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# Differentiation

## Definitions for Differentiation

Function: ANTID (G,X,U(X))
  A routine for evaluating integrals of expressions involving an arbitrary unspecified function and its derivatives. It may be used by LOAD(ANTID); , after which, the function ANTIDIFF may be used. E.g. ANTIDIFF(G,X,U(X)); where G is the expression involving U(X) (U(X) arbitrary) and its derivatives, whose integral with respect to X is desired. The functions NONZEROANDFREEOF and LINEAR are also defined, as well as ANTID. ANTID is the same as ANTIDIFF except that it returns a list of two parts, the first part is the integrated part of the expression and the second part of the list is the non-integrable remainder.

Function: ANTIDIFF -
  See ANTID.

property: ATOMGRAD
  - the atomic gradient property of an expression. May be set by GRADEF.

Function: ATVALUE (form, list, value)
  enables the user to assign the boundary value value to form at the points specified by list.

      (C1) ATVALUE(F(X,Y),[X=0,Y=1],A**2)$
          The form must be a function, f(v1,v2,...), or a derivative,

  DIFF(f(v1,v2,...),vi,ni,vj,nj,...) in which the functional arguments explicitly appear (ni is the order of differentiation with respect vi). The list of equations determine the "boundary" at which the value is given; list may be a list of equations, as above, or a single equation, vi = expr. The symbols @1, @2,... will be used to represent the functional variables v1,v2,... when atvalues are displayed. PRINTPROPS([f1, f2,...], ATVALUE) will display the atvalues of the functions f1,f2,... as specified in previously given uses of the ATVALUE function. If the list contains just one element then the element can be given without being in a list. If a first argument of ALL is given then atvalues for all functions which have them will be displayed. Do EXAMPLE(ATVALUE); for an example.

Function: CARTAN -
  The exterior calculus of differential forms is a basic tool of differential geometry developed by Elie Cartan and has important applications in the theory of partial differential equations. The present implementation is due to F.B. Estabrook and H.D. Wahlquist. The program is self-explanatory and can be accessed by doing batch("cartan"); which will give a description with examples.

Function: DEL (v)
  This is the differential of the variable v. This commonly occurs when DIFF is called without the dependent variable being given. Then the differential of the expression is computed. For example:

      (C1) diff(x);

      (D1) 				    DEL(x)
      (C2) diff(x^2);

      (D2) 				  2 x DEL(x)
      (C3) diff(sin(x^2));

      				       2
      (D3) 			      2 x COS(x ) DEL(x)

Function: DELTA (t)
  This is the Dirac Delta function. Currently only LAPLACE knows about the DELTA function:

      (C1) LAPLACE(DELTA(T-A)*SIN(B*T),T,S);
      Is A positive, negative or zero?
      POS;
                                                - A S
      (D1)                           SIN(A B) %E

Variable: DEPENDENCIES
  default: [] - the list of atoms which have functional dependencies (set up by the DEPENDS or GRADEF functions). The command DEPENDENCIES has been replaced by the DEPENDS command. Do DESCRIBE(DEPENDS);

Function: DEPENDS (funlist1,varlist1,funlist2,varlist2,...)
  declares functional dependencies for variables to be used by DIFF.

      DEPENDS([F,G],[X,Y],[R,S],[U,V,W],U,T)

  informs DIFF that F and G depend on X and Y, that R and S depend on U,V, and W, and that U depends on T. The arguments to DEPENDS are evaluated. The variables in each funlist are declared to depend on all the variables in the next varlist. A funlist can contain the name of an atomic variable or array. In the latter case, it is assumed that all the elements of the array depend on all the variables in the succeeding varlist. Initially, DIFF(F,X) is 0; executing DEPENDS(F,X) causes future differentiations of F with respect to X to give dF/dX or Y (if DERIVABBREV:TRUE).

      (C1) DEPENDS([F,G],[X,Y],[R,S],[U,V,W],U,T);
      (D1)           [F(X, Y), G(X, Y), R(U, V, W), S(U, V, W), U(T)]
      (C2) DEPENDENCIES;
      (D2)           [F(X, Y), G(X, Y), R(U, V, W), S(U, V, W), U(T)]
      (C3) DIFF(R.S,U);
                                     dR           dS
      (D3)                           -- . S + R . --
                                     dU           dU

  Since MACSYMA knows the chain rule for symbolic derivatives, it takes advantage of the given dependencies as follows:

      (C4) DIFF(R.S,T);
                                 dR dU             dS dU
      (D4)                      (-- --) . S + R . (-- --)
                                 dU dT             dU dT
      If we set
      (C5) DERIVABBREV:TRUE;
      (D5)                                 TRUE
      then re-executing the command C4, we obtain
      (C6) "C4;
      (D6)                      (R  U ) . S + R . (S  U )
                                  U  T              U  T

  To eliminate a previously declared dependency, the REMOVE command can be used. For example, to say that R no longer depends on U as declared in C1, the user can type

      REMOVE(R,DEPENDENCY)

  This will eliminate all dependencies that may have been declared for R.

      (C7) REMOVE(R,DEPENDENCY);
      (D7)                                 DONE
      (C8) "C4;
      (D8)                             R . (S  U )
                                        U    T

  CAVEAT: DIFF is the only MACSYMA command which uses DEPENDENCIES information. The arguments to INTEGRATE, LAPLACE, etc. must be given their dependencies explicitly in the command, e.g., INTEGRATE(F(X),X).

Variable: DERIVABBREV
  default: [FALSE] if TRUE will cause derivatives to display as subscripts.

Function: DERIVDEGREE (exp, dv, iv)
  finds the highest degree of the derivative of the dependent variable dv with respect to the independent variable iv occuring in exp.

      (C1) 'DIFF(Y,X,2)+'DIFF(Y,Z,3)*2+'DIFF(Y,X)*X**2$
      (C2) DERIVDEGREE(%,Y,X);
      (D2)                           2

Function: DERIVLIST (var1,...,vark)
  causes only differentiations with respect to the indicated variables, within the EV command.

Variable: DERIVSUBST
  default: [FALSE] - controls non-syntactic substitutions such as

      SUBST(X,'DIFF(Y,T),'DIFF(Y,T,2));

  If DERIVSUBST is set to true, this gives 'DIFF(X,T).

Function: DIFF (exp, v1, n1, v2, n2, ...)
  DIFF differentiates exp with respect to each vi, ni times. If just the first derivative with respect to one variable is desired then the form DIFF(exp,v) may be used. If the noun form of the function is required (as, for example, when writing a differential equation), 'DIFF should be used and this will display in a two dimensional format. DERIVABBREV[FALSE] if TRUE will cause derivatives to display as subscripts. DIFF(exp) gives the "total differential", that is, the sum of the derivatives of exp with respect to each of its variables times the function DEL of the variable. No further simplification of DEL is offered.

      (C1) DIFF(EXP(F(X)),X,2);
                                   2
                            F(X)  d             F(X)  d         2
      (D1)                %E     (--- F(X)) + %E     (-- (F(X)))
                                    2                 dX
                                  dX
      (C2) DERIVABBREV:TRUE$
      (C3) 'INTEGRATE(F(X,Y),Y,G(X),H(X));
                                      H(X)
                                     /
                                     [
      (D3)                           I     F(X, Y) dY
                                     ]
                                     /
                                      G(X)
      (C4) DIFF(%,X);
                  H(X)
                 /
                 [
      (D4)       I     F(X, Y)  dY + F(X, H(X)) H(X)  - F(X, G(X)) G(X)
                 ]            X                     X                  X
                 /
                  G(X)

  For the tensor package, the following modifications have been incorporated: 1) the derivatives of any indexed objects in exp will have the variables vi appended as additional arguments. Then all the derivative indices will be sorted. 2) the vi may be integers from 1 up to the value of the variable DIMENSION[default value: 4]. This will cause the differentiation to be carried out wrt the vith member of the list COORDINATES which should be set to a list of the names of the coordinates, e.g., [x,y,z,t]. If COORDINATES is bound to an atomic variable, then that variable subscripted by vi will be used for the variable of differentiation. This permits an array of coordinate names or subscripted names like X[1], X[2],... to be used. If COORDINATES has not been assigned a value, then the variables will be treated as in 1) above.

special symbol: DIFF
  [flag] for ev causes all differentiations indicated in exp to be performed.

Function: DSCALAR (function)
  applies the scalar d'Alembertian to the scalar function.

      (C41) DEPENDENCIES(FIELD(R));
      (D41)                           [FIELD(R)]
      (C42) DSCALAR(FIELD);
      (D43)
          -M
        %E  ((FIELD  N - FIELD  M + 2 FIELD   ) R + 4 FIELD )
                   R  R       R  R         R R             R

      - -----------------------------------------------------
                                   2 R

Function: 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.

Function: GENDIFF
  Sometimes DIFF(E,X,N) can be reduced even though N is symbolic.

      batch("gendif")$

  and you can try, for example,

      DIFF(%E^(A*X),X,Q)

  by using GENDIFF rather than DIFF. Unevaluable items come out quoted. Some items are in terms of "GENFACT", which see.

Function: GRADEF (f(x1, ..., xn), g1, ..., gn)
  defines the derivatives of the function f with respect to its n arguments. That is, df/dxi = gi, etc. If fewer than n gradients, say i, are given, then they refer to the first i arguments of f. The xi are merely dummy variables as in function definition headers and are used to indicate the ith argument of f. All arguments to GRADEF except the first are evaluated so that if g is a defined function then it is invoked and the result is used. Gradients are needed when, for example, a function is not known explicitly but its first derivatives are and it is desired to obtain higher order derivatives. GRADEF may also be used to redefine the derivatives of MACSYMA's predefined functions (e.g. GRADEF(SIN(X),SQRT(1-SIN(X)**2)) ). It is not permissible to use GRADEF on subscripted functions. GRADEFS is a list of the functions which have been given gradients by use of the GRADEF command (i.e. GRADEF(f(x1, ..., xn), g1, ..., gn)). PRINTPROPS([f1,f2,...],GRADEF) may be used to display the gradefs of the functions f1,f2,.. GRADEF(a,v,exp) may be used to state that the derivative of the atomic variable a with respect to v is exp. This automatically does a DEPENDS(a,v). PRINTPROPS([a1,a2,...],ATOMGRAD) may be used to display the atomic gradient properties of a1,a2,...

Variable: GRADEFS
  default: [] - a list of the functions which have been given gradients by use of the GRADEF command (i.e. GRADEF(f(x1, ..., xn), g1, ..., gn).)

Function: LAPLACE (exp, ovar, lvar)
  takes the Laplace transform of exp with respect to the variable ovar and transform parameter lvar. Exp may only involve the functions EXP, LOG, SIN, COS, SINH, COSH, and ERF. It may also be a linear, constant coefficient differential equation in which case ATVALUE of the dependent variable will be used. These may be supplied either before or after the transform is taken. Since the initial conditions must be specified at zero, if one has boundary conditions imposed elsewhere he can impose these on the general solution and eliminate the constants by solving the general solution for them and substituting their values back. Exp may also involve convolution integrals. Functional relationships must be explicitly represented in order for LAPLACE to work properly. That is, if F depends on X and Y it must be written as F(X,Y) wherever F occurs as in LAPLACE('DIFF(F(X,Y),X),X,S). LAPLACE is not affected by DEPENDENCIES set up with the DEPENDS command.

      (C1) LAPLACE(%E**(2*T+A)*SIN(T)*T,T,S);
                           A
                       2 %E  (S - 2)
      (D1)            ---------------
                              2     2
                      ((S - 2)  + 1)

Function: UNDIFF (exp)
  returns an expression equivalent to exp but with all derivatives of indexed objects replaced by the noun form of the DIFF function with arguments which would yield that indexed object if the differentiation were carried out. This is useful when it is desired to replace a differentiated indexed object with some function definition and then carry out the differentiation by saying EV(...,DIFF).

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