Documentation.nb

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 "\nThe new user should first read the Read Me First file, followed by the \
Installation Guide, then Quick Start, Working Tutorial notebook, and then \
this file.\n\nThis package was written using Mathematica version 10, then 11 \
and 12, so it should work in all of these versions. The Palette is the best \
source of documentation for this package as well as the center piece for \
entering equations. The best way to get started is to examine and play with \
the palette as described below. Doing this should make you comfortable and \
ready to see example computations in the Working Tutorial notebook.\n\n Open \
a new notebook*. If the palette is not open, select it from the Palette menu. \
Click on various items to see what they do. Hover over items to see tooltips. \
\n\nStart by selecting your ",
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 " (or use the default) and then click on the \[OpenCurlyDoubleQuote]Needs\
\[CloseCurlyDoubleQuote] statement in the palette to enable the GA package \
for this notebook. You\[CloseCurlyQuote]ll need to do this for each new \
notebook you open. \n\nIn the 2nd section of the palette, ",
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 ", click on ",
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 ". Hover over the 6 operators (geometric, wedge/exterior, scalar, and dot \
products and right and left contractions) until you find the small diamond \
that represents geometric product, and click it. Then click on ",
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 " again and press \[EnterKey] on the number pad or Shift-\[EnterKey] on the \
key pad to evaluate the expression. Try other examples and use the \
wedge(a.k.a. exterior) and dot operators to examine the results. Observe that \
the tooltips explain how to enter the operators from the keyboard. If you \
find the operator symbols too small to see, click on a magnification icon at \
the bottom of the palette. Zoom back when you wish to restore a smaller \
palette.\n\nCaution. Always use parentheses in your operations. For example, ",
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 ") is associative so you are safe when you enter x \[Diamond] y \[Diamond] \
z, but not when you enter x \[CenterDot] y \[CenterDot] z,  x ",
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 " z, etc. For convenience, all the operators have been given default \
definitions mirroring the following example:\n\n\tx \[CenterDot] y \
\[CenterDot] z = (x \[CenterDot] y) \[CenterDot] z\n\tx \[CenterDot] y \
\[CenterDot] z \[CenterDot] w = ( (x \[CenterDot] y) \[CenterDot] z) \
\[CenterDot] w\n\nJust remember that in general (x \[CenterDot] y) \
\[CenterDot] z \[NotEqual] x \[CenterDot]( y \[CenterDot] z). \n\nAnd, you \
should use parentheses when you mix operators like x \[CenterDot] y ",
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 " z.\n\nThe blade, clif , and vector operators in the Typing Aids section \
are pretty straight-forward except that at this time (2017) there seems not \
to be a consistent vocabulary for terms like multivector, n-vector, and \
blade. For example, is a 2-vector a vector in 2-space, or is it a bivector, \
or a grade 2 multivector? The vocabulary used in this package is illustrated \
in the spreadsheet named Multivector Terminology and is also explained in the \
tooltips.\n\nRotors are used to perform rotations in any number of \
dimensions. They are used in Clifford algebras and there are examples in the \
Working Tutorial notebook.\n\nThe last 2 items in this section are Complex \
numbers and Quaternions. The complex numbers are the Clifford subalgebra {a + \
b i} with i = ",
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 " and a and b are reals. Quaternions are the Clifford subalgebra of elements \
{a + b i + c j + d k} where a, b, c, & d are real numbers and i, j, and k are \
as explained in the palette (remember to hover your mouse to reveal \
tooltips). The GA quaternions are a left-handed system unlike the standard \
quaternions that are right-handed. (Left-handed is the correct, natural \
definition.) GA operations that are appropriate (such as InverseG and NormG) \
for complex numbers or quaternions will also work when restricted to these \
subalgebras. To use this package to perform complex multiplication, simply \
type e1e2 where you would normally type i. Similarly, for Quaternions, type \
e2e3 for i, -",
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 " , etc. in order to enter your equations using i, j, and k. In order to see \
your output in terms of i, j, and k, append the rule below to your last \
calculation.\n\n   i=.; j=.; k=.;    (to make sure you \
haven\[CloseCurlyQuote]t previously assigned values to i, j, or k)\n   \
ruleQ={ ",
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 " //.ruleQ\n\nFYI. Octonions are non-associative and so cannot be \
represented using geometric products.\n\nThere are a few things I wish to \
clarify. First, unlike other GA packages, this package uses standard notation \
for subscripts. It was a LOT of work to program this way, and I was \
discouraged from doing so by several of the Mathematica gurus, but I believe \
it is up to the programmer to make a package friendly for the user rather \
than making the work easy for the programmer.\n\nThe second thing is the \
matter of dimension. Other than computation time, there is no constraint on \
dimension. The odd thing that I wish to point out is that there is no need \
for the user to specify dimension except in rare cases such as the Hodge \
Dual. It might seem that the dimension n would be required input for, say, \
the wedge product of two multivectors (or clifs, as I prefer to call them). \
The message is to not worry about the dimension of your space. Just enter \
your clifs and the operations take care of dimension for you. \n\nA 3rd item \
is that of basis. For vectors, the basis used is clear: ",
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 " , \[Ellipsis] But, for grade 2 I do not find any need to define which of  ",
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 "  or ",
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 "  is in the basis. When the palette generates a general multivector it uses \
the Mathematica default of numerical order. That is, it uses ",
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 ". But this package doesn\[CloseCurlyQuote]t use or manipulate bases. It \
simply computes things like ",
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 " so bases are never mentioned. \n\nI apologize for not using the standard \
symbols for left and right contractions, but those symbols are not set up in \
Mathematica for use as binary operators. It is possible to utilize the \
correct symbols but is a lot of work. \n\nFinally, the geometric product is \
usually represented by juxtaposition but the \[Diamond] operator is used in \
this package since Mathematica already uses juxtaposition for regular \
multiplication.\n\nIn the 3rd section of the palette, ",
 StyleBox["Typing Aids, Multivector Generators",
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 ", you may need to click the triangle to expand the section. (Click it again \
to hide the section.) In the 1st column enter a command by selecting it. In \
the workbook, press \[EnterKey]. The output should be the same as the \
corresponding entry in the right-hand column of the palette. Now modify some \
parameters of the formula you just inputted, say, change the letter a to the \
letter b, or dimension 3 to dimension 4, and press \[EnterKey] again. Next, \
select something from column 2. This generates the same output, but now it is \
provided in the input area for you to use, saving you from having to copy and \
paste the output. Finally, hoover over the commands in both columns to get \
more information. These functions have been provided to hopefully simplify \
typing by providing inputs that you modify rather than having to type all \
inputs from scratch.\n\nThe 4rd section of the palette contains the main \
Geometric Algebra (GA) operations. The Geometric Product can be entered using \
the first command in this section, or by using the \[Diamond] symbol, \
entering from the palette manually typing. Some texts restrict the Geometric \
Product to homogenous multivectors; that is, multivectors whose terms are all \
of the same grade. These texts then discuss \[OpenCurlyDoubleQuote]extensions\
\[CloseCurlyDoubleQuote], or define other terms, to discuss the natural \
extensions of geometric products to non-homogeneous multivectors. This \
Mathematica package simply uses the term Geometric Product to cover products \
of any two (or more) multivectors, whether or not homogeneous.\n\nThe same \
thing goes for Wedge, Dot, and Scalar products and Contractions. The user can \
enter homogeneous or non-homogeneous multivectors. The Wedge Product of 2 \
multivectors is computed by taking the terms of the Geometric Product whose \
grade is the sum of the grades of its factors. That is,\n\n\tA ",
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 "         (Scalar Product)\n\nThere are several different definitions given \
in current literature for the Hodge Dual so I have provided a couple of Hodge \
functions to allow the user freedom of choice. The equations defining the two \
I have chosen are shown in the tooltips (hover the mouse over the palette). \
The first definition is consistent with the implicit definition that the \
Hodge Dual is the unique operator that satisfies clif2 ",
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 "( clif \[CenterDot] clif2 ) \[Diamond] i  for all multivectors clif2 and \
where i is the pseudoscalar. By using this package I was able to \
experimentally verify that the simple explicit definition I use in function \
HodgeDualG satisfies this definition. The HodgeDual2G definition does NOT \
satisfy the implicit equations but does have nice geometric properties and \
seems to be pretty commonly used. Hover your mouse over the palette to view \
these definitions in the tooltips.\n\nGorm is basically the square of the \
norm. The Working Tutorial notebook illustrates both of these. \n\nInverseG \
computes the inverse of a multivector. If the inverse does not exist, an \
error message is given and a value of 0 is returned.\n\nThe reverse of ",
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preceded by the appropriate \[NotEqual]1. For example, GormG[ ",
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 ".\n\nDefinitions of the terms can either be found in the tooltips and by \
examining the source code. The source code organization mirrors the \
organization in the toolbar.\n\nMost of the operators in the 5th section, ",
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 ", facilitate extracting portions of a multivector. ConstantG will pick out \
the constant term or terms, if any, in a multivector. Similarly, FreeTermG \
will pick out the non-constant term or terms, if any. \n\npSliceG will pick \
out the grade p term(s), if any, of a multivector. EijTermG is even more \
selective. It will pick out only the terms, if any, that have ",
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 ", etc. \n\nExpandG and CollectG help simplify multivector expressions \
similarly to Mathematica\[CloseCurlyQuote]s Expand and Collect commands \
except these are blade-aware, able to expand and collect with respect to ",
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 " terms. \n\nReduceG is used internally to reduce squares of basis elements. \
In the event that the user creates his own products, not using the GA \
operators, ReduceG can be used to reduce the squares according to the GA \
initialization specified by the user in step 1. c\n\nMaxDimG will find the \
highest basis subscript among the terms of a multivector. \n\nIt is not \
necessary to revert to lists in order to perform most operations. This \
package allows straight-forward operation on the multivectors themselves \
using standard mathematical notation. However, it can be convenient to use \
lists for complex operations where one needs to keep the terms of a \
multivector in a particular order in order to operate on them. (Mathematica, \
has its own inimitable and mostly-uncontrollable order.) Thus, the last \
section of the palette,",
 StyleBox[" List Operations and Support",
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 ", is list-related. The top two operators quickly switch between a \
multivector and its corresponding list of term, a more difficult task than it \
would at first seem. \n\nThe 2nd row generates two lists from a multivector. \
SubscriptListG provides a list of the subscripts of the blades of the \
multivector terms. For example, SubscriptListG[ 2 + ",
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 " }. Both lists can come in handy. \n\nGradeListG generates a list of \
integers that represent the grades of the terms in a multivector. The order \
of the grades matches the order of the terms in ClifListG [clif]. That allows \
you to perform (or check) GA operations on each terms of a clif.\n\nThe last \
item is a Signature operator that extends Mathematica\[CloseCurlyQuote]s \
signature function. Mathematica\[CloseCurlyQuote]s operator will find the \
signature of a list of, say, integers as long as the integers are all \
distinct. But, in GA we often as not deal with duplicate integers such as in ",
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 ". SignatureG will find the signature for { 2, 3, 1, 3 } whereas Signature \
will not. I believe, actually, that Signature G will find the signature for \
any class of items for which Mathematica\[CloseCurlyQuote]s Signature \
function works but I have not extensively tested this. \n\nSignature is a \
measure of the number of pairwise transpositions of adjacent terms required \
to put the list in natural order. An odd number of transpositions reverses \
sign and an even number preserves the sign. It is used to determine the \
correct sign during antisymmetrization operations. For example, ",
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 " terms in your output, you should quit the kernel using the Evaluation \
menu, close and reopen the palette, and reset your GA initialization settings \
in the palette. \n\n   2. Until such time as the author might implement \
operator precedence, it is necessary to put parentheses around your \
multivectors when using the binary operator symbols (see 1st section of \
palette). For example, you must enter (2 ",
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\[CapitalKappa]. So is DotPrdtG [ ",
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 "\n\nThis package contains only a very basic set of GA operations, but other \
GA operations can easily be built upon them. The geometric product is \
somewhat complicated to program because it must handle antisymmetrization. \
Most GA operations are easy to implement once the geometric product is \
defined. Thus, the benefits of this package are:\n\n   1. It uses natural \
mathematical notation for subscripts and standard wedge and dot symbols, not \
requiring you to learn new notation. (Both geometric product and Mathematica\
\[CloseCurlyQuote]s product use a space. To distinguish them, we use a \
diamond for geometrical product.)\n   2. Operations are performed naturally, \
not by converting back and forth to lists (though lists are supported)\n   3. \
The function definitions are generally very short, self-documented, and easy \
to follow, thus easily modified or extended\n   4. The palette greatly \
simplifies typing input, entering sample multivectors that can quickly be \
edited \n   5. Because symbolic coefficients are handled (as well as \
numeric), it is easy to generate generic formulas and to test hypotheses such \
as whether an identity works in dimensions higher than, say, 3 or whether an \
identity can be expanded from vectors to blades or beyond.\n   6. The numeric \
capabilities allow you to instantly carry out computations that would require \
an extensive amount of time and tedium to do manually\n  7. One can quickly \
configure, or switch between, spatial dimensions and space and space-time.\n  \
8. Except for printouts (which can be pages long in higher dimensions) most \
operations compute instantly even in higher dimensions. The definitions are \
written entirely without inefficient \[OpenCurlyDoubleQuote]for loops\
\[CloseCurlyDoubleQuote] and other such constructs.\n\nFinally, the author is \
retired and built this package working in a vacuum as a way of teaching \
himself both GA and Mathematica. If you are using this package, consider \
yourself a beta-tester. Let me know of any bugs you find. I might find time \
to correct them. I have thoroughly tested - + + + and somewhat tested + - - - \
Clifford algebras. Contact me at uvw@sbcglobal.net.\n\n\n* You can use the \
notebook as is or else implement a private notebook context. Private cell \
contexts have not been tested but are likely compatible. The issue is that \
the author has had to take great care to manage the context of the symbol e \
used in the basis ",
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 " This is because only the multivector symbols, like A,  are passed to the \
package yet the package manipulates the basis elements contained in A. Were \
the package \[OpenCurlyDoubleQuote]e\[CloseCurlyDoubleQuote] to have a \
different context than the notebook \[OpenCurlyDoubleQuote]e\
\[CloseCurlyDoubleQuote], those manipulations would fail.\n\n\n"
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