Cylindrical and spherical coordinate systems

X.B.1.3 Cylindrical and spherical coordinate systems

In sections X.B.1.1 and X.B.1.2, one presented vector and tensor components in Cartesian coordinate systems. The same definition is also worthy in curvilinear coordinate systems. However, the director vectors depend on the point on which the vector or tensor is attached. Conventionally, one decides that the director vectors are chosen tangent to the coordinate lines and are of unit length.

For example, for a cylindrical coordinate system, the position $p$ of a point depends on three coordinates $r$ , $z$ and $θ$ . So one has:

p = p (r, z, θ) .

Then three tangent vectors $g_{i}$ are obtained by deriving the position $p$ wrt coordinates:

\begin{align} g_{1} & = g_{r} = \frac{\partial p}{\partial r}, \\ g_{2} & = g_{θ} = \frac{\partial p}{\partial θ}, \\ g_{3} & = g_{z} = \frac{\partial p}{\partial z} . \end{align}

Finally, the three tangent vectors are normalized as follows:

e_{i} = \frac{g_{i}}{∥g_{i}∥} .

This process to define base vectors at any point can be generalized to all curvilinear coordinate systems. However, the cylindrical and spherical coordinate systems have a peculiarity: at a given point, the three base vectors $e_{i}$ are mutually orthogonal. This is not a general characteristic of curvilinear coordinate systems.

The orthogonality property of the coordinate systems one uses in FeResPost simplifies the transformation of components from one coordinate system to another. Indeed such transformations reduce to transformations between Cartesian coordinate systems. There is only one difficulty in this process: to calculate the base vectors at every point.

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