Joel E. Tohline
tohline@rouge.phys.lsu.edu
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The information contained in this appendix can be found in a very large number of published references. We have drawn the following primarily from the presentation in Appendix 1.B of Binney and Tremaine (1987); hence, if clarification is needed of any of the following, Binney and Tremaine should be consulted first.
On a few accompanying pages, we have detailed the specific form that this operator takes in Cartesian, cylindrical, or spherical coordinates.


The
accompanying application permits you to determine the gradient
of virtually any analytically expressible scalar function G(x),
in any one of fourteen
different orthogonal coordinate systems, utilizing
the symbolic manipulation capabilities of
Mathematica^{¨}.
While this very general, interactive tool is potentially very powerful,
it is strongly recommended that you not try to use it
until you have had some experience using the
related, but less general, applications that have been taylored for
Cartesian,
cylindrical, or
spherical coordinates.

Consider any vector function F(x) with orthogonal components (F_{1}, F_{2}, F_{3}); the divergence of this function is usually written as Ñ×F. The general (curvilinear coordinate) form of the divergence operator that is valid in any orthogonal coordinate system is:
[Equation VI.M.11]
On a few accompanying pages, we have detailed the specific form that this operator takes in Cartesian, cylindrical, or spherical coordinates.


The
accompanying application permits you to determine the divergence
of virtually any analytically expressible vector function F(x),
in any one of fourteen
different orthogonal coordinate systems, utilizing
the symbolic manipulation capabilities of
Mathematica^{¨}.
While this very general, interactive tool is potentially very powerful,
it is strongly recommended that you not try to use it
until you have had some experience using the
related, but less general, applications that have been taylored for
Cartesian,
cylindrical, or
spherical coordinates.

Consider any scalar function G(x); the Laplacian of this function is:
[Equation VI.M.12]
On a few accompanying pages, we have detailed the specific form that this operator takes in Cartesian, cylindrical, or spherical coordinates.


The
accompanying application permits you to determine the Laplacian
of virtually any analytically expressible scalar function G(x),
in any one of fourteen
different orthogonal coordinate systems, utilizing
the symbolic manipulation capabilities of
Mathematica^{¨}.
While this very general, interactive tool is potentially very powerful,
it is strongly recommended that you not try to use it
until you have had some experience using the
related, but less general, applications that have been taylored for
Cartesian,
cylindrical, or
spherical coordinates.

When dealing with any fluid dynamical system, it is extremely important that you understand the physical relationship between the the total and partial time derivatives that is implied by this mathematical expression. If you don't understand the difference between partial and total time derivatives in this context, read the accompanying discussion of Lagrangian vs. Eulerian representations.
As detailed, for example, in Part I (page 26; equation 1.3.6) of Morse and Feshbach (1953), given the definitions of x_{i} and h_{i} for a specified coordinate system, the various coefficients in this expression can be evaluated according to the following tabulated expressions:
Evaluating: (¶_{xj}e_{i})  
¶_{x1}

¶_{x2}

¶_{x3}



e_{1}

 e_{2} (1/h_{2})¶_{x2}h_{1}
+  e_{3} (1/h_{3})¶_{x3}h_{1} 
e_{2} (1/h_{1})¶_{x1}h_{2}

e_{3} (1/h_{1})¶_{x1}h_{3}

e_{2}

e_{1} (1/h_{2})¶_{x2}h_{1}

 e_{3} (1/h_{3})¶_{x3}h_{2}
+  e_{1} (1/h_{1})¶_{x1}h_{2} 
e_{3} (1/h_{2})¶_{x2}h_{3}

e_{3}

e_{1} (1/h_{3})¶_{x3}h_{1}

e_{2} (1/h_{3})¶_{x3}h_{2}

 e_{1} (1/h_{1})¶_{x1}h_{3}
+  e_{2} (1/h_{2})¶_{x2}h_{3} 
[Equation VI.M.16]
Hence, we deduce that for any curvilinear coordinate system,
De_{1}  =  e_{2}A + e_{3}B, 
De_{2}  =   e_{1}A + e_{3}C, 
De_{3}  =   e_{1}B  e_{2}C, 
[Equation VI.M.17]
where:  
A  º [Dx_{2}] (1/h_{1}) ¶_{x1}h_{2}  [Dx_{1}] (1/h_{2}) ¶_{x2}h_{1} 
B  º [Dx_{3}] (1/h_{1}) ¶_{x1}h_{3}  [Dx_{1}] (1/h_{3}) ¶_{x3}h_{1} 
C  º [Dx_{3}] (1/h_{2}) ¶_{x2}h_{3}  [Dx_{2}] (1/h_{3}) ¶_{x3}h_{2} 
As an example, given the definition of the position vector [VI.M.7] x in any curvilinear coordinate system, the above relations must be utilized when determining how to express the vector velocity (v º dx/dt) or the vector acceleration (a º d^{2}x/dt^{2}) in that coordinate system. 

On a few accompanying pages, we have detailed the derived expressions for v and a in Cartesian, cylindrical, and spherical coordinates.
When "D" operates on any scalar function G(x), determining how to handle the v×Ñ "convective operator" on the righthandside is straightforward. Specifically, for any orthogonal coordinate system,
[Equation VI.M.19]
However, it is less obvious what form the convective operator should take when "D" operates on a vector function. In general, when v×Ñ operates on the vector F(x), the result is a vector whose j^{th} component takes the following form:
[Equation VI.M.20]
On a few accompanying pages, we have detailed the specific form that this operator takes in Cartesian, cylindrical, and spherical coordinates.
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