Covariant and Contravariant Components

Relevant parts to questions...

• Using $A=A^i{\mathbf{e}}_i=A_i{\mathbf{e}}^i$ to switch between expansions.
• Computing components via dot products: $A^i=\mathbf{A}\cdot {\mathbf{e}}^i$ and $A_i=\mathbf{A}\cdot {\mathbf{e}}_i$.
• Using the Metric Tensor to convert between them: $A_i=g_{ik}{A}^k$, $A^i=g^{ik}{A}_k$.
• Remembering upper indices = contravariant, lower indices = covariant.

In a generalised (possibly non-orthonormal) coordinate system, every vector $\mathbf{A}$ has two equally-valid expansions - one against the original basis $\{{\mathbf{e}}_i\}$ and one against its dual basis $\{{\mathbf{e}}^i\}$, defined by ${\mathbf{e}}_i\cdot {\mathbf{e}}^k={\delta }_i^k$:

$$\boxed{\mathbf{A}=A^i{\mathbf{e}}_i=A_i{\mathbf{e}}^i}$$

• Contravariant components $A^i$::the coefficients in the expansion along ${\mathbf{e}}_i$.
• Covariant components $A_i$::the coefficients in the expansion along ${\mathbf{e}}^i$.

Because of the dual-basis identity, the individual components can be pulled out by a single dot product:

$$A^i=\mathbf{A}\cdot {\mathbf{e}}^i,A_i=\mathbf{A}\cdot {\mathbf{e}}_i$$

Converting Between the Two

The Metric Tensor performs the conversion:

$$\boxed{A_i=g_{ik}{A}^k,A^i=g^{ik}{A}_k}$$

$g_{ik}$ lowers a contravariant index to a covariant one; $g^{ik}$ raises the reverse direction (see Index Raising and Lowering).

Properties

• Orthonormal-basis shortcut::$g_{ik}={\delta }_{ik}$, so $A_i=A^i$. In Cartesian coordinates the distinction disappears - this is why earlier module material ignored the upper/lower split.
• Transformation rules differ::$A_i^{\prime }=L_{i^{\prime }}^j{A}_j$ (covariant) and $A^{\prime i}=L_j^{i^{\prime }}{A}^j$ (contravariant). Contravariant pieces transform with the inverse of the basis matrix.
• Inner products are type-aware::$\mathbf{A}\cdot \mathbf{B}=A^i{B}_i=A_i{B}^i=g_{ik}{A}^i{B}^k=g^{ik}{A}_i{B}_k$ - you can always pair one raised with one lowered.
• Can’t add mismatched types::$A_i+B^i$ is not a tensor (see Tensor Transformation Rule): one term transforms with $L$, the other with $L^{-1}$.

Applications

1. Writing vector operations cleanly in any coordinate system, by pairing one raised with one lowered index.
2. Identifying tensorial objects: sums must preserve index position; contractions require an upper index paired with a lower one.
3. Angle between vectors::$\cos (\mathbf{A},\mathbf{B})=\frac{A^i{B}_i}{\sqrt{A^i{A}_i}\sqrt{B^i{B}_i}}$.

Finding both kinds of components

Given basis ${\mathbf{e}}_1=(2,0,1{)}^T,{\mathbf{e}}_2=(1,3,0{)}^T,{\mathbf{e}}_3=(0,0,4{)}^T$ and vector $\mathbf{J}=(8,0,4{)}^T$ in the enclosing Cartesian frame:

Covariant (dot with originals): $J_1=\mathbf{J}\cdot {\mathbf{e}}_1=20$, $J_2=8$, $J_3=16$.

Contravariant (dot with duals ${\mathbf{e}}^1=(1/2,-1/6,0),{\mathbf{e}}^2=(0,-1/3,0),{\mathbf{e}}^3=(-1/8,1/24,1/4)$): $J^1=4$, $J^2=0$, $J^3=0$.

$\boxed{(J_1,J_2,J_3)=(20,8,16),(J^1,J^2,J^3)=(4,0,0)}$.

Upper index: original basis. Lower index: dual basis. Metric bridges both.