Covariant and Contravariant Components of a Vector
Lets consider any VectorA having components A1,A2,...,An given in terms of Basis Vector Sete1, e2, ... , en (as given in equation (1) below)
A=A1e1+A2e2+...+Anen ...(1)
The Components of VectorA (i.e. A1,A2,...,An) are also called Contravariant Components of Vector A.
The VectorA can also be given in terms of Basis Vector Sete′1, e′2, ... , e′n
such that the Basis Vector Matrices formed by the Basis Vector Setse1, e2, ... , en and e′1, e′2, ... , e′n are Dual Matrices of each other.
(as given in equation (2) below)
A=A′1e′1+A′2e′2+...+A′ne′n ...(2)
The Components of VectorA given in terms of Dual Basis Vector Sete′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 e1, e2, ... , en form an Orthonormal Vector Set, then the Basis Vector Setse1, e2, ... , en 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.
The Covariant Components of VectorA (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
In the equation (3) above , the First Matrix in the Matrix Product is the Metric Tensor corresponding to Basis Vector Sete1, e2, ... , en.
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 VectorA (i.e. A1,A2,...,An) 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
In the equation (4) above , the First Matrix in the Matrix Product is the Metric Tensor corresponding to Basis Vector Sete′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 Sete1, e2, ... , en on Both Sides we get
From equations (6) and (7) we get that the Metric Tensor for any Basis Vector Set e1, e2, ... , en is Inverse of Metric Tensor of its corresponding Dual Basis Vector Set e′1, e′2, ... , e′n.
Multiplying a Basis Vector Matrix with the Inverse of its Metric Tensor gives its corresponding Dual Basis Vector Matrix as given in the following