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Covariant and Contravariant Components of a Vector

1. Lets consider any Vector \(A\) having components \(A_1, A_2, ..., A_n\) given in terms of Basis Vector Set \(e_1\), \(e_2\), ... , \(e_n\) (as given in equation (1) below)
   
   \(A=A_1e_1 + A_2e_2 + ... + A_ne_n\)   ...(1)
   
   The Components of Vector \(A\) (i.e. \(A_1, A_2, ..., A_n\)) are also called Contravariant Components of Vector \(A\).
   
   The Vector \(A\) can also be given in terms of Basis Vector Set \(e_1'\), \(e_2'\), ... , \(e_n'\) such that the Basis Vector Matrices formed by the Basis Vector Sets \(e_1\), \(e_2\), ... , \(e_n\) and \(e_1'\), \(e_2'\), ... , \(e_n'\) are Dual Matrices of each other. (as given in equation (2) below)
   
   \(A=A_1'e_1' + A_2'e_2' + ... + A_n'e_n'\)   ...(2)
   
   The Components of Vector \(A\) given in terms of Dual Basis Vector Set \(e_1'\), \(e_2'\), ... , \(e_n'\) (i.e. \(A_1', A_2', ..., A_n'\) as given in equation (2) above) are called Covariant Components of Vector \(A\).
   
   Please note that if Basis Vectors \(e_1\), \(e_2\), ... , \(e_n\) form an Orthonormal Vector Set, then the Basis Vector Sets \(e_1\), \(e_2\), ... , \(e_n\) and \(e_1'\), \(e_2'\), ... , \(e_n'\) are same. Under such condition the Covariant Components of Vector \(A\) are Same as it's Contravariant Components.
2. The Covariant Components of Vector \(A\) (i.e. \(A_1', A_2', ..., A_n'\)) can be found out by Calculating the Dot Product of Vector \(A\) given in terms of it's Contravariant Components with it's coresponding Basis Vectors as follows
   
   \(A_1'=e_1\cdot A=A_1(e_1.e_1) + A_2(e_1.e_2) + ... + A_n(e_1.e_n)\)
   \(A_2'=e_2\cdot A=A_1(e_2.e_1) + A_2(e_2.e_2) + ... + A_n(e_2.e_n)\)
   \(\vdots\)
   \(A_n'=e_n\cdot A=A_1(e_n.e_1) + A_2(e_n.e_2) + ... + A_n(e_n.e_n)\)
   
   The above set of equations can also be written in form of Matrix Equation as follows
   
   \(\begin{bmatrix}A_1'\\A_2'\\\vdots\\A_n'\end{bmatrix}=\begin{bmatrix}e_1.e_1 & e_1.e_2 & \cdots & e_1.e_n\\e_2.e_1 & e_2.e_2 & \cdots & e_2.e_n\\ \vdots & \vdots & \ddots & \vdots \\ e_n.e_1 & e_n.e_2 & \cdots & e_n.e_n\end{bmatrix}\begin{bmatrix}A_1\\A_2\\\vdots\\A_n\end{bmatrix}\)  ...(3)
   
   In the equation (3) above , the First Matrix in the Matrix Product is the Metric Tensor corresponding to Basis Vector Set \(e_1\), \(e_2\), ... , \(e_n\).
   
   This implies that Column Matrix consisting of Covariant Components of any Vector can be obtained by Pre-Multiplying the Column Matrix consisting of it's Contravariant Components with the corresponding Metric Tensor.
   
   Also, the Contravariant Components of Vector \(A\) (i.e. \(A_1, A_2, ..., A_n\)) can be found out by Calculating the Dot Product of Vector \(A\) given in terms of it's Covariant Components with it's coresponding Basis Vectors as follows
   
   \(A_1=e_1' \cdot A=A_1'(e_1'.e_1') + A_2'(e_1'.e_2') + ... + A_n'(e_1'.e_n')\)
   \(A_2=e_2' \cdot A=A_1'(e_2'.e_1') + A_2'(e_2'.e_2') + ... + A_n'(e_2'.e_n')\)
   \(\vdots\)
   \(A_n=e_n' \cdot A=A_1'(e_n'.e_1') + A_2'(e_n'.e_2') + ... + A_n'(e_n'.e_n')\)
   
   The above set of equations can also be written in form of Matrix Equation as follows
   
   \(\begin{bmatrix}A_1\\A_2\\\vdots\\A_n\end{bmatrix}=\begin{bmatrix}e_1'.e_1' & e_1'.e_2' & \cdots & e_1'.e_n'\\e_2'.e_1' & e_2'.e_2' & \cdots & e_2'.e_n'\\ \vdots & \vdots & \ddots & \vdots \\ e_n'.e_1' & e_n'.e_2' & \cdots & e_n'.e_n'\end{bmatrix}\begin{bmatrix}A_1'\\A_2'\\\vdots\\A_n'\end{bmatrix}\)  ...(4)
   
   In the equation (4) above , the First Matrix in the Matrix Product is the Metric Tensor corresponding to Basis Vector Set \(e_1'\), \(e_2'\), ... , \(e_n'\).
   
   This implies that Column Matrix consisting of Contravariant Components of any Vector can be obtained by Pre-Multiplying the Column Matrix consisting of it's Covariant Components with the corresponding Metric Tensor.
   
   Now, Pre-Multiplying equation (3) above with Inverse of the Metric Tensor for Basis Vector Set \(e_1\), \(e_2\), ... , \(e_n\) on Both Sides we get
   
   \(\begin{bmatrix}A_1\\A_2\\\vdots\\A_n\end{bmatrix}=\begin{bmatrix}e_1.e_1 & e_1.e_2 & \cdots & e_1.e_n\\e_2.e_1 & e_2.e_2 & \cdots & e_2.e_n\\ \vdots & \vdots & \ddots & \vdots \\ e_n.e_1 & e_n.e_2 & \cdots & e_n.e_n\end{bmatrix}^{-1}\begin{bmatrix}A_1'\\A_2'\\\vdots\\A_n'\end{bmatrix}\)  ...(5)
   
   From equations (4) and (5) above we get
   
   \(\begin{bmatrix}e_1.e_1 & e_1.e_2 & \cdots & e_1.e_n\\e_2.e_1 & e_2.e_2 & \cdots & e_2.e_n\\ \vdots & \vdots & \ddots & \vdots \\ e_n.e_1 & e_n.e_2 & \cdots & e_n.e_n\end{bmatrix}^{-1}= \begin{bmatrix}e_1'.e_1' & e_1'.e_2' & \cdots & e_1'.e_n'\\e_2'.e_1' & e_2'.e_2' & \cdots & e_2'.e_n'\\ \vdots & \vdots & \ddots & \vdots \\ e_n'.e_1' & e_n'.e_2' & \cdots & e_n'.e_n'\end{bmatrix}\)  ...(6)
   
   And therefore
   
   \(\begin{bmatrix}e_1'.e_1' & e_1'.e_2' & \cdots & e_1'.e_n'\\e_2'.e_1' & e_2'.e_2' & \cdots & e_2'.e_n'\\ \vdots & \vdots & \ddots & \vdots \\ e_n'.e_1' & e_n'.e_2' & \cdots & e_n'.e_n'\end{bmatrix}^{-1}= \begin{bmatrix}e_1.e_1 & e_1.e_2 & \cdots & e_1.e_n\\e_2.e_1 & e_2.e_2 & \cdots & e_2.e_n\\ \vdots & \vdots & \ddots & \vdots \\ e_n.e_1 & e_n.e_2 & \cdots & e_n.e_n\end{bmatrix}\)  ...(7)
   
   From equations (6) and (7) we get that the Metric Tensor for any Basis Vector Set \(e_1\), \(e_2\), ... , \(e_n\) is Inverse of Metric Tensor of its corresponding Dual Basis Vector Set \(e_1'\), \(e_2'\), ... , \(e_n'\).
3. Multiplying a Basis Vector Matrix with the Inverse of its Metric Tensor gives its corresponding Dual Basis Vector Matrix as given in the following
   
   \(\begin{bmatrix}e_1' & e_2' & \cdots & e_n'\end{bmatrix} =\begin{bmatrix}e_1 & e_2 & \cdots & e_n\end{bmatrix}\begin{bmatrix}e_1.e_1 & e_1.e_2 & \cdots & e_1.e_n\\e_2.e_1 & e_2.e_2 & \cdots & e_2.e_n\\ \vdots & \vdots & \ddots & \vdots \\ e_n.e_1 & e_n.e_2 & \cdots & e_n.e_n\end{bmatrix}^{-1}\)  ...(8)
   
   \(\begin{bmatrix}e_1 & e_2 & \cdots & e_n\end{bmatrix} =\begin{bmatrix}e_1' & e_2' & \cdots & e_n'\end{bmatrix}\begin{bmatrix}e_1'.e_1' & e_1'.e_2' & \cdots & e_1'.e_n'\\e_2'.e_1' & e_2'.e_2' & \cdots & e_2'.e_n'\\ \vdots & \vdots & \ddots & \vdots \\ e_n'.e_1' & e_n'.e_2' & \cdots & e_n'.e_n'\end{bmatrix}^{-1}\)  ...(9)
4. You can use the Covariant Vector Components Calculator to calculate Covariant Components of a Vector.