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Covariant and Contravariant Components of a Vector

  1. Lets consider any Vector A having components A1,A2,...,An given in terms of Basis Vector Set e1, e2, ... , en (as given in equation (1) below)

    A=A1e1+A2e2+...+Anen   ...(1)

    The Components of Vector A (i.e. A1,A2,...,An) 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 e1, 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 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 e1, e2, ... , en form an Orthonormal Vector Set, then the Basis Vector Sets e1, 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.
  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=e1⋅A=A1(e1.e1)+A2(e1.e2)+...+An(e1.en)
    A′2=e2⋅A=A1(e2.e1)+A2(e2.e2)+...+An(e2.en)
    ⋮
    A′n=en⋅A=A1(en.e1)+A2(en.e2)+...+An(en.en)

    The above set of equations can also be written in form of Matrix Equation as follows

    [A′1A′2⋮A′n]=[e1.e1e1.e2⋯e1.ene2.e1e2.e2⋯e2.en⋮⋮⋱⋮en.e1en.e2⋯en.en][A1A2⋮An]  ...(3)

    In the equation (3) above , the First Matrix in the Matrix Product is the Metric Tensor corresponding to Basis Vector Set e1, 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 Vector A (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

    A1=e′1⋅A=A′1(e′1.e′1)+A′2(e′1.e′2)+...+A′n(e′1.e′n)
    A2=e′2⋅A=A′1(e′2.e′1)+A′2(e′2.e′2)+...+A′n(e′2.e′n)
    ⋮
    An=e′n⋅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

    [A1A2⋮An]=[e′1.e′1e′1.e′2⋯e′1.e′ne′2.e′1e′2.e′2⋯e′2.e′n⋮⋮⋱⋮e′n.e′1e′n.e′2⋯e′n.e′n][A′1A′2⋮A′n]  ...(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 e1, e2, ... , en on Both Sides we get

    [A1A2⋮An]=[e1.e1e1.e2⋯e1.ene2.e1e2.e2⋯e2.en⋮⋮⋱⋮en.e1en.e2⋯en.en]−1[A′1A′2⋮A′n]  ...(5)

    From equations (4) and (5) above we get

    [e1.e1e1.e2⋯e1.ene2.e1e2.e2⋯e2.en⋮⋮⋱⋮en.e1en.e2⋯en.en]−1=[e′1.e′1e′1.e′2⋯e′1.e′ne′2.e′1e′2.e′2⋯e′2.e′n⋮⋮⋱⋮e′n.e′1e′n.e′2⋯e′n.e′n]  ...(6)

    And therefore

    [e′1.e′1e′1.e′2⋯e′1.e′ne′2.e′1e′2.e′2⋯e′2.e′n⋮⋮⋱⋮e′n.e′1e′n.e′2⋯e′n.e′n]−1=[e1.e1e1.e2⋯e1.ene2.e1e2.e2⋯e2.en⋮⋮⋱⋮en.e1en.e2⋯en.en]  ...(7)

    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.
  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

    [e′1e′2⋯e′n]=[e1e2⋯en][e1.e1e1.e2⋯e1.ene2.e1e2.e2⋯e2.en⋮⋮⋱⋮en.e1en.e2⋯en.en]−1  ...(8)

    [e1e2⋯en]=[e′1e′2⋯e′n][e′1.e′1e′1.e′2⋯e′1.e′ne′2.e′1e′2.e′2⋯e′2.e′n⋮⋮⋱⋮e′n.e′1e′n.e′2⋯e′n.e′n]−1  ...(9)
  4. You can use the Covariant Vector Components Calculator to calculate Covariant Components of a Vector.
Related Calculators
Covariant Vector Components Calculator
Related Topics
Gramian Matrix / Gram Matrix / Metric Tensor,    Dual of Vector/Matrix,    Introduction to Vector Algebra
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