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diff --git a/src/input/skew.input.pamphlet b/src/input/skew.input.pamphlet new file mode 100644 index 00000000..4c86808f --- /dev/null +++ b/src/input/skew.input.pamphlet @@ -0,0 +1,113 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input skew.input} +\author{The Axiom Team} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{License} +<<license>>= +--Copyright The Numerical Algorithms Group Limited 1994. +@ +<<*>>= +<<license>> +-- make sure that LALG, EAB, ANTISYM are loaded +-- +)cl all + +-- We will look at the deRham complex of Euclidean 3-space and use +-- coordinates (x,y,z). +lv:List Symbol := [x,y,z] + +-- Next is our ring of functions. We can have functions of any +-- number of variables, but since we've chosen to work with ordinary +-- Euclidean 3-space, an expression like f(x,t,r,y,u,z) will be treated +-- as a parameterized function of (x,y,z) and will be considered to be +-- constant in the variables t,r,u. We choose expressions with integer +-- coefficients in this example. +macro coefRing == Integer +R := Expression coefRing + +-- The declaration for the deRham complex takes arguments a ring coefRing +-- and a list of variables (lv is of type List Symbol). +der := DERHAM(coefRing,lv) + +-- here are some functions chosen at random. +f:R:=x**2*y*z-5*x**3*y**2*z**5 +g:R:=z**2*y*cos(z)-7*sin(x**3*y**2)*z**2 +h:R:=x*y*z-2*x**3*y*z**2 + +-- The multiplicative basis elements for the exterior algebra over R are +-- defined here. +dx :der := generator(1) +dy :der := generator(2) +dz :der := generator(3) +-- A nice alternate for the assignments above is +[dx,dy,dz] := [generator(i)$der for i in 1..3] + +-- Now some 1-forms chosen at random. +alpha:der := f*dx + g*dy + h*dz +beta:der := cos(tan(x*y*z)+x*y*z)*dx + x*dy + +-- we know that exteriorDifferential^2 = 0, let's see that: +exteriorDifferential alpha +exteriorDifferential % + +-- exteriorDifferential is long, let's shorten that. +macro exD == exteriorDifferential + +-- we know that exD is a (graded) derivation, let's see that: +gamma := alpha * beta + +delta := exD gamma +-- need the "-" because alpha is a 1-form and 1 is odd. +epsilon := exD(alpha)*beta - alpha * exD(beta) + +delta - epsilon + +-- We define some operators. +a:BOP := operator('a) +b:BOP := operator('b) +c:BOP := operator('c) + +-- Now some indeterminate one and two forms. +alpha := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz +beta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz + +-- the "gradient". +totalDifferential(a(x,y,z))$der +-- the "curl". +exD alpha +-- the "divergence". +exD beta + +-- Note that the deRham complex is an algebra with 1. +id:der := 1 + +-- Now some parameterized functions (and fomrs -- left as an exercise). +-- Note how the system keeps track of where your coordinate functions +-- are located in expressions. By multiplying the expressions below by +-- 1 in the deRham complex, we automatically convert them to 0-forms, +-- i.e., functions on our space. +g1:der := a([x,t,y,u,v,z,e]) * id +h1:der := a([x,y,x,t,x,z,y,r,u,x]) * id + +exD g1 +exD h1 + +-- Now note that we can find the coefficient of some basis term in +-- any form (the basis in this case consists of the 8 forms +-- 1, dx, dx, dz, dx dy, dx dz, dy dz, dx dy dz. +coefficient(gamma, dx*dy) +coefficient(gamma, id) +coefficient(g1,id) +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} |