Plotting Vector Fields for the Basis Vectors of Different Coordinate Systems

Question

In short, I am hoping to plot out vector fields for the basis vectors of my elliptic cylindrical coordinate system.

For more information on the coordinate system I mentioned, Wolfram|Alpha has a site that quickly provides it: https://mathworld.wolfram.com/EllipticCylindricalCoordinates.html

The basis vectors for my coordinate system are defined below.

$$\hat{\mu} = \frac{1}{a \sqrt{\sinh^2 {\mu} + \sin^2 {v}}} (a \sinh{\mu} \cos{v} \; \hat{i} + a \cosh{\mu} \sin{v} \; \hat{j})$$

$$\hat{v} = \frac{1}{a \sqrt{\sinh^2 {\mu} + \sin^2 {v}}} (-a \cosh{\mu} \sin{v} \; \hat{i} + a \sinh{\mu} \cos{v} \; \hat{j})$$

$$\hat{z} = 1 \hat{z}$$

Despite knowing my basis vectors, though, creating vector field plots in the $x$, $y$ plane for my basis vectors has been giving me a really hard time.

I realize that Wolfram provides a tutorial on the topic (i.e. https://reference.wolfram.com/language/howto/PlotAVectorField.html), however, plotting a different coordinate system (rather than Cartesian) has proven to be quite difficult as I am a novice at Mathematica.

Therefore, any assistance to help me figure out how to plot out the vector fields would be greatly appreciated. Thank you very much for reading this!

Answer 1

Let's start with the definition of the Elliptic Cylindrical coordinates, that tells us how they map to cartesian coordinates:

\[CurlyPhi] = {\[FormalA] Cosh[u] Cos[v], \[FormalA] Sinh[u] Sin[v], w}

Honestly, i mainly used [FormalA] here, because the function TransformedField that we'll use later uses it by default for the free parameter a of the coordinate system and i haven't figured out yet how to specify a different value, so this choice of naming saves us one symbol replacement later.

Alternatively, instead of giving the definition manually we could also conveniently use

\[CurlyPhi] = CoordinateTransformData[
                "EllipticCylindrical" -> "Cartesian",
                "Mapping",
                {u, v, w}
              ]

, which does the same.

Now, since we want to have the basis vectors and not just the point mappings, we need the jacobian with respect to {u,v,w} which gives us the unnormalized basis vectors:

jacobian = D[\[CurlyPhi], {{u, v, w}}];
jacobian // MatrixForm

Now we can use the builtin function TransformedField to reexpress the jacobian in cartesian coordinates for us:

basisunnormalized = 
  Simplify[
    TransformedField[
      "EllipticCylindrical" -> "Cartesian", 
      Transpose[jacobian],
      {u, v, w} -> {x, y, z}
    ],
    \[FormalA] > 0
  ]

Note that we transposed the jacobian, so that each of the three list elements holds a basis-vector (corresponding to the direction of increasing u, v and w).

We can already plot this

GraphicsRow[
  VectorPlot[#, {x, -2, 2}, {y, -2, 2}] & /@
    (basisunnormalized[[{1, 2}, {1, 2}]] /. \[FormalA] -> 1)
]

or normalize the basis vectors if needed.

Edit:

The choice to start with the jacobian was mainly based on its simpler structure compared to the normalized basis vectors, which gives simpler (plain rational fraction without trigonometric functions) expressions after invoking TransformedField. But with some extra care we can also work with the normalized basis vectors and in the process skip a lot of manual work.

The basis you gave in the questions can either be computed via normalizing our jacobian

basisnormalized = Transpose[
  Simplify[
    Normalize /@ Transpose[jacobian],
    u > 0 && v > 0 && \[FormalA] > 0
  ]
]

or directly via

basisnormalized = CoordinateTransformData[
  "EllipticCylindrical" -> "Cartesian", 
  "OrthonormalBasisRotation",
  {u, v, w}
]

giving

basisnormalized //MatrixForm

Invoking TransformedField now already works but can't get rid of the trigonometric functions Cos[2 v] and Cosh[2 u] which contain double angles. By using TrigExpand we can convert them to powers of single angle trigonometric expressions, which TransformedField has an easier time with:

basisnormalized2 = basisnormalized /. {a : (Cos | Cosh)[2 _] :> TrigExpand[a]}

or in an easier readable form

basisnormalized2 = basisnormalized /. {
  Cos[2 v] -> Cos[v]^2 - Sin[v]^2,
  Cosh[2 u] -> Cosh[u]^2 + Sinh[u]^2
};

results in

basisnormalized2 //MatrixForm

which we can use to get an expression in terms of x, y and z without trigonometric functions:

basisnormalizedcartesian = Simplify[
  TransformedField[
    "EllipticCylindrical" -> "Cartesian",
    basisnormalized2,
    {u, v, w} -> {x, y, z}
  ],
  \[FormalA] > 0
]