Orthogonality relation

Next, we shall prove the following orthogonality relation

$$\nabla x^i \cdot \frac{\partial \mathbf{r}}{\partial x^j} = \delta_{i j},$$ (106)

where $\delta_{i j}$ is the Kronical delta function. [Proof: The left-hand side of Eq. (106) can be written as

$$\nabla x^i \cdot \frac{\partial \mathbf{r}}{\partial x^j} = \left[ \left( \frac{\partial x^i}{\partial x} \right) \hat{\mathb... ...^j} \hat{\mathbf{z}} + z \frac{\partial \hat{\mathbf{z}}}{\partial x^j} \right]$$

$$= \left[ \left( \frac{\partial x^i}{\partial x} \right) \hat{\mathb... ...^j} \hat{\mathbf{y}} + \frac{\partial z}{\partial x^j} \hat{\mathbf{z}} \right]$$

$$= \frac{\partial x^i}{\partial x} \frac{\partial x}{\partial x^j} +... ...\partial x^j} + \frac{\partial x^i}{\partial z} \frac{\partial z}{\partial x^j}$$

$$= \frac{\partial x^i}{\partial x^j}$$

$$= \delta_{i j},$$

where the second equality is due to

$$\frac{\partial \hat{\mathbf{x}}}{\partial x^j} = 0, \frac{\partia... ...hbf{y}}}{\partial x^j} = 0, \frac{\partial \hat{\mathbf{z}}}{\partial x^j} = 0,$$ (107)

since $\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}$ are constant vectors independent of spatial location.]

It can be proved that $\nabla x^i$ is a contravariant vector while $\partial \mathbf{r}/ \partial x^i$ is a covariant vector (I do not prove this). The orthogonality relation in Eq. (106) is fundamental to the theory of general coordinates. (The cylindrical coordinate system $(R, \phi, Z)$ is an example of general coordinates. As an exercise, we can verify that the cylindrical coordinates have the above property. In this case, $x = x_1 \cos x_2$, $y = x_1 \sin x_2$, $z = x_3$, where $x_1 \equiv R$, $x_2 \equiv \phi$, $x_3 \equiv Z$.) The orthogonality relation allows one to write the covariant basis vectors in terms of contravariant basis vectors and vice versa. For example, the orthogonality relation tells that $\partial \mathbf{r}/ \partial x^1$ is orthogonal to $\nabla x^2$ and $\nabla x^3$, thus, $\partial \mathbf{r}/ \partial x^1$ can be written as

$$\frac{\partial \mathbf{r}}{\partial x^1} = A \nabla x^2 \times \nabla x^3,$$ (108)

where $A$ is a unknown variable to be determined. To determine $A$, dotting Eq. (108) by $\nabla x^1$, and using the orthogonality relation again, we obtain

$$1 = A (\nabla x^2 \times \nabla x^3) \cdot \nabla x^1,$$ (109)

which gives

$$A = \frac{1}{(\nabla x^2 \times \nabla x^3) \cdot \nabla x^1}$$

$$= \mathcal{J}$$ (110)

Thus $\partial \mathbf{r}/ \partial x^1$ is written, in terms of $\nabla x^1$, $\nabla x^2$, and $\nabla x^3$, as

$$\frac{\partial \mathbf{r}}{\partial x^1} =\mathcal{J} \nabla x^2 \times \nabla x^3 .$$ (111)

Similarly, we obtain

$$\frac{\partial \mathbf{r}}{\partial x^2} =\mathcal{J} \nabla x^3 \times \nabla x^1$$ (112)

and

$$\frac{\partial \mathbf{r}}{\partial x^3} =\mathcal{J} \nabla x^1 \times \nabla x^2 .$$ (113)

Equations (111)-(113) can be generally written

$$\frac{\partial \mathbf{r}}{\partial x^i} =\mathcal{J} \nabla x^j \times \nabla x^k,$$ (114)

where $(i, j, k)$ represents the cyclic order in the variables $(x^1, x^2, x^3)$. Equation (114) expresses the covariant basis vectors in terms of the contravariant basis vectors. On the other hand, from Eq. (111)-(113), we obtain

$$\nabla x^i =\mathcal{J}^{- 1} \frac{\partial \mathbf{r}}{\partial x^j} \times \frac{\partial \mathbf{r}}{\partial x^k},$$ (115)

which expresses the contravariant basis vectors in terms of the covariant basis vectors.