
     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.
