Transformation

Description

Vector

Again I consider the radius vector:

$$\vectorcompbasis,$$

and its small displacement (which is known as the line segment):

$$d\vec{r} = \sum_i dx^i \vec{e}_i.$$

By using

$$dx^i = \sum_j \pder{x^i}{X^j} dX^j,$$

I have

$$\begin{split}d\vec{r} & = \sum_i \left( \sum_j \pder{x^i}{X^j} dX^j \right) \vec{e}_i \\ & = \sum_{ij} \pder{x^i}{X^j} dX^j \vec{e}_i \\ & = \sum_j dX^j \left( \sum_i \pder{x^i}{X^j} \vec{e}_i \right).\end{split}$$

Since the same line segment can be written as

$$d\vec{r} = \sum_j dX^j \vec{E}_j,$$

I obtain

$$\fromctog.$$

Similarly, I have

$$\fromgtoc.$$

The following relations for the normalised vector are obvious from the results.

$$\vec{\hat{E}}_i = \frac{1}{H_i} \sum_j \pder{x^j}{X^i} \vec{e}_j,$$

$$\vec{e}_i = \sum_j H_j \pder{X^j}{x^i} \vec{\hat{E}}_j.$$

Component

Assigning the relation

$$\fromgtoc$$

to

$$\vec{u} = \sum_i u^i \vec{e}_i,$$

and assigning the other relation

$$\fromctog$$

to

$$\vec{u} = \sum_i U^i \vec{E}_i$$

yield

$$\vec{u} = \sum_i \left( \sum_j u^j \pder{X^i}{x^j} \right) \vec{E}_i$$

and

$$\vec{u} = \sum_i \left( \sum_j U^j \pder{x^i}{X^j} \right) \vec{e}_i,$$

respectively.

By taking the inner product, the following relations are obtained:

$$U^i = \sum_j \pder{X^i}{x^j} u^j,$$

$$u^i = \sum_j \pder{x^i}{X^j} U^j.$$

The normalised relations are

$$\hat{U}^i = H_i \sum_j \pder{X^i}{x^j} u^j,$$

$$u^i = \sum_j \frac{1}{H_j} \pder{x^i}{X^j} \hat{U}^j.$$

Transformation matrix

Taking the inner product of the relation

$$\fromctog$$

and $\vec{e}_k$ gives

$$\begin{split}\vec{E}_i \cdot \vec{e}_k & = \sum_j \pder{x^j}{X^i} \vec{e}_j \cdot \vec{e}_k \\ & = \sum_j \pder{x^j}{X^i} \delta_{jk} \\ & = \pder{x^k}{X^i},\end{split}$$

while the inner product between the relation

$$\fromgtoc$$

and $\vec{E}_k$ yields

$$\begin{split}\vec{e}_i \cdot \vec{E}_k & = \sum_j \pder{X^j}{x^i} \vec{E}_j \cdot \vec{E}_k \\ & = \sum_j \pder{X^j}{x^i} H_j H_k \delta_{jk} \\ & = \pder{X^k}{x^i} H_k H_k,\end{split}$$

where the orthogonality

$$\orthogonal$$

is adopted.

By comparing these two relations (note that the indices are dummy and thus are interchangeable), I obtain

$$\jacobiconv,$$

giving the relation of the transformation matrix and its inversed one.

Jacobian determinant

I define the determinant of the transformation matrix $J$ as

$$J \equiv \prod_i H_i.$$

Example

• Cylindrical coordinates
• Rectilinear coordinates
• Application