Example: ( $\psi , \theta , \zeta$) coordinates

Suppose $(\psi , \theta , \zeta )$ is an arbitrary general coordinate system. Then the covariant basis vectors are $\nabla \psi$, $\nabla \theta$, and $\nabla \zeta$, and the corresponding contravariant basis vectors are written as

$$\frac{\partial \mathbf{r}}{\partial \psi} =\mathcal{J} \nabla \theta \times \nabla \zeta,$$ (116)

$$\frac{\partial \mathbf{r}}{\partial \theta} =\mathcal{J} \nabla \zeta \times \nabla \psi,$$ (117)

and

$$\frac{\partial \mathbf{r}}{\partial \zeta} =\mathcal{J} \nabla \psi \times \nabla \theta .$$ (118)

In Einstein notation, contravariant basis vectors are denoted with upper indices as

$$\mathbf{e}^{\psi} \equiv \nabla \psi ; \mathbf{e}^{\theta} \equiv \nabla \theta ; \mathbf{e}^{\zeta} \equiv \nabla \zeta .$$ (119)

and the covariant basis vectors are denoted with low indices as

$$\mathbf{e}_{\psi} \equiv \frac{\partial \mathbf{r}}{\partial \psi} ; \mathbf{e}_{\theta} \equiv \frac{\partial \mathbf{r}}{\partial \theta} ; \mathbf{e}_{\zeta} \equiv \frac{\partial \mathbf{r}}{\partial \zeta} .$$

Then the orthogonality relation, Eq. (106), is written as

$$\mathbf{e}^{\alpha} \cdot \mathbf{e}_{\beta} = \delta_{\alpha \beta} .$$ (120)

Then, in term of the contravairant basis vectors, $\mathbf{A}$ is written

$$\mathbf{A}= A_{\psi} \mathbf{e}^{\psi} + A_{\theta} \mathbf{e}^{\theta} + A_{\zeta} \mathbf{e}^{\zeta},$$ (121)

where $A_{\psi} =\mathbf{A} \cdot \mathbf{e}_{\psi}$, $A_{\theta} =\mathbf{A} \cdot \mathbf{e}_{\theta}$, and $A_{\zeta} =\mathbf{A} \cdot \mathbf{e}_{\zeta}$. Similarly, in term of the covariant basis vectors, $\mathbf{A}$ is written

$$\mathbf{A}= A^{\psi} \mathbf{e}_{\psi} + A^{\theta} \mathbf{e}_{\theta} + A^{\zeta} \mathbf{e}_{\zeta},$$ (122)

where $A^{\psi} =\mathbf{A} \cdot \mathbf{e}^{\psi}$, $A^{\theta} =\mathbf{A} \cdot \mathbf{e}^{\theta}$, and $A^{\zeta} =\mathbf{A} \cdot \mathbf{e}^{\zeta}$.

[In passing, we note that $\Psi \equiv A_{\phi} R$ is the covariant toroidal component of $\mathbf{A}$ in cylindrical coordinates $(R, \phi, Z)$. The proof is as follows. Note that the covariant form of $\mathbf{A}$ should be expressed in terms of the contravariant basis vector ($\nabla R$, $\nabla \phi$, and $\nabla Z$), i.e.,

$$\mathbf{A}= A_1 \nabla R + A_2 \nabla \phi + A_3 \nabla Z.$$ (123)

where $A_2$ is the covariant toroidal component of $\mathbf{A}$. To obtain $A_2$, we take scalar product of Eq. (123) with $\partial \mathbf {r}/ \partial \phi$ and use the orthogonality relation (106), which gives

$$\mathbf{A} \cdot \frac{\partial \mathbf{r}}{\partial \phi} = A_2 .$$ (124)

In cylindrical coordinates $(R, \phi, Z)$, the location vector is written as

$$\mathbf{r} (R, Z, \phi) = R \hat{\mathbf{e}}_R (\phi) + Z \hat{\mathbf{e}}_Z .$$ (125)

Using this, we obtain

$$\frac{\partial \mathbf{r}}{\partial \phi} = R \hat{\mathbf{e}}_{\phi},$$ (126)

Use Eq. (126) in Eq. (124) giving

$$A_2 = A_{\phi} R,$$ (127)

which indicates that $\Psi = A_{\phi} R$ is the covariant toroidal component of the vector potential.]