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This package may be broken. Please use the VECT package on
the SHARE directory. - JPG 6/5/78
This is a package of vector operators designed to make vector
operations convenient. Several vector operators are included.
The coordinate system may be specified by the user. Included in
the system are Cartesian, cylindrical and spherical coordinates in
three dimensions. Rectangular and polar coordinates are included
in two dimensions. The user is able to specify a new system to be
used, such as elliptical, in either two or three dimensions. A
function is included to change from any coordinate system to another
predefined system, or to another new system. This system is designed
only to handle orthogonal curvilinear coordinate systems.
The vectors used in the system may be either lists or
one-dimensional matrices. When an operation on a vector returns a
vector, then a one-dimensional matrix is returned if the arguments
are matrices. If any argument is a list, then a list is returned.
When an operation on a scalar returns a vector, then the value of
the global variable "vector" determines the form of the answer.
When vector has a value of "list", a list is returned. Otherwise
a one-dimensional matrix is returned. When a two-dimensional sys-
tem is specified, a scalar may be used where a vector is required,
in which case the scalar is considered to be a vector pointing out
of the plane.
Future Extensions:
Currently only three pre-defined three-dimensional coordinate
systems are included. Eight additional systems that lead to sep-
arable forms for the wave equation are: elliptic cylinder, prolate
spheroidal, oblate spheroidal, parabolic-cylinder, parabolic, coni-
cal, ellipsoidal, and paraboloidal.
1. Description of Global Variables.
Global Initial
Variable Value Explanation
-------- ------- -----------
coordsystem cartesian Coordinate system to be used.
dimension 3 Number of dimensions of coordinate
system.
coordvar [x, y, z] Variables varying along axes of
coordinate system being used.
scalefactor [1, 1, 1] Scale factors associated with
coordinate system being used.
vector list Vectors to be used are lists, rather
than one-dimensional matrices.
2. Description of Pre-defined Coordinate Systems.
Coordsystem Dimension Coordvar Scalefactor
----------- --------- -------- -----------
rectangular 2 [x, y] [1, 1]
polar 2 [r, th] [1, r]
cartesian 3 [x, y, z] [1, 1, 1]
cylindrical 3 [r, ph, z] [1, r, 1]
spherical 3 [r, th, ph] [1, r, r*SIN(th)]
coordsystem(arg) Specifies that coordinate system arg is to be set
up to be used by the vector operators in this package.
When arg is one of the values possible in the preced-
ing table for the global variable coordsystem, the
specified coordinate system is set up. All the global
variables are changed to new values as indicated by
the table.
3. Specification of Other Coordinate Systems.
A) First Method.
Coordsystem(arg) specifies that coordinate system arg is to
be set up. When arg is not one of the pre-defined systems,
then the program asks the user to specify the new values of
the global variables. Thus any coordinate system may be
specified by the user.
B) Second Method.
The user may also set the values of the global variables di-
rectly. This may be useful in, for example, a BATCH file
if the user does not desire to type the new coordinate sys-
tem each time the program is BATCHed. The variable
coordsystem should be set to a name corresponding to the
coordinate system being specified. The global variables
dimension, coordvar and scalefactor must be set to appropriate
values. Note that just setting coordsystem does not cause
other global variables to be changed.
4. Description of Vector Operators Available.
Operation Explanation
--------- -----------
vectorp(a) A predicate that returns TRUE if a is a matrix or a
list. Thus A is characterized by both magnitude and
direction, quantities computed from the components of
a.
a cross b Cross product of arguments a and b.
grad s Gradient of scalar s, or the vector with a magnitude
and direction of the maximum directional derivative.
div v Divergence of vector v, or the scalar product of the
vector v with the outward drawn normal.
curl a Curl of vector a, or the vector product of vector a
with the outward drawn normal.
laplacian a The Laplacian of a, or the divergence of the gradient
of a. If a is a scalar, then the result is a scalar.
The Laplacian of vector a is defined as
grad div a - curl curl a.
v dotdel b Directional derivative of vector v in the direction of
b. Result is a scalar if b is a scalar, and is a
vector if b is a vector. Christoffel symbols are
used if defined.
christoffel Defines the Christoffel symbols for use with the
directional derivative, dotdel.
curlgrad s Curl of the gradient of scalar a, which is zero.
graddiv v Gradient of the divergence of vector v.
divcurl v Divergence of the curl of vector v, which is zero.
curlcurl v Curl of the curl of vector v.
This system has been written by Martin Cole. Suggested improve-
ments may be incorporated into this package. Any comments or sug-
gestions should be mailed to MSC@MIT-MC.
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