## The Structure, Stability, and Dynamics of Self-Gravitating Systems

Joel E. Tohline
tohline@rouge.phys.lsu.edu

## System of Cartesian Coordinates

DEFINITIONS
x1 º x
x2 º y
x3 º z
h1 = 1
h2 = 1
h3 = 1

Position Vector
x = e1 x1 + e2 x2 + e3 x3 = i x + j y + k z

 ¶x1 ¶x2 ¶x3 Evaluating: (¶xjei) e1 0 0 0 e2 0 0 0 e3 0 0 0

Derived Expressions
de1/dt = di /dt = 0
de2/dt = dj /dt = 0
de3/dt = dk /dt = 0
v º dx/dt
= i (dx/dt) + j (dy/dt) + k (dz/dt)
a º dv/dt
= i (d2x /dt2) + j (d2y /dt2) + k (d2z /dt2)

### Operators (in Cartesian Coordinates)

Ñ = ix + jy + kz

Mathematica¨ Application

Developed by:
David Sherfesee
July, 1997

This application permits you to determine the gradient of virtually any analytically expressible scalar function G(x) in Cartesian coordinates, utilizing the symbolic manipulation capabilities of Mathematica¨.

### Enter Desired Function Expression:

In the space provided, type in the analytical expression for the scalar function of interest, G, in terms of the Cartesian coordinates x, y, z. (For example, when you first load this HTML page, you will find in the space provided the Mathematica String Equivalent of the function G = 1/r, written in Cartesian coordinates.)

G(x,y,z) =

### Evaluate ÑG:

<== Press this button.
If you are uncertain how to deal with or interpret the idiosyncrasies of Mathematica's input/output formats, read the accompanying page of "Mathematica hints." In the present implementation of this application, you must limit your input string to 50 characters, including embedded blanks.

### Divergence:

Ñ×F = xFx + yFy + zFz

Mathematica¨ Application

Developed by:
David Sherfesee
July, 1997

This application permits you to determine the divergence of virtually any analytically expressible vector function F(x) in cartesian coordinates, utilizing the symbolic manipulation capabilities of Mathematica¨.

### Enter Desired Function Expression:

In the spaces provided, type in the analytical expression for the three components of the vector function of interest, F, in terms of the cartesian coordinates x,y,z. (For example, when you first load this HTML page, you will find in the space provided the Mathematica String Equivalent of the vector,

F = i x2 + j y2 + k z2,
written in cartesian coordinates.)

 F(x,y,z) = i + j + k

### Evaluate Ñ×F:

<== Press this button.

### Laplacian:

Ñ2G = x(xG) + y(yG) + z(zG)

Mathematica¨ Application

Developed by:
David Sherfesee
July, 1997

This application permits you to determine the Laplacian of virtually any analytically expressible scalar function G(x) in cartesian coordinates, utilizing the symbolic manipulation capabilities of Mathematica¨.

### Enter Desired Function Expression:

In the space provided, type in the analytical expression for the scalar function of interest, G, in terms of the cartesian coordinates x, y, z. (For example, when you first load this HTML page, you will find in the space provided the Mathematica String Equivalent of the function G = 1/r, written in cartesian coordinates.)

G(x,y,z) =

### Evaluate Ñ2G:

<== Press this button.

 (v×Ñ)F = i [ vx¶xFx + vy¶yFx + vz¶zFx ] + j [ vx¶xFy + vy¶yFy + vz¶zFy ] + k [ vx¶xFz + vy¶yFz + vz¶zFz ]