Scale factors for the Oblate Spheroidal Coordiante system

Question

I need the scale factors for the Oblate Spheroidal Coordiante system. Using Wolfram Mathematica Mathematica, where a is any constant and is the elliptical focus point. The code returns the following result:

\begin{array}{l} \frac{a \sqrt{\cos (2 \eta )+\cosh (2 \xi )}}{\sqrt{2}} \\ \frac{a \sqrt{\cos (2 \eta )+\cosh (2 \xi )}}{\sqrt{2}} \\ a \sin (\eta ) \cosh (\xi ) \\ \end{array}

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"ξ">0 ,0<"η"<π ,-π<"φ"≤π

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But this result is different from what was published in the MathWorld Encyclopedia

Also contradicts what is published at Field Theory Handbook DOI: 10.1007/978-3-642-83243-7

Can anyone comment which is correct and if there might be a bug?

UPDATE 1

to @Jiaxin Zhong Here is the math code returning various parameters including scale factors.

 CoordinateTransformData[{{"OblateSpheroidal", {\[FormalA]}}, 3} -> 
  "Cartesian", "Mapping"]

CoordinateTransformData[{{"OblateSpheroidal", 1}, 3} -> 
  "Cartesian", "Mapping"]

toOblatefromSpherical = 
  CoordinateTransformData["Cartesian" -> {{"OblateSpheroidal", 1}, 3},
    "Mapping"];
%[{1, Pi/4, Pi/6}] // Simplify

fromOblatetoSpherical = 
  CoordinateTransformData[{{"OblateSpheroidal", 1}, 3} -> "Cartesian",
    "Mapping"];
param = CoordinateChartData[{{"OblateSpheroidal", {\[FormalA]}}, 
    "Euclidean", 3}, "StandardCoordinateNames"];
param // Column
(*sph=fromOblatetoSpherical@param//Column;*)
sph = 
 fromOblatetoSpherical@param

Thread[% == {x, y, z}] // Simplify // MatrixForm

sph2 = Simplify[sph /. x_String :> ToExpression[x]] // MatrixForm

CoordinateChartData[{{"OblateSpheroidal", {\[FormalA]}}, "Euclidean", 
   3}, "CoordinateRangeAssumptions"]@param

CoordinateChartData[{{"OblateSpheroidal", {a}}, "Euclidean", 3}, 
   "ScaleFactors", param] /. x_String :> ToExpression[x] // Column

CoordinateChartData[{{"OblateSpheroidal", {a}}, "Euclidean", 3}, 
   "Metric", param] /. x_String :> ToExpression[x] // MatrixForm

Answer 1

They all can be true because there are kinds of definitions of oblate coordinates.

The best way to check the scale factors depends on the definition of oblate coordinates you adopt!

There exist 3 canonical definitions of oblate spheroidal coordinates. Each of them can depict a oblate spheroid as the 3 coordinates (such as $\eta,\xi,\varphi$) varying in its domain. You may check here for details: Oblate spheroidal coordinates@Wikipedia.

Different definitions serve different specific problems.

For example, the following definition is one of my favorite: (called Definition $(\eta,\xi,\varphi)$)

Let $0\leq\xi<\infty, -1\leq\eta\leq1, 0\leq\varphi<2\pi$. $$ \begin{cases} x=a\left[(1-\eta^2)(1+\xi^2)\right]^{1/2}\cos\varphi,\\ y=a\left[(1-\eta^2)(1+\xi^2)\right]^{1/2}\sin\varphi,\\ z=a\eta\xi. \end{cases} $$

I don't know how you get your answer on Mathematica, but I am wondering there may be some typos on the scale factors you post.

You may check the scale factors according to the definition.

Definition. The scale factors of $(\eta,\xi,\varphi)$ over $(x,y,z)$ are: $$\begin{cases} h_\eta=\sqrt{\left(\dfrac{\partial x}{\partial\eta}\right)^2+\left(\dfrac{\partial y}{\partial\eta}\right)^2+\left(\dfrac{\partial z}{\partial\eta}\right)^2},\\ h_\xi=\sqrt{\left(\dfrac{\partial x}{\partial\xi}\right)^2+\left(\dfrac{\partial y}{\partial\xi}\right)^2+\left(\dfrac{\partial z}{\partial\xi}\right)^2},\\ h_\varphi=\sqrt{\left(\dfrac{\partial x}{\partial\varphi}\right)^2+\left(\dfrac{\partial y}{\partial\varphi}\right)^2+\left(\dfrac{\partial z}{\partial\varphi}\right)^2}. \end{cases}$$

Therefore, the corresponding scale factors of the example mentioned above are:

$$ \begin{cases} h_\eta=a\sqrt{\dfrac{\xi^2+\eta^2}{1-\eta^2}},\\ h_\xi=a\sqrt{\dfrac{\xi^2+\eta^2}{\xi^2+1}},\\ h_\varphi=a\sqrt{(1-\eta^2)(\xi^2+1)}. \end{cases} $$