@@ -1,422 +1,422 @@
-\documentstyle[11pt,reduce]{article}
-\date{}
-\title{A Vector Algebra and Calculus Package for REDUCE}
-\author{David Harper \\
-Astronomy Unit \\
-Queen Mary and Westfield College \\
-University of London \\
-Mile End Road \\
-London E1 4NS \\
-England \\[0.05in]
-Electronic mail: {\it adh@star.qmw.ac.uk}}
-\begin{document}
-\maketitle
-
-\index{AVECTOR package}
-
-\section{Introduction}
-
-This package \footnote{Reference: Computer Physics Communications,
-{\bf 54}, 295-305 (1989)}
-is written in RLISP (the LISP meta-language) and is
-intended for use with REDUCE 3.4. \index{vector algebra} It provides
-REDUCE with the ability to perform vector algebra using the same
-notation as scalar algebra.  The basic algebraic operations are
-supported, as are differentiation and integration of vectors with
-respect to scalar variables, cross product and dot product, component
-manipulation and application of scalar functions ({\em e.g.} cosine)
-to a vector to yield a vector result.
-
-A set of vector calculus operators are provided for use with any
-orthogonal curvilinear coordinate system. These operators are
-gradient, divergence, curl and del-squared (Laplacian).  The Laplacian
-operator can take scalar or vector arguments.
-
-Several important coordinate systems are pre-defined and can be
-invoked by name. It is also possible to create new coordinate systems
-by specifying the names of the coordinates and the values of the scale
-factors.
-
-\section{Vector declaration and initialisation}
-
-Any name may be declared to be a vector, provided that it has
-not previously been declared as a matrix or an array. To
-declare a list of names to be vectors use the VEC command:
-\index{VEC command}
-\begin{verbatim}
-   VEC A,B,C;
-\end{verbatim}
-declares the variables {\tt A}, {\tt B} and {\tt C} to be vectors.
-If they have already been assigned (scalar) values, these will be lost.
-
-When a vector is declared using the {\tt VEC} command, it does not
-have an initial value.
-
-If a vector value is assigned to a scalar variable, then that
-variable will automatically be declared as a vector and the
-user will be notified that this has happened.
-
-\index{AVEC function}
-A vector may be initialised using the {\tt AVEC} function which
-takes three scalar arguments and returns a vector made up
-from those scalars. For example
-\begin{verbatim}
-   A := AVEC(A1, A2, A3);
-\end{verbatim}
-sets the components of the vector {\tt A} to {\tt A1}, {\tt A2} and {\tt A3}.
-
-\section{Vector algebra}
-
-(In the examples which follow, {\tt V}, {\tt V1}, {\tt V2} {\em etc}
-are assumed to be vectors while {\tt S}, {\tt S1}, {\tt S2} etc are scalars.)
-
-\index{+ ! vector} \index{- ! vector} \index{* ! vector} \index{/ ! vector}
-The scalar algebra operators +,-,* and / may be used with
-vector operands according to the rules of vector algebra.
-Thus multiplication and division of a vector by a scalar
-are both allowed, but it is an error to multiply or
-divide one vector by another.
-
-\begin{tabular}{l l}
-{\tt V := V1 + V2 - V3;} & Addition and subtraction \\
-{\tt V := S1*3*V1;} & Scalar multiplication \\
-{\tt V := V1/S;} & Scalar division \\
-{\tt V := -V1;} & Negation \\
-\end{tabular}
-
-\index{DOT ! vector} \index{Dot product} \index{CROSS ! vector}
-\index{cross product}
-\noindent Vector multiplication is carried out using the infix
-operators {\tt DOT} and {\tt CROSS}. These are defined to have
-higher precedence than scalar multiplication and
-division.
-
-\begin{tabular}{l l}
-{\tt V := V1 CROSS V2;} & Cross product \\
-{\tt S := V1 DOT V2;} & Dot product \\
-{\tt V := V1 CROSS V2 + V3;} & \\
-{\tt V := (V1 CROSS V2) + V3;} & \\
-\end{tabular}
-
-The last two expressions are equivalent due to the precedence of
-the {\tt CROSS} operator.
-
-\index{VMOD operator}
-The modulus of a vector may be calculated using the {\tt VMOD} operator.
-
-\begin{verbatim}
-   S := VMOD V;
-\end{verbatim}
-
-A unit vector may be generated from any vector using the {\tt VMOD}
-operator.
-
-\begin{verbatim}
-   V1 := V/(VMOD V);
-\end{verbatim}
-
-Components may be extracted from any vector using index notation
-in the same way as an array.
-
-\begin{tabular}{l l}
-{\tt V := AVEC(AX, AY, AZ);} & \\
-{\tt V(0);} & yields AX \\
-{\tt V(1);} & yields AY \\
-{\tt V(2);} & yields AZ \\
-\end{tabular}
-
-It is also possible to set values of individual components. Following
-from above:
-
-\begin{verbatim}
-   V(1) := B;
-\end{verbatim}
-
-The vector {\tt V} now has components {\tt AX}, {\tt B}, {\tt AZ}.
-
-\index{vector ! differentiation} \index{vector ! integration}
-\index{differentiation ! vector} \index{differentiation ! vector}
-Vectors may be used as arguments in the differentiation and
-integration routines in place of the dependent expression.
-
-\begin{tabular}{l l}
-{\tt V := AVEC(X**2, SIN(X), Y);} & \\
-{\tt DF(V,X);} & yields (2*X, COS(X), 0) \\
-{\tt INT(V,X);} & yields (X**3/3, -COS(X), Y*X) \\
-\end{tabular}
-
-Vectors may be given as arguments to monomial functions such as {\tt
-SIN}, {\tt LOG} and {\tt TAN}. The result is a vector obtained by
-applying the function component-wise to the argument vector.
-
-\begin{tabular}{l l}
-{\tt V := AVEC(A1, A2, A3);} & \\
-{\tt SIN(V);} & yields (SIN(A1), SIN(A2), SIN(A3)) \\
-\end{tabular}
-
-\section{ Vector calculus}
-
-\index{DIV ! operator} \index{divergence ! vector field}
-\index{GRAD ! operator} \index{gradient ! vector field}
-\index{CURL ! operator} \index{curl ! vector field}
-\index{DELSQ ! operator} \index{Laplacian ! vector field}
-The vector calculus operators div, grad and curl are recognised.
-The Laplacian operator is also available and may be applied to
-scalar and vector arguments.
-
-\begin{tabular}{l l}
-{\tt V := GRAD S;} & Gradient of a scalar field \\
-{\tt S := DIV V;} & Divergence of a vector field \\
-{\tt V := CURL V1;} & Curl of a vector field \\
-{\tt S := DELSQ S1;} & Laplacian of a scalar field \\
-{\tt V := DELSQ V1;} & Laplacian of a vector field \\
-\end{tabular}
-
-These operators may be used in any orthogonal curvilinear coordinate
-system. The user may alter the names of the coordinates and the values
-of the scale factors. Initially the coordinates are {\tt X}, {\tt Y}
-and {\tt Z} and the scale factors are all unity.
-
-\index{COORDS vector} \index{HFACTORS scale factors}
-There are two special vectors : {\tt COORDS} contains the names
-of the coordinates in the current system and {\tt HFACTORS}
-contains the values of the scale factors.
-
-\index{COORDINATES operator}
-The coordinate names may be changed using the {\tt COORDINATES}
-operator.
-
-\begin{verbatim}
-   COORDINATES R,THETA,PHI;
-\end{verbatim}
-
-This command changes the coordinate names to {\tt R}, {\tt THETA} and
-{\tt PHI}.
-
-\index{SCALEFACTORS operator}
-The scale factors may be altered using the {\tt SCALEFACTORS} operator.
-
-\begin{verbatim}
-   SCALEFACTORS(1,R,R*SIN(THETA));
-\end{verbatim}
-
-This command changes the scale factors to {\tt 1}, {\tt R} and {\tt R
-SIN(THETA)}.
-
-Note that the arguments of {\tt SCALEFACTORS} must be enclosed in
-parentheses. This is not necessary with {\tt COORDINATES}.
-
-
-When vector differential operators are applied to an expression,
-the current set of coordinates are used as the independent
-variables and the scale factors are employed in the calculation.
-(See, for example, Batchelor G.K. 'An Introduction to Fluid
-Mechanics', Appendix 2.)
-
-
-\index{"!*CSYSTEMS global (AVECTOR)}
-Several coordinate systems are pre-defined and may be invoked by
-name. To see a list of valid names enter
-
-\begin{verbatim}
-   SYMBOLIC !*CSYSTEMS;
-\end{verbatim}
-
-and REDUCE will respond with something like
-
-\begin{verbatim}
-   (CARTESIAN SPHERICAL CYLINDRICAL)
-\end{verbatim}
-
-\index{GETCSYSTEM command}
-To choose a coordinate system by name, use the command {\tt GETCSYSTEM}.
-
-To choose the Cartesian coordinate system :
-\begin{verbatim}
-   GETCSYSTEM 'CARTESIAN;
-\end{verbatim}
-\index{PUTCSYSTEM command}
-
-Note the quote which prefixes the name of the coordinate system. This
-is required because {\tt GETCSYSTEM} (and its complement {\tt
-PUTCSYSTEM}) is a {\tt SYMBOLIC} procedure which requires a literal
-argument.
-
-REDUCE responds by typing a list of the coordinate names in that
-coordinate system. The example above would produce the response
-
-\begin{verbatim}
-   (X Y Z)
-\end{verbatim}
-
-whilst
-
-\begin{verbatim}
-   GETCSYSTEM 'SPHERICAL;
-\end{verbatim}
-
-would produce
-
-\begin{verbatim}
-   (R THETA PHI)
-\end{verbatim}
-
-Note that any attempt to invoke a coordinate system is subject to the
-same restrictions as the implied calls to {\tt COORDINATES} and {\tt
-SCALEFACTORS}.  In particular, {\tt GETCSYSTEM} fails if any of the
-coordinate names has been assigned a value and the previous coordinate
-system remains in effect.
-
-A user-defined coordinate system can be assigned a name using the
-command {\tt PUTCSYSTEM}. It may then be re-invoked at a later stage using
-{\tt GETCSYSTEM}.
-
-\example\index{AVECTOR package ! example}
-
-We define a general coordinate system with coordinate names {\tt
-X},{\tt Y},{\tt Z} and scale factors {\tt H1},{\tt H2},{\tt H3} :
-
-\begin{verbatim}
-   COORDINATES X,Y,Z;
-   SCALEFACTORS(H1,H2,H3);
-   PUTCSYSTEM 'GENERAL;
-\end{verbatim}
-
-This system may later be invoked by entering
-
-\begin{verbatim}
-   GETCSYSTEM 'GENERAL;
-\end{verbatim}
-
-\section{Volume and Line Integration}
-
-Several functions are provided to perform volume and line integrals.
-These operate in any orthogonal curvilinear coordinate system and
-make use of the scale factors described in the previous section.
-
-
-Definite integrals of scalar and vector expressions may be calculated
-using the {\tt DEFINT} function.
-
-\example\index{AVECTOR package ! example}
-
-\index{DEFINT function} \index{integration ! definite (simple)}
-\index{definite integration (simple)}
-\noindent To calculate the definite integral of $\sin(x)^2$ between 0 and
-2$\pi$ we enter
-
-\begin{verbatim}
-   DEFINT(SIN(X)**2,X,0,2*PI);
-\end{verbatim}
-
-This function is a simple extension of the {\tt INT} function taking
-two extra arguments, the lower and upper bounds of integration
-respectively.
-
-\index{VOLINTEGRAL function} \index{integration ! volume}
-Definite volume integrals may be calculated using the {\tt
-VOLINTEGRAL} function whose syntax is as follows :
-
-\noindent {\tt VOLINTEGRAL}({\tt integrand}, vector {\tt lower-bound},
-vector {\tt upper-bound});
-
-\example\index{AVECTOR package ! example}
-
-\noindent In spherical polar coordinates we may calculate the volume of a
-sphere by integrating unity over the range $r$=0 to {\tt RR}, $\theta$=0 to
-{\tt PI}, $\phi$=0 to 2*$\pi$ as follows :
-
-\begin{tabular}{l l}
-{\tt VLB := AVEC(0,0,0);} & Lower bound \\
-{\tt VUB := AVEC(RR,PI,2*PI);} & Upper bound in $r, \theta, \phi$
- respectively \\
-{\tt VOLINTORDER := (0,1,2);} & The order of integration \\
-{\tt VOLINTEGRAL(1,VLB,VUB);} & \\
-\end{tabular}
-
-\index{VOLINTORDER vector}
-Note the use of the special vector {\tt VOLINTORDER} which controls
-the order in which the integrations are carried out. This vector
-should be set to contain the number 0, 1 and 2 in the required order.
-The first component of {\tt VOLINTORDER} contains the index of the
-first integration variable, the second component is the index of the
-second integration variable and the third component is the index of
-the third integration variable.
-
-\example\index{AVECTOR package ! example}
-
-Suppose we wish to calculate the volume of a right circular cone. This
-is equivalent to integrating unity over a conical region with the
-bounds:
-
-\begin{tabular}{l l}
-z = 0 to H  & (H = the height of the cone) \\
-r = 0 to pZ & (p = ratio of base diameter to height) \\
-phi = 0 to 2*PI & \\
-\end{tabular}
-
-We evaluate the volume by integrating a series of infinitesimally thin
-circular disks of constant z-value. The integration is thus performed
-in the order : d($\phi$) from 0 to $2\pi$, dr from 0 to p*Z, dz from 0 to H.
-The order of the indices is thus 2, 0, 1.
-
-\begin{verbatim}
-   VOLINTORDER := AVEC(2,0,1);
-   VLB := AVEC(0,0,0);
-   VUB := AVEC(P*Z,H,2*PI);
-   VOLINTEGRAL(1,VLB,VUB);
-\end{verbatim}
-
-(At this stage, we replace {\tt P*H} by {\tt RR}, the base radius of the cone,
-to obtain the result in its more familiar form.)
-
-
-\index{LINEINT function} \index{DEFLINEINT function}
-\index{integration ! line} \index{line integrals}
-Line integrals may be calculated using the {\tt LINEINT} and {\tt DEFLINEINT}
-functions. Their general syntax is
-
-\noindent {\tt LINEINT}({\tt vector-function}, {\tt vector-curve},
-{\tt variable});
-
-\noindent{\tt DEFLINENINT}({\tt vector-function}, {\tt vector-curve},
-{\tt variable}, {\tt lower-bound}, {\tt upper-bound});
-
-\noindent where
-\begin{description}
-\item[{\tt vector-function}] is any vector-valued expression;
-\item[{\tt vector-curve}] is a vector expression which describes the path of
-integration in terms of the independent variable;
-\item[{\tt variable}] is the independent variable;
-\item[{\tt lower-bound}]
-\item[{\tt upper-bound}] are the bounds of integration in terms of the
-independent variable.
-\end{description}
-
-\example\index{AVECTOR package ! example}
-
-In spherical polar coordinates, we may integrate round a line of
-constant theta (`latitude') to find the length of such a line. The
-vector function is thus the tangent to the `line of latitude', (0,0,1)
-and the path is {\tt (0,LAT,PHI)} where {\tt PHI} is the independent
-variable. We show how to obtain the definite integral {\em i.e.} from
-$\phi=0$ to $2 \pi$ :
-\begin{verbatim}
-DEFLINEINT(AVEC(0,0,1),AVEC(0,LAT,PHI),PHI,0,2*PI);
-\end{verbatim}
-
-\section{Defining new functions and procedures}
-
-Most of the procedures in this package are defined in symbolic mode
-and are invoked by the REDUCE expression-evaluator when a vector
-expression is encountered. It is not generally possible to define
-procedures which accept or return vector values in algebraic mode.
-This is a consequence of the way in which the REDUCE interpreter
-operates and it affects other non-scalar data types as well : arrays
-cannot be passed as algebraic procedure arguments, for example.
-
-\section{Acknowledgements}
-
-This package was written whilst the author was the U.K. Computer
-Algebra Support Officer at the University of Liverpool Computer Laboratory.
-\end{document}
+\documentstyle[11pt,reduce]{article}
+\date{}
+\title{A Vector Algebra and Calculus Package for REDUCE}
+\author{David Harper \\
+Astronomy Unit \\
+Queen Mary and Westfield College \\
+University of London \\
+Mile End Road \\
+London E1 4NS \\
+England \\[0.05in]
+Electronic mail: {\it adh@star.qmw.ac.uk}}
+\begin{document}
+\maketitle
+
+\index{AVECTOR package}
+
+\section{Introduction}
+
+This package \footnote{Reference: Computer Physics Communications,
+{\bf 54}, 295-305 (1989)}
+is written in RLISP (the LISP meta-language) and is
+intended for use with REDUCE 3.4. \index{vector algebra} It provides
+REDUCE with the ability to perform vector algebra using the same
+notation as scalar algebra.  The basic algebraic operations are
+supported, as are differentiation and integration of vectors with
+respect to scalar variables, cross product and dot product, component
+manipulation and application of scalar functions ({\em e.g.} cosine)
+to a vector to yield a vector result.
+
+A set of vector calculus operators are provided for use with any
+orthogonal curvilinear coordinate system. These operators are
+gradient, divergence, curl and del-squared (Laplacian).  The Laplacian
+operator can take scalar or vector arguments.
+
+Several important coordinate systems are pre-defined and can be
+invoked by name. It is also possible to create new coordinate systems
+by specifying the names of the coordinates and the values of the scale
+factors.
+
+\section{Vector declaration and initialisation}
+
+Any name may be declared to be a vector, provided that it has
+not previously been declared as a matrix or an array. To
+declare a list of names to be vectors use the VEC command:
+\index{VEC command}
+\begin{verbatim}
+   VEC A,B,C;
+\end{verbatim}
+declares the variables {\tt A}, {\tt B} and {\tt C} to be vectors.
+If they have already been assigned (scalar) values, these will be lost.
+
+When a vector is declared using the {\tt VEC} command, it does not
+have an initial value.
+
+If a vector value is assigned to a scalar variable, then that
+variable will automatically be declared as a vector and the
+user will be notified that this has happened.
+
+\index{AVEC function}
+A vector may be initialised using the {\tt AVEC} function which
+takes three scalar arguments and returns a vector made up
+from those scalars. For example
+\begin{verbatim}
+   A := AVEC(A1, A2, A3);
+\end{verbatim}
+sets the components of the vector {\tt A} to {\tt A1}, {\tt A2} and {\tt A3}.
+
+\section{Vector algebra}
+
+(In the examples which follow, {\tt V}, {\tt V1}, {\tt V2} {\em etc}
+are assumed to be vectors while {\tt S}, {\tt S1}, {\tt S2} etc are scalars.)
+
+\index{+ ! vector} \index{- ! vector} \index{* ! vector} \index{/ ! vector}
+The scalar algebra operators +,-,* and / may be used with
+vector operands according to the rules of vector algebra.
+Thus multiplication and division of a vector by a scalar
+are both allowed, but it is an error to multiply or
+divide one vector by another.
+
+\begin{tabular}{l l}
+{\tt V := V1 + V2 - V3;} & Addition and subtraction \\
+{\tt V := S1*3*V1;} & Scalar multiplication \\
+{\tt V := V1/S;} & Scalar division \\
+{\tt V := -V1;} & Negation \\
+\end{tabular}
+
+\index{DOT ! vector} \index{Dot product} \index{CROSS ! vector}
+\index{cross product}
+\noindent Vector multiplication is carried out using the infix
+operators {\tt DOT} and {\tt CROSS}. These are defined to have
+higher precedence than scalar multiplication and
+division.
+
+\begin{tabular}{l l}
+{\tt V := V1 CROSS V2;} & Cross product \\
+{\tt S := V1 DOT V2;} & Dot product \\
+{\tt V := V1 CROSS V2 + V3;} & \\
+{\tt V := (V1 CROSS V2) + V3;} & \\
+\end{tabular}
+
+The last two expressions are equivalent due to the precedence of
+the {\tt CROSS} operator.
+
+\index{VMOD operator}
+The modulus of a vector may be calculated using the {\tt VMOD} operator.
+
+\begin{verbatim}
+   S := VMOD V;
+\end{verbatim}
+
+A unit vector may be generated from any vector using the {\tt VMOD}
+operator.
+
+\begin{verbatim}
+   V1 := V/(VMOD V);
+\end{verbatim}
+
+Components may be extracted from any vector using index notation
+in the same way as an array.
+
+\begin{tabular}{l l}
+{\tt V := AVEC(AX, AY, AZ);} & \\
+{\tt V(0);} & yields AX \\
+{\tt V(1);} & yields AY \\
+{\tt V(2);} & yields AZ \\
+\end{tabular}
+
+It is also possible to set values of individual components. Following
+from above:
+
+\begin{verbatim}
+   V(1) := B;
+\end{verbatim}
+
+The vector {\tt V} now has components {\tt AX}, {\tt B}, {\tt AZ}.
+
+\index{vector ! differentiation} \index{vector ! integration}
+\index{differentiation ! vector} \index{differentiation ! vector}
+Vectors may be used as arguments in the differentiation and
+integration routines in place of the dependent expression.
+
+\begin{tabular}{l l}
+{\tt V := AVEC(X**2, SIN(X), Y);} & \\
+{\tt DF(V,X);} & yields (2*X, COS(X), 0) \\
+{\tt INT(V,X);} & yields (X**3/3, -COS(X), Y*X) \\
+\end{tabular}
+
+Vectors may be given as arguments to monomial functions such as {\tt
+SIN}, {\tt LOG} and {\tt TAN}. The result is a vector obtained by
+applying the function component-wise to the argument vector.
+
+\begin{tabular}{l l}
+{\tt V := AVEC(A1, A2, A3);} & \\
+{\tt SIN(V);} & yields (SIN(A1), SIN(A2), SIN(A3)) \\
+\end{tabular}
+
+\section{ Vector calculus}
+
+\index{DIV ! operator} \index{divergence ! vector field}
+\index{GRAD ! operator} \index{gradient ! vector field}
+\index{CURL ! operator} \index{curl ! vector field}
+\index{DELSQ ! operator} \index{Laplacian ! vector field}
+The vector calculus operators div, grad and curl are recognised.
+The Laplacian operator is also available and may be applied to
+scalar and vector arguments.
+
+\begin{tabular}{l l}
+{\tt V := GRAD S;} & Gradient of a scalar field \\
+{\tt S := DIV V;} & Divergence of a vector field \\
+{\tt V := CURL V1;} & Curl of a vector field \\
+{\tt S := DELSQ S1;} & Laplacian of a scalar field \\
+{\tt V := DELSQ V1;} & Laplacian of a vector field \\
+\end{tabular}
+
+These operators may be used in any orthogonal curvilinear coordinate
+system. The user may alter the names of the coordinates and the values
+of the scale factors. Initially the coordinates are {\tt X}, {\tt Y}
+and {\tt Z} and the scale factors are all unity.
+
+\index{COORDS vector} \index{HFACTORS scale factors}
+There are two special vectors : {\tt COORDS} contains the names
+of the coordinates in the current system and {\tt HFACTORS}
+contains the values of the scale factors.
+
+\index{COORDINATES operator}
+The coordinate names may be changed using the {\tt COORDINATES}
+operator.
+
+\begin{verbatim}
+   COORDINATES R,THETA,PHI;
+\end{verbatim}
+
+This command changes the coordinate names to {\tt R}, {\tt THETA} and
+{\tt PHI}.
+
+\index{SCALEFACTORS operator}
+The scale factors may be altered using the {\tt SCALEFACTORS} operator.
+
+\begin{verbatim}
+   SCALEFACTORS(1,R,R*SIN(THETA));
+\end{verbatim}
+
+This command changes the scale factors to {\tt 1}, {\tt R} and {\tt R
+SIN(THETA)}.
+
+Note that the arguments of {\tt SCALEFACTORS} must be enclosed in
+parentheses. This is not necessary with {\tt COORDINATES}.
+
+
+When vector differential operators are applied to an expression,
+the current set of coordinates are used as the independent
+variables and the scale factors are employed in the calculation.
+(See, for example, Batchelor G.K. 'An Introduction to Fluid
+Mechanics', Appendix 2.)
+
+
+\index{"!*CSYSTEMS global (AVECTOR)}
+Several coordinate systems are pre-defined and may be invoked by
+name. To see a list of valid names enter
+
+\begin{verbatim}
+   SYMBOLIC !*CSYSTEMS;
+\end{verbatim}
+
+and REDUCE will respond with something like
+
+\begin{verbatim}
+   (CARTESIAN SPHERICAL CYLINDRICAL)
+\end{verbatim}
+
+\index{GETCSYSTEM command}
+To choose a coordinate system by name, use the command {\tt GETCSYSTEM}.
+
+To choose the Cartesian coordinate system :
+\begin{verbatim}
+   GETCSYSTEM 'CARTESIAN;
+\end{verbatim}
+\index{PUTCSYSTEM command}
+
+Note the quote which prefixes the name of the coordinate system. This
+is required because {\tt GETCSYSTEM} (and its complement {\tt
+PUTCSYSTEM}) is a {\tt SYMBOLIC} procedure which requires a literal
+argument.
+
+REDUCE responds by typing a list of the coordinate names in that
+coordinate system. The example above would produce the response
+
+\begin{verbatim}
+   (X Y Z)
+\end{verbatim}
+
+whilst
+
+\begin{verbatim}
+   GETCSYSTEM 'SPHERICAL;
+\end{verbatim}
+
+would produce
+
+\begin{verbatim}
+   (R THETA PHI)
+\end{verbatim}
+
+Note that any attempt to invoke a coordinate system is subject to the
+same restrictions as the implied calls to {\tt COORDINATES} and {\tt
+SCALEFACTORS}.  In particular, {\tt GETCSYSTEM} fails if any of the
+coordinate names has been assigned a value and the previous coordinate
+system remains in effect.
+
+A user-defined coordinate system can be assigned a name using the
+command {\tt PUTCSYSTEM}. It may then be re-invoked at a later stage using
+{\tt GETCSYSTEM}.
+
+\example\index{AVECTOR package ! example}
+
+We define a general coordinate system with coordinate names {\tt
+X},{\tt Y},{\tt Z} and scale factors {\tt H1},{\tt H2},{\tt H3} :
+
+\begin{verbatim}
+   COORDINATES X,Y,Z;
+   SCALEFACTORS(H1,H2,H3);
+   PUTCSYSTEM 'GENERAL;
+\end{verbatim}
+
+This system may later be invoked by entering
+
+\begin{verbatim}
+   GETCSYSTEM 'GENERAL;
+\end{verbatim}
+
+\section{Volume and Line Integration}
+
+Several functions are provided to perform volume and line integrals.
+These operate in any orthogonal curvilinear coordinate system and
+make use of the scale factors described in the previous section.
+
+
+Definite integrals of scalar and vector expressions may be calculated
+using the {\tt DEFINT} function.
+
+\example\index{AVECTOR package ! example}
+
+\index{DEFINT function} \index{integration ! definite (simple)}
+\index{definite integration (simple)}
+\noindent To calculate the definite integral of $\sin(x)^2$ between 0 and
+2$\pi$ we enter
+
+\begin{verbatim}
+   DEFINT(SIN(X)**2,X,0,2*PI);
+\end{verbatim}
+
+This function is a simple extension of the {\tt INT} function taking
+two extra arguments, the lower and upper bounds of integration
+respectively.
+
+\index{VOLINTEGRAL function} \index{integration ! volume}
+Definite volume integrals may be calculated using the {\tt
+VOLINTEGRAL} function whose syntax is as follows :
+
+\noindent {\tt VOLINTEGRAL}({\tt integrand}, vector {\tt lower-bound},
+vector {\tt upper-bound});
+
+\example\index{AVECTOR package ! example}
+
+\noindent In spherical polar coordinates we may calculate the volume of a
+sphere by integrating unity over the range $r$=0 to {\tt RR}, $\theta$=0 to
+{\tt PI}, $\phi$=0 to 2*$\pi$ as follows :
+
+\begin{tabular}{l l}
+{\tt VLB := AVEC(0,0,0);} & Lower bound \\
+{\tt VUB := AVEC(RR,PI,2*PI);} & Upper bound in $r, \theta, \phi$
+ respectively \\
+{\tt VOLINTORDER := (0,1,2);} & The order of integration \\
+{\tt VOLINTEGRAL(1,VLB,VUB);} & \\
+\end{tabular}
+
+\index{VOLINTORDER vector}
+Note the use of the special vector {\tt VOLINTORDER} which controls
+the order in which the integrations are carried out. This vector
+should be set to contain the number 0, 1 and 2 in the required order.
+The first component of {\tt VOLINTORDER} contains the index of the
+first integration variable, the second component is the index of the
+second integration variable and the third component is the index of
+the third integration variable.
+
+\example\index{AVECTOR package ! example}
+
+Suppose we wish to calculate the volume of a right circular cone. This
+is equivalent to integrating unity over a conical region with the
+bounds:
+
+\begin{tabular}{l l}
+z = 0 to H  & (H = the height of the cone) \\
+r = 0 to pZ & (p = ratio of base diameter to height) \\
+phi = 0 to 2*PI & \\
+\end{tabular}
+
+We evaluate the volume by integrating a series of infinitesimally thin
+circular disks of constant z-value. The integration is thus performed
+in the order : d($\phi$) from 0 to $2\pi$, dr from 0 to p*Z, dz from 0 to H.
+The order of the indices is thus 2, 0, 1.
+
+\begin{verbatim}
+   VOLINTORDER := AVEC(2,0,1);
+   VLB := AVEC(0,0,0);
+   VUB := AVEC(P*Z,H,2*PI);
+   VOLINTEGRAL(1,VLB,VUB);
+\end{verbatim}
+
+(At this stage, we replace {\tt P*H} by {\tt RR}, the base radius of the cone,
+to obtain the result in its more familiar form.)
+
+
+\index{LINEINT function} \index{DEFLINEINT function}
+\index{integration ! line} \index{line integrals}
+Line integrals may be calculated using the {\tt LINEINT} and {\tt DEFLINEINT}
+functions. Their general syntax is
+
+\noindent {\tt LINEINT}({\tt vector-function}, {\tt vector-curve},
+{\tt variable});
+
+\noindent{\tt DEFLINENINT}({\tt vector-function}, {\tt vector-curve},
+{\tt variable}, {\tt lower-bound}, {\tt upper-bound});
+
+\noindent where
+\begin{description}
+\item[{\tt vector-function}] is any vector-valued expression;
+\item[{\tt vector-curve}] is a vector expression which describes the path of
+integration in terms of the independent variable;
+\item[{\tt variable}] is the independent variable;
+\item[{\tt lower-bound}]
+\item[{\tt upper-bound}] are the bounds of integration in terms of the
+independent variable.
+\end{description}
+
+\example\index{AVECTOR package ! example}
+
+In spherical polar coordinates, we may integrate round a line of
+constant theta (`latitude') to find the length of such a line. The
+vector function is thus the tangent to the `line of latitude', (0,0,1)
+and the path is {\tt (0,LAT,PHI)} where {\tt PHI} is the independent
+variable. We show how to obtain the definite integral {\em i.e.} from
+$\phi=0$ to $2 \pi$ :
+\begin{verbatim}
+DEFLINEINT(AVEC(0,0,1),AVEC(0,LAT,PHI),PHI,0,2*PI);
+\end{verbatim}
+
+\section{Defining new functions and procedures}
+
+Most of the procedures in this package are defined in symbolic mode
+and are invoked by the REDUCE expression-evaluator when a vector
+expression is encountered. It is not generally possible to define
+procedures which accept or return vector values in algebraic mode.
+This is a consequence of the way in which the REDUCE interpreter
+operates and it affects other non-scalar data types as well : arrays
+cannot be passed as algebraic procedure arguments, for example.
+
+\section{Acknowledgements}
+
+This package was written whilst the author was the U.K. Computer
+Algebra Support Officer at the University of Liverpool Computer Laboratory.
+\end{document}