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VECTOR3D
The vector3d package adds functions for doing vector algebra and calculus
in orthogonal coordinate systems in three dimensions. It provides the
vector or cross product and the differential operators grad, div, curl,
and laplacian.
This package provides an alternative to vect and vector. Unlike those
packages, vector3d does not require vectors to be declared nonscalar,
does not require the coordinate transformation to be given explicitly,
does not need express() to display the components, and does not need an
array of simplification options. Usually, expand() is enough to collapse
complex chains of vector operators into recognizable form.
On the other hand, vector3d does not do as much as the other packages.
The operators that work on vectors do not work with abstract vectors;
they must be defined as three element lists even if the list elements
are unbound. It does assume but not check that the coordinate system
is orthogonal but works just from the scale factors. Therefore,
conversion from one coordinate system to another is not available.
Potentials and Christoffel symbols are not implemented either.
Definitions for vector3d
coordsys(sys)
sets the orthogonal coordinate system to be used, default cartesian.
If sys is not one of cartesian, cylindrical, or spherical, then the
user is asked for the names of the three coordinate variables and for
the three scale factors. The coordinate variables for the already
defined systems are [x,y,z] for cartesian, [r,ph,z] for cylindrical,
and [r,th,ph] for spherical. Note that ph is the polar angle in this
spherical polar system. There is nothing stopping you from setting the
scale factors scalef:[ , , ] and coordinate variables coordvar:[ , , ]
yourself and not using this function.
cross(a,b)
returns the vector or cross product of the two vectors a and b.
curl(u)
returns the vector curl (del cross u) of the vector u, each component of
which is a function of the coordinates. If it is not given explicitly,
do something like
u : [ux,uy,uz]; depends([ux,uy,uz],[x,y,z]);
to establish it.
div(u)
returns the scalar divergence (del dot u) of the vector u.
grad(f)
returns the vector gradient (del f) of the scalar f, which is a function
of the coordinates. Use depends([f],[x,y,z]), for example, to establish
the functional dependence if it is not given explicitly already.
laplacian(u)
If u is a scalar function of the coordinates, then the scalar laplacian
(div(grad(u)) is returned. If u is a vector function, then the vector
laplacian (del squared u) is returned.
a . b, a + b, a * s
The dot product of vectors a and b returns a scalar, and is available
outside this package. The vector sum of two vectors and the product
of a vector and a scalar are likewise available and compatible with the
other operators.
Walter Eastes (eastes@infinet.com)
June, 2004
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