NOTE: The current version of VECT is the one due to Stoutemyer from
the summer of 1977. It will be replaced soon by a package which
handles vectors and dyadics (as soon as I clean up a few remaining
details and write a new version of this usage file).
Michael C. Wirth (MCW), 12/18/78
BATCH file VECT > DSK SHARE contains a vector analysis package,
DEMO file VECT DEMO DSK SHARE contains a corresponding demonstration, and
BATCH file VECT ORTH DSK SHARE contains definitions of various orthogonal
curvilinear coordinate systems.
Send questions or comments to David R. Stoutemyer, Electrical
Engineering Department, University of Hawaii, Honolulu, Hawaii 96822,
(login name: STOUTE).
The vector analysis package can combine and simplify symbolic
expressions including dot products and cross products, together with
the gradient, divergence, curl, and Laplacian operators. The
distribution of these operators over sums or products is under
user control, as are various other expansions, including expansion
into components in any specific orthogonal coordinate systems.
There is also a capability for deriving the scalar or vector potential
of a field.
To establish indeterminate1, indeterminate2, ... as vector entities,
type
DECLARE([indeterminate1, indeterminate2, ...], NONSCALAR) $
Vectors can also be represented as lists of components.
"." is the dot-product operator, "~" is the cross-product operator,
GRAD is the gradient operator, DIV is the divergence operator, CURL
is the curl or rotation operator, and LAPLACIAN is DIV GRAD.
Most non-controversial simplifications are automatic. For additional
simplification, there is a function which can be used in the form
VECTORSIMP(vectorexpression)
This function employs additional non-controversial simplifications,
together with various optional expansions according to the settings
of the following global flags:
EXPANDALL,
EXPANDDOT,
EXPANDDOTPLUS,
EXPANDCROSS,
EXPANDCROSSPLUS,
EXPANDCROSSCROSS,
EXPANDGRAD,
EXPANDGRADPLUS,
EXPANDGRADPROD,
EXPANDDIV,
EXPANDDIVPLUS,
EXPANDDIVPROD,
EXPANDCURL,
EXPANDCURLPLUS,
EXPANDCURLCURL,
EXPANDLAPLACIAN,
EXPANDLAPLACIANPLUS,
EXPANDLAPLACIANPROD.
The PLUS suffix refers to employing additivity or distributivity.
The PROD suffix refers to the expansion for an operand that is any
kind of product. EXPANDCROSSCROSS refers to replacing p~(q~r)
with (p.r)*q-(p.q)*r, and EXPANDCURLCURL refers to replacing
CURL CURL p with GRAD DIV p + DIV GRAD p. EXPANDCROSS=TRUE has the
same effect as EXPANDCROSSPLUS=EXPANDCROSSCROSS=TRUE, etc. Two other
flags, EXPANDPLUS and EXPANDPROD, have the same effect as setting all
similarly suffixed flags true. When TRUE, another flag named
EXPANDLAPLACIANTODIVGRAD, replaces the LAPLACIAN operator with the
composition DIV GRAD. All of these flags are initially FALSE.
For convenience, all of these flags have been declared EVFLAG.
For orthogonal curvilinear coordinates, the global variables
COORDINATES, DIMENSION, SF, and SFPROD are set by the function
invocation
SCALEFACTORS(coordinatetransform) $
Here coordinatetransform evaluates to the form
[[expression1, expression2, ...], indeterminate1, indeterminat2, ...],
where indeterminate1, indeterminate2, etc. are the curvilinear
coordinate variables and where a set of rectangular Cartesian
components is given in terms of
the curvilinear coordinates by [expression1, expression2, ...]. COORDINATES
is set to the vector [indeterminate1, indeterminate2, ...], and
DIMENSION is set to the length of this vector. SF[1], SF[2], ...,
SF[DIMENSION] are set to the coordinate scale factors, and SFPROD is
set to the product of these scale factors. Initially, COORDINATES is
[X, Y, Z], DIMENSION is 3, and SF[1]=SF[2]=SF[3]=SFPROD=1,
corresponding to 3-dimensional rectangular Cartesian coordinates.
To expand an expression into physical components in the current
coordinate system, there is a function with usage of the form
EXPRESS(expression).
The result uses the noun form of any derivatives arising from
expansion of the vector differential operators.
To force evaluation of these derivatives,
the built-in EV function can be used together with the DIFF evflag,
after using the built-in DEPENDS function to establish any new
implicit dependencies.
The scalar potential of a given gradient vector, in the current
coordinate system, is returned as the result of
POTENTIAL(givengradient).
The calculation makes use of the global variable POTENTIALZEROLOC,
which must be NONLIST or of the form [indeterminatej=expressionj,
indeterminatek=expressionk, ...], the former being equivalent to
the nonlist expression
for all right-hand sides in the latter. The indicated right-hand
sides are used as the lower limit of integration. The success of
the integrations may depend upon their values and order.
POTENTIALZEROLOC is initially set to 0.
The vector potential of a given curl vector, in the current
coordinate system, is returned as the result of
VECTORPOTENTIAL(givencurl).
POTENTIALZEROLOC has a similar role as for POTENTIAL, but the order
of the left-hand sides of the equations must be a cyclic permutation
of the coordinate variables.
EXPRESS, POTENTIAL, and VECTORPOTENTIAL can have a second argument
like the argument of SCALEFACTORS, causeing an corresponding
invocation of SCALEFACTORS before the other computations.
Warning: The VECT package DECLAREs "." to be a commutative operator.