The Cartesian coordinate system uses the coordinate axis vectors
The coordinate axis of the cylindrical coordinate system is dependent on the position vector
We have already derived the unit vectors of the cylindrical coordinate systems in Eq. 7.23 and Eq. 7.24. We have also learned that the z-coordinate is identical to the coordinate of the Cartesian coordinate system. We found
As stated, the unit vectors depend on the unit vectors of the Cartesian coordinate system. In order to derive the velocity vector
Please note that we have to use the chain rule of differentiation here (see Eq. 3.14). As you can see, the derivatives can be written as functions of the coordinate axes. Using these we can now formulate the position vector
where we have used Eq. 7.54, Eq. 7.55, and Eq. 7.56 to convert the time-derivative of the coordinate axes into functions of the axes only. Eq. 7.57, Eq. 7.58, and Eq. 7.59 are the sought position, velocity, and acceleration vectors in the cylindrical coordinate system.
Given the fact that polar coordinates (see Fig. 7.4c) are a special case of cylindrical coordinates, we can note down the position, velocity, and acceleration vectors by copying the vectors from the cylindrical coordinates while ignoring the z-component. We therefore find from Eq. 7.57, Eq. 7.58, and Eq. 7.59
We have already derived the unit vectors of the spherical coordinate system (see Fig. 7.4d) given by Eq. 7.40, Eq. 7.42, and Eq. 7.41 as
where we again formulate the time-derivatives that we will require for finding the position, velocity, and acceleration vectors
The last expression that results in Eq. 7.65 is not intuitive and the replacing of
Eq. 7.66, Eq. 7.67, and Eq. 7.68 are the sought vectors for position
The Jacobian matrix or simply Jacobian is a matrix which is required for the conversion of surface and volume integrals from one coordinate system to another. For a vector
As you can see, the Jacobian matrix sums up all the changes of each component of the vector
We will begin with the Jacobian matrix for the conversion of Cartesian to cylindrical coordinates. We have already determined the vector (see Eq. 7.22) in Cartesian coordinates that is
Now we build the Jacobian for this vector given according to Eq. 7.69 by
As stated, the Jacobian matrix allows us to express a relationship between the infinitesimal vectors
One can easily confirm that this calculus is correct by performing the matrix/vector multiplication which yields the total differentials according to Eq. 3.12. If the inverse transformation is to be made, the inverse Jacobian matrix J− 1 must be used. Without actually deriving the matrix (which is straightforward, but a little tedious) the inverse Jacobi matrix J− 1 is given by
We will discuss examples of matrix inversion in section 25.2.2. The inverse Jacobian is used to transfer the infinitesimal vector
which is, again, simply the total differential (see Eq. 3.12). Inverse matrices can only be constructed if the determinant of the matrix is nonzero (see the following section on the Jacobian determinant for a quick review of what exactly the determinant of a matrix is). There exist several methods for constructing the inverse matrix of a mix with nonzero determinant, e.g., the Gauß-Jordan1 elimination (see section 25.2.2). In case of the Jacobian constructed for our matrix, the inverse matrix is given by Eq. 7.72. As stated, Jacobian and inverse Jacobian are used to transfer the infinitesimal vectors between the two coordinate systems according to
If you perform the matrix multiplication J·J− 1 you would find the unit matrix.
Jacobian Determinant. Besides the Jacobian matrix, we will also require the Jacobian determinant which is given by
The Jacobian determinant is used extensively during integral transformation as it is an important component of the Gauss’s theorem that we discussed in section 7.2.1. Returning to the Jacobian found for the transformation of Cartesian to cylindrical coordinates (see Eq. 7.70), the Jacobian determinant can be calculated to be
So in fact the Jacobian matrix of Eq. 7.70 has a nonzero determinant and thus the inverse can be constructed. It is given by Eq. 7.72.
Again, we can simplify the derivation of the Jacobian and the inverse Jacobian for the conversion of Cartesian to polar coordinates by noting that polar coordinates are merely a special case of cylindrical coordinates. Both matrices are 2 × 2 matrices given by
and
Jacobian Determinant. The Jacobian determinant is given by
We will now derive the Jacobian matrix for the conversion of Cartesian to spherical coordinates using the position vector (see Eq. 7.78) in Cartesian coordinates given by
Now we built the Jacobian for this vector given according to Eq. 7.69 given by
The inverse matrix is given by
Jacobian Determinant. The determinant of Eq. 7.79 is given by
In this section, we will convert the most common operators into cylindrical, polar, and spherical coordinates. As you will see, transferring operators from one coordinate system to another is not difficult, but it involves quite a bit of rewriting of partial and total differentials. The operators which we require are the gradient, the divergence, the curl, and the Laplace operator. We will also transform the material derivative. We introduced these operators in section 7.1.3.
We will begin by converting the gradient of the scalar ψ defined as
We already discussed this operator for Cartesian coordinates where ψ (x, y, z) is a function of x, y, and z. When converted to a function of r, φ, and z we obtain ψ (r, φ, z). Consequently we need to use the total differentials given by
As you can see, we require a couple of partial derivatives that we can obtain from the inverse Jacobian (see Eq. 7.72) from which we find
Inserting these partial differentials into Eq. 7.83, Eq. 7.84, and Eq. 7.85 we find
which we can reinsert into Eq. 7.82 to find
In the last step, we need to express the basis vectors
Eq. 7.90 gives the gradient in cylindrical coordinates.
Next we derive the divergence
in cylindrical coordinates. Here we have used Eq. 7.32 to convert the basis vectors. The divergence is given as
where we need to transfer the partial differentials with respect to x, y, and z to partial differentials of r, φ, and z. For this we find
We have derived a couple of additional partial differentials which we do not require at this point. However, we will require them when deriving the curl in cylindrical coordinates. Using the derived partial differentials we can rewrite Eq. 7.92 to
Eq. 7.93 gives the divergence in cylindrical coordinates with Eq. 7.94 being a slightly more compact notation.
In the next step, we will derive the curl in cylindrical coordinates. For this we use the vector
From Eq. 7.5 we know the curl is defined as
where we have replaced the basis vectors
from which we can reassemble the terms we require for Eq. 7.95 as
If we reinsert these equations into Eq. 7.95 we obtain