frac{du_3}{dt}} }dt\)   (since \(g\) is Symmetric \(g_{12}=g_{21}\), \(g_{13}=g_{31}\) and \(g_{23}=g_{32}\)) ...(7)