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(Appendix E:   Kinetic Energy Components)
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<math>~\iiint  \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}^2 ~dx ~dy ~dz</math>
<math>~\iiint  \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}^2 ~dx ~dy ~dz</math>
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<math>~\iiint  \biggl\{
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\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 y^2
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- 2\biggl[ \frac{a^2}{a^2 + b^2}\biggr] 
 +
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \zeta_3 yz
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+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 z^2
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Revision as of 21:01, 5 August 2020


Contents

Steady-State 2nd-Order Tensor Virial Equations

By satisfying all six — not necessarily unique — components of the Second-Order Tensor Virial Equation, the entire set of Riemann Ellipsoids can be determined.

Whitworth's (1981) Isothermal Free-Energy Surface
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Here we are only interested in determining the equilibrium conditions of uniform-density ellipsoids that have semi-axes, ~a_1, a_2, a_3.

General Coefficient Expressions

As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, ~(a_1,a_2,a_3), of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:


~A_1


~=

~2\biggl(\frac{a_2}{a_1}\biggr)\biggl(\frac{a_3}{a_1}\biggr)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,


~A_3


~=


~2\biggl(\frac{a_2}{a_1}\biggr) \biggl[  \frac{(a_2/a_1) \sin\theta - (a_3/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,


~A_2


~=

~2 - (A_1+A_3) \, ,

where, ~F(\theta,k) and ~E(\theta,k) are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

~\theta = \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr)

      and      

~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .

[ EFE, Chapter 3, §17, Eq. (32) ]

Adopted (Internal) Velocity Field

EFE (p. 130) states that the … kinematical requirement, that the motion ~(\vec{u}), associated with ~\vec{\zeta}, preserves the ellipsoidal boundary, leads to the following expressions for its components:

~u_1

~=

~- \biggl[ \frac{a_1^2}{a_1^2 + a_2^2}\biggr] \zeta_3 x_2 + \biggl[ \frac{a_1^2}{a_1^2+a_3^2}\biggr] \zeta_2 x_3 \, ,

~u_2

~=

~- \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 x_3 + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x_1 \, ,

~u_3

~=

~- \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x_1 + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 x_2 \, .

[ EFE, Chapter 7, §47, Eq. (1) ]

Equilibrium Expressions

[EFE §11(b), p. 22] Under conditions of a stationary state, [the tensor virial equation] gives,

~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij}

~=

~- \delta_{ij}\Pi \, .

[This] provides six integral relations which must obtain whenever the conditions are stationary.

When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2nd-order TVE takes on the more general form:

~0

~=

~
2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi 
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
\, .

[ EFE, Chapter 2, §12, Eq. (64) ]

EFE (p. 57) also shows that … The potential energy tensor … for a homogeneous ellipsoid is given by

~\frac{\mathfrak{W}_{ij}}{\pi G\rho}

~=

~-2A_i I_{ij} \, ,

[ EFE, Chapter 3, §22, Eq. (128) ]

where

~I_{ij}

~=

~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,

[ EFE, Chapter 3, §22, Eq. (129) ]

is the moment of inertia tensor.

The Three Diagonal Elements

For ~i = j = 1, we have,

~0

~=

~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi 
+ \Omega^2 I_{11} - \Omega_1\Omega_k I_{k1} + 2\epsilon_{1lm}\Omega_m \int_V \rho u_lx_1 dx

 

~=

~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi + \Omega^2 I_{11} 
- \Omega_1^2I_{11} 
+ 2 \Omega_3 \int_V \rho u_2x_1 dx
- 2\Omega_2 \int_V \rho u_3x_1 dx

 

~=

~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi 
+( \Omega_2^2 + \Omega_3^2) I_{11} 
+ 2 \Omega_3 \int_V \rho u_2x_1 dx
- 2\Omega_2 \int_V \rho u_3x_1 dx

Similarly, for ~i = j = 2,

~0

~=

~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi 
+ \Omega^2 I_{22} - \Omega_2\Omega_k I_{k2} + 2\epsilon_{2lm}\Omega_m \int_V \rho u_lx_2 dx

 

~=

~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi 
+ (\Omega_1^2 + \Omega_3^2) I_{22} + 2\Omega_1 \int_V \rho u_3x_2 dx
- 2\Omega_3 \int_V \rho u_1x_2 dx

and, for ~i=j=3,

~0

~=

~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi 
+ \Omega^2 I_{33} - \Omega_3\Omega_k I_{k3} + 2\epsilon_{3lm}\Omega_m \int_V \rho u_lx_3 dx

 

~=

~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi 
+ (\Omega_1^2 + \Omega_2^2) I_{33}  + 2\Omega_2 \int_V \rho u_1x_3 dx
- 2\Omega_1 \int_V \rho u_2 x_3 dx

The Six Off-Diagonal Elements

Notice that the off-diagonal components of both ~I_{ij} and ~\mathfrak{W}_{ij} are zero. Hence, the equilibrium expression that is dictated by each off-diagonal component of the 2nd-order TVE is,

~0

~=

~
2 \mathfrak{T}_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
\, .

For example — as is explicitly illustrated on p. 130 of EFE — for ~i=2 and ~j=3,

~0

~=

~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 dx}
- 2\Omega_3 \int_V \rho u_1x_3 dx \, ,

[ EFE, Chapter 7, §47, Eq. (3) ]

whereas for ~i=3 and ~j=2,

~0

~=

~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
- 2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2 dx}
\, .

[ EFE, Chapter 7, §47, Eq. (4) ]

Given our adoption of a uniform-density configuration whose surface has a precisely ellipsoidal shape and, along with it, our adoption of the above specific prescription for the internal velocity field, ~\vec{u}, we recognize that,

~\int_V \rho u_i x_j dx

~=

~0

      if    ~i = j \, .
[ EFE, Chapter 7, §47, Eq. (5) ]

This has allowed us to set to zero one of the integrals in each of these last two expressions. In what follows, we will benefit from recognizing, as well, that,

~\mathfrak{T}_{32}

~=

~\mathfrak{T}_{23}

~=

~\frac{1}{2} \int_V \rho v_2 v_3 d^3x \, .


Adding this pair of governing expressions we obtain,

~0

~=

~
\biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} 
- 2\Omega_3 \int_V \rho u_1x_3 dx \biggr]
+
\biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
\biggr]

 

~=

~4 \mathfrak{T}_{23} - \Omega_2\Omega_3(I_{22}+ I_{33} )
+
2 \int_V \rho u_1 (\Omega_2 x_2 - \Omega_3 x_3) dx \, ;

[ EFE, Chapter 7, §47, Eq. (6) ]

and subtracting the pair gives,

~0

~=

~
\biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} 
- 2\Omega_3 \int_V \rho u_1x_3 dx \biggr]
-
\biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
\biggr]

 

~=

~
\Omega_2\Omega_3 (I_{22} - I_{33} )
- 2 \int_V \rho u_1 ( \Omega_2 x_2 + \Omega_3 x_3) dx \, .

[ EFE, Chapter 7, §47, Eq. (7) ]

Various Degrees of Simplification

Riemann S-Type Ellipsoids

Describe …

Jacobi and Dedekind Ellipsoids

Describe …

Maclaurin Spheroids

Describe …

Appendices:  Various Integrals Over Ellipsoid Volume

Throughout this set of appendices, we work with a uniform-density ellipsoid whose surface is defined by the expression,

~1

~=

~
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \, .

Appendix A:  Volume

Here we seek to find the volume of the ellipsoid via the Cartesian integral expression,

~V

~=

~
\iiint  dx ~dy ~dz \, .

Preliminaries

First, we will integrate over ~x and specify the integration limits via the expression,

~x_\ell

~\equiv

~
a\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \, ;

second, we will integrate over ~z and specify the integration limits via the expression,

~z_\ell

~\equiv

~
c\biggl[ 1 - \frac{y^2}{b^2} \biggr]^{1 / 2} \, ;

third, we will integrate over ~y and set the limits of integration as ~\pm b.

Carry Out the Integration

Following thestrategy that has just been outlined, we have,

~V

~=

~
\iint  dy ~dz \int_{-x_\ell}^{+x_\ell} dx
=
\iint  dy ~dz \biggl[ x \biggr]_{-x_\ell}^{+x_\ell}
= 
2\int dy \int x_\ell ~dz

 

~=

~
2a\int dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz
=
\frac{2a}{c} \int dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz

 

~=

~
\frac{2a}{c} \int \frac{dy}{2}  \biggl[ z\sqrt{ z_\ell^2- z^2 } + z_\ell^2 \sin^{-1} \biggl( \frac{z}{|z_\ell |} \biggr) \biggr]_{-z_\ell}^{+z_\ell}

 

~=

~
\frac{2a}{c} \int \biggl[ z_\ell \cancelto{0}{\sqrt{ z_\ell^2- z_\ell^2 }} + z_\ell^2 \sin^{-1} \biggl(1\biggr) \biggr] dy
=
\frac{2a}{c} \int \biggl[ \frac{\pi}{2} z_\ell^2 \biggr] dy

 

~=

~
\pi a c \int_{-b}^{+b} \biggl( 1 - \frac{y^2}{b^2} \biggr)  dy
=
\pi a c  \biggl[ y - \frac{y^3}{3b^2}  \biggr]_{-b}^{+b}

 

~=

~
\frac{4\pi}{3} \cdot a b c\, .

Appendix B:  Coriolis Component u1x2

~\iiint  [u_1 y] ~dx ~dy ~dz

~=

~
\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} y ~dx ~dy ~dz

 

~=

~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \iiint y^2 ~dx ~dy ~dz
+
 \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint yz ~dx ~dy ~dz

 

~=

~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int dz \int_{-x_\ell}^{+x_\ell} dx 
+
 \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z ~dz \int_{-x_\ell}^{+x_\ell} dx

 

~=

~
- \biggl[ \frac{2a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int x_\ell dz  
+
 \biggl[ \frac{2a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~x_\ell ~dz

 

~=

~
- \biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz  
+
 \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} ~dz

 

~=

~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz  
~+~
\frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int_{-z_\ell}^{+z_\ell} z~\biggl[ z_\ell^2 - z^2 \biggr]^{1 / 2} ~dz

 

~=

~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z \sqrt{z_\ell^2 - z^2} + z_\ell^2 \sin^{-1}\biggl(\frac{z}{|z_\ell |}\biggr) \biggr\}_{-z_\ell}^{+z_\ell}  
~-~
\frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \cdot \frac{1}{3} \biggl\{ \biggl[ z_\ell^2 - z^2 \biggr]^{3 / 2} \biggr\}_{-z_\ell}^{+z_\ell}

 

~=

~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z_\ell^2 \sin^{-1}\biggl(\frac{z}{|z_\ell |}\biggr) \biggr\}_{-z_\ell}^{+z_\ell}  
=
- \pi a~c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int_{-b}^b y^2 \biggl[1 - \frac{y^2}{b^2}  \biggr] dy

 

~=

~
- \pi ac\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{y^3}{3} - \frac{y^5}{5b^2}  \biggr]_{-b}^{+b}  
=
- 2\pi a b^3 c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{2}{15}  \biggr]  
=
- \frac{4\pi abc}{3} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{b^2}{5}  \biggr]

 

~=

~
- \frac{I_{22}}{\rho} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3   \, .

[ EFE, Chapter 7, §47, p. 130, Eq. (9a) ]

Appendix C:  Coriolis Component u1x3

Here we will additionally make use of the integration limits,

~y_\ell^2

~\equiv

~b^2 \biggl(1 - \frac{z^2}{c^2}\biggr) \, .

Integration over the relevant Coriolis component gives,

~\iiint  [u_1 z] ~dx ~dy ~dz

~=

~
\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} z ~dx ~dy ~dz

 

~=

~-
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3\iiint \cancelto{0}{y  z ~dx ~dy ~dz}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint  z^2 ~dx ~dy ~dz

 

~=

~
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \int_{-x_\ell}^{+x_\ell} dx

 

~=

~
2a\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \biggl\{ \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \biggr\}

 

~=

~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int_{-y_\ell}^{+y_\ell} \biggl[ y_\ell^2 - y^2 \biggr]^{1 / 2} dy

 

~=

~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \cdot \frac{1}{2}\biggl\{ y \sqrt{y_\ell^2 - y^2} + y_\ell^2 \sin^{-1}\biggr( \frac{y}{|y_\ell |} \biggr)\biggr\}_{-y_\ell}^{+y_\ell}

 

~=

~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2  \biggl\{ \frac{\pi}{2} y_\ell^2 \biggr\} dz
=
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2  \biggl\{ 1 - \frac{z^2}{c^2} \biggr\} dz

 

~=

~
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{z^3}{3} - \frac{z^5}{5c^2} \biggr\}_{-c}^{+c}
=
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{1}{3} - \frac{1}{5} \biggr\}2c^3
=
\frac{4 \pi a b c}{3}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{c^2}{5} \biggr\}

 

~=

~+
~\frac{I_{33}}{\rho} \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \, .

[ EFE, Chapter 7, §47, p. 130, Eq. (9b) ]


Appendix D:   The Other Four Coriolis Components

It follows that,

~\iiint  [u_2 x] ~dx ~dy ~dz

~=

~
\iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z} + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x \biggr\} x ~dx ~dy ~dz

 

~=

~
+~\frac{I_{11}}{\rho}\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 \, ;

and,

~\iiint  [u_2 z] ~dx ~dy ~dz

~=

~
\iiint \biggl\{ - \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z + \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x} \biggr\} z ~dx ~dy ~dz

 

~=

~
-~\frac{I_{33}}{\rho} \biggl[\frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \, ;

and,

~\iiint  [u_3 x] ~dx ~dy ~dz

~=

~
\iiint \biggl\{ - \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x + \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y} \biggr\} x ~dx ~dy ~dz

 

~=

~
-~\frac{I_{11}}{\rho} \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \, ;

and,

~\iiint  [u_3 y] ~dx ~dy ~dz

~=

~
\iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x} + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y \biggr\} y ~dx ~dy ~dz

 

~=

~
+~\frac{I_{22}}{\rho} \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1  \, .


Appendix E:   Kinetic Energy Components

Looking first at the diagonal elements, we have,

~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{11} = \int_V  u_1 u_1 d^3x

~=

~\iiint  [u_1^2] ~dx ~dy ~dz

 

~=

~\iiint  \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}^2 ~dx ~dy ~dz

 

~=

~\iiint  \biggl\{
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 y^2 
- 2\biggl[ \frac{a^2}{a^2 + b^2}\biggr]  
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \zeta_3 yz 
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 z^2 
\biggr\} ~dx ~dy ~dz

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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