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# 19. Differentiation

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## 19.1 Definitions for Differentiation

Function: antid (expr, x, u(x))

  Returns a two-element list, such that an antiderivative of expr with respect to x can be constructed from the list. The expression expr may contain an unknown function u and its derivatives.

  Let L, a list of two elements, be the return value of `antid`. Then `L[1] + 'integrate (L[2], x)` is an antiderivative of expr with respect to x.

  When `antid` succeeds entirely, the second element of the return value is zero. Otherwise, the second element is nonzero, and the first element is nonzero or zero. If `antid` cannot make any progress, the first element is zero and the second nonzero.

  `load ("antid")` loads this function. The `antid` package also defines the functions `nonzeroandfreeof` and `linear`.

  `antid` is related to `antidiff` as follows. Let L, a list of two elements, be the return value of `antid`. Then the return value of `antidiff` is equal to `L[1] + 'integrate (L[2], x)` where x is the variable of integration.

  Examples:

      (%i1) load ("antid")$
      (%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                                  z(x)  d
      (%o2)                y(x) %e     (-- (z(x)))
                                        dx
      (%i3) a1: antid (expr, x, z(x));
                             z(x)      z(x)  d
      (%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                             dx
      (%i4) a2: antidiff (expr, x, z(x));
                                  /
                           z(x)   [   z(x)  d
      (%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                                  ]         dx
                                  /
      (%i5) a2 - (first (a1) + 'integrate (second (a1), x));
      (%o5)                           0
      (%i6) antid (expr, x, y(x));
                                   z(x)  d
      (%o6)             [0, y(x) %e     (-- (z(x)))]
                                         dx
      (%i7) antidiff (expr, x, y(x));
                        /
                        [        z(x)  d
      (%o7)             I y(x) %e     (-- (z(x))) dx
                        ]              dx
                        /

Function: antidiff (expr, x, u(x))

  Returns an antiderivative of expr with respect to x. The expression expr may contain an unknown function u and its derivatives.

  When `antidiff` succeeds entirely, the resulting expression is free of integral signs (that is, free of the `integrate` noun). Otherwise, `antidiff` returns an expression which is partly or entirely within an integral sign. If `antidiff` cannot make any progress, the return value is entirely within an integral sign.

  `load ("antid")` loads this function. The `antid` package also defines the functions `nonzeroandfreeof` and `linear`.

  `antidiff` is related to `antid` as follows. Let L, a list of two elements, be the return value of `antid`. Then the return value of `antidiff` is equal to `L[1] + 'integrate (L[2], x)` where x is the variable of integration.

  Examples:

      (%i1) load ("antid")$
      (%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                                  z(x)  d
      (%o2)                y(x) %e     (-- (z(x)))
                                        dx
      (%i3) a1: antid (expr, x, z(x));
                             z(x)      z(x)  d
      (%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                             dx
      (%i4) a2: antidiff (expr, x, z(x));
                                  /
                           z(x)   [   z(x)  d
      (%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                                  ]         dx
                                  /
      (%i5) a2 - (first (a1) + 'integrate (second (a1), x));
      (%o5)                           0
      (%i6) antid (expr, x, y(x));
                                   z(x)  d
      (%o6)             [0, y(x) %e     (-- (z(x)))]
                                         dx
      (%i7) antidiff (expr, x, y(x));
                        /
                        [        z(x)  d
      (%o7)             I y(x) %e     (-- (z(x))) dx
                        ]              dx
                        /

Property: atomgrad

  `atomgrad` is the atomic gradient property of an expression. This property is assigned by `gradef`.

Function: atvalue (expr, [x_1 = a_1, ..., x_m = a_m], c)
Function: atvalue (expr, x_1 = a_1, c)

  Assigns the value c to expr at the point `x = a`. Typically boundary values are established by this mechanism.

  expr is a function evaluation, `f(x_1, ..., x_m)`, or a derivative, `diff (f(x_1, ..., x_m), x_1, n_1, ..., x_n, n_m)` in which the function arguments explicitly appear. n_i is the order of differentiation with respect to x_i.

  The point at which the atvalue is established is given by the list of equations `[x_1 = a_1, ..., x_m = a_m]`. If there is a single variable x_1, the sole equation may be given without enclosing it in a list.

  `printprops ([f_1, f_2, ...], atvalue)` displays the atvalues of the functions `f_1, f_2, ...` as specified by calls to `atvalue`. `printprops (f, atvalue)` displays the atvalues of one function f. `printprops (all, atvalue)` displays the atvalues of all functions for which atvalues are defined.

  The symbols `@1`, `@2`, ... represent the variables x_1, x_2, ... when atvalues are displayed.

  `atvalue` evaluates its arguments. `atvalue` returns c, the atvalue.

  Examples:

      (%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
                                      2
      (%o1)                          a
      (%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
      (%o2)                        @2 + 1
      (%i3) printprops (all, atvalue);
                                      !
                        d             !
                       --- (f(@1, @2))!       = @2 + 1
                       d@1            !
                                      !@1 = 0

                                           2
                                f(0, 1) = a

      (%o3)                         done
      (%i4) diff (4*f(x,y)^2 - u(x,y)^2, x);
                        d                          d
      (%o4)  8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
                        dx                         dx
      (%i5) at (%, [x = 0, y = 1]);
                                               !
                    2              d           !
      (%o5)     16 a  - 2 u(0, 1) (-- (u(x, y))!            )
                                   dx          !
                                               !x = 0, y = 1

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 `cartan` package implements the functions `ext_diff` and `lie_diff`, along with the operators `~` (wedge product) and `|` (contraction of a form with a vector.) Type `demo (tensor)` to see a brief description of these commands along with examples.

  `cartan` was implemented by F.B. Estabrook and H.D. Wahlquist.

Function: del (x)

  `del (x)` represents the differential of the variable x.

  `diff` returns an expression containing `del` if an independent variable is not specified. In this case, the return value is the so-called "total differential".

  Examples:

      (%i1) diff (log (x));
                                   del(x)
      (%o1)                        ------
                                     x
      (%i2) diff (exp (x*y));
                           x y              x y
      (%o2)            x %e    del(y) + y %e    del(x)
      (%i3) diff (x*y*z);
      (%o3)         x y del(z) + x z del(y) + y z del(x)

Function: delta (t)

  The Dirac Delta function.

  Currently only `laplace` knows about the `delta` function.

  Example:

      (%i1) laplace (delta (t - a) * sin(b*t), t, s);
      Is  a  positive, negative, or zero?

      p;
                                         - a s
      (%o1)                   sin(a b) %e

System variable: dependencies

  Default value: `[]`

  `dependencies` is the list of atoms which have functional dependencies, assigned by `depends` or `gradef`. The `dependencies` list is cumulative: each call to `depends` or `gradef` appends additional items.

  See `depends` and `gradef`.

Function: depends (f_1, x_1, ..., f_n, x_n)

  Declares functional dependencies among variables for the purpose of computing derivatives. In the absence of declared dependence, `diff (f, x)` yields zero. If `depends (f, x)` is declared, `diff (f, x)` yields a symbolic derivative (that is, a `diff` noun).

  Each argument f_1, x_1, etc., can be the name of a variable or array, or a list of names. Every element of f_i (perhaps just a single element) is declared to depend on every element of x_i (perhaps just a single element). If some f_i is the name of an array or contains the name of an array, all elements of the array depend on x_i.

  `diff` recognizes indirect dependencies established by `depends` and applies the chain rule in these cases.

  `remove (f, dependency)` removes all dependencies declared for f.

  `depends` returns a list of the dependencies established. The dependencies are appended to the global variable `dependencies`. `depends` evaluates its arguments.

  `diff` is the only Maxima command which recognizes dependencies established by `depends`. Other functions (`integrate`, `laplace`, etc.) only recognize dependencies explicitly represented by their arguments. For example, `integrate` does not recognize the dependence of `f` on `x` unless explicitly represented as `integrate (f(x), x)`.

      (%i1) depends ([f, g], x);
      (%o1)                     [f(x), g(x)]
      (%i2) depends ([r, s], [u, v, w]);
      (%o2)               [r(u, v, w), s(u, v, w)]
      (%i3) depends (u, t);
      (%o3)                        [u(t)]
      (%i4) dependencies;
      (%o4)      [f(x), g(x), r(u, v, w), s(u, v, w), u(t)]
      (%i5) diff (r.s, u);
                               dr           ds
      (%o5)                    -- . s + r . --
                               du           du

      (%i6) diff (r.s, t);
                            dr du           ds du
      (%o6)                 -- -- . s + r . -- --
                            du dt           du dt

      (%i7) remove (r, dependency);
      (%o7)                         done
      (%i8) diff (r.s, t);
                                      ds du
      (%o8)                       r . -- --
                                      du dt

Option variable: derivabbrev

  Default value: `false`

  When `derivabbrev` is `true`, symbolic derivatives (that is, `diff` nouns) are displayed as subscripts. Otherwise, derivatives are displayed in the Leibniz notation `dy/dx`.

Function: derivdegree (expr, y, x)

  Returns the highest degree of the derivative of the dependent variable y with respect to the independent variable x occuring in expr.

  Example:

      (%i1) 'diff (y, x, 2) + 'diff (y, z, 3) + 'diff (y, x) * x^2;
                               3     2
                              d y   d y    2 dy
      (%o1)                   --- + --- + x  --
                                3     2      dx
                              dz    dx
      (%i2) derivdegree (%, y, x);
      (%o2)                           2

Function: derivlist (var_1, ..., var_k)

  Causes only differentiations with respect to the indicated variables, within the `ev` command.

Option variable: derivsubst

  Default value: `false`

  When `derivsubst` is `true`, a non-syntactic substitution such as `subst (x, 'diff (y, t), 'diff (y, t, 2))` yields `'diff (x, t)`.

Function: diff (expr, x_1, n_1, ..., x_m, n_m)
Function: diff (expr, x, n)
Function: diff (expr, x)
Function: diff (expr)

  Returns the derivative or differential of expr with respect to some or all variables in expr.

  `diff (expr, x, n)` returns the n'th derivative of expr with respect to x.

  `diff (expr, x_1, n_1, ..., x_m, n_m)` returns the mixed partial derivative of expr with respect to x_1, ..., x_m. It is equivalent to `diff (... (diff (expr, x_m, n_m) ...), x_1, n_1)`.

  `diff (expr, x)` returns the first derivative of expr with respect to the variable x.

  `diff (expr)` returns the total differential of expr, that is, the sum of the derivatives of expr with respect to each its variables times the differential `del` of each variable. No further simplification of `del` is offered.

  The noun form of `diff` is required in some contexts, such as stating a differential equation. In these cases, `diff` may be quoted (as `'diff`) to yield the noun form instead of carrying out the differentiation.

  When `derivabbrev` is `true`, derivatives are displayed as subscripts. Otherwise, derivatives are displayed in the Leibniz notation, `dy/dx`.

  Examples:

      (%i1) diff (exp (f(x)), x, 2);
                           2
                    f(x)  d               f(x)  d         2
      (%o1)       %e     (--- (f(x))) + %e     (-- (f(x)))
                            2                   dx
                          dx
      (%i2) derivabbrev: true$
      (%i3) 'integrate (f(x, y), y, g(x), h(x));
                               h(x)
                              /
                              [
      (%o3)                   I     f(x, y) dy
                              ]
                              /
                               g(x)
      (%i4) diff (%, x);
             h(x)
            /
            [
      (%o4) 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 expr will have the variables x_i appended as additional arguments. Then all the derivative indices will be sorted.

  (2) The x_i 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 with respect to the x_i'th 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 x_i 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

  When `diff` is present as an `evflag` in call to `ev`, all differentiations indicated in `expr` are carried out.

Function: dscalar (f)

  Applies the scalar d'Alembertian to the scalar function f.

  `load ("ctensor")` loads this function.

Function: express (expr)

  Expands differential operator nouns into expressions in terms of partial derivatives. `express` recognizes the operators `grad`, `div`, `curl`, `laplacian`. `express` also expands the cross product `~`.

  Symbolic derivatives (that is, `diff` nouns) in the return value of express may be evaluated by including `diff` in the `ev` function call or command line. In this context, `diff` acts as an `evfun`.

  `load ("vect")` loads this function.

  Examples:

      (%i1) load ("vect")$
      (%i2) grad (x^2 + y^2 + z^2);
                                    2    2    2
      (%o2)                  grad (z  + y  + x )
      (%i3) express (%);
             d    2    2    2   d    2    2    2   d    2    2    2
      (%o3) [-- (z  + y  + x ), -- (z  + y  + x ), -- (z  + y  + x )]
             dx                 dy                 dz
      (%i4) ev (%, diff);
      (%o4)                    [2 x, 2 y, 2 z]
      (%i5) div ([x^2, y^2, z^2]);
                                    2   2   2
      (%o5)                   div [x , y , z ]
      (%i6) express (%);
                         d    2    d    2    d    2
      (%o6)              -- (z ) + -- (y ) + -- (x )
                         dz        dy        dx
      (%i7) ev (%, diff);
      (%o7)                    2 z + 2 y + 2 x
      (%i8) curl ([x^2, y^2, z^2]);
                                     2   2   2
      (%o8)                   curl [x , y , z ]
      (%i9) express (%);
             d    2    d    2   d    2    d    2   d    2    d    2
      (%o9) [-- (z ) - -- (y ), -- (x ) - -- (z ), -- (y ) - -- (x )]
             dy        dz       dz        dx       dx        dy
      (%i10) ev (%, diff);
      (%o10)                      [0, 0, 0]
      (%i11) laplacian (x^2 * y^2 * z^2);
                                        2  2  2
      (%o11)                laplacian (x  y  z )
      (%i12) express (%);
               2                2                2
              d     2  2  2    d     2  2  2    d     2  2  2
      (%o12)  --- (x  y  z ) + --- (x  y  z ) + --- (x  y  z )
                2                2                2
              dz               dy               dx
      (%i13) ev (%, diff);
                            2  2      2  2      2  2
      (%o13)             2 y  z  + 2 x  z  + 2 x  y
      (%i14) [a, b, c] ~ [x, y, z];
      (%o14)                [a, b, c] ~ [x, y, z]
      (%i15) express (%);
      (%o15)          [b z - c y, c x - a z, a y - b x]

Function: gradef (f(x_1, ..., x_n), g_1, ..., g_m)
Function: gradef (a, x, expr)

  Defines the partial derivatives (i.e., the components of the gradient) of the function f or variable a.

  `gradef (f(x_1, ..., x_n), g_1, ..., g_m)` defines `df/dx_i` as g_i, where g_i is an expression; g_i may be a function call, but not the name of a function. The number of partial derivatives m may be less than the number of arguments n, in which case derivatives are defined with respect to x_1 through x_m only.

  `gradef (a, x, expr)` defines the derivative of variable a with respect to x as expr. This also establishes the dependence of a on x (via `depends (a, x)`).

  The first argument `f(x_1, ..., x_n)` or a is quoted, but the remaining arguments g_1, ..., g_m are evaluated. `gradef` returns the function or variable for which the partial derivatives are defined.

  `gradef` can redefine the derivatives of Maxima's built-in functions. For example, `gradef (sin(x), sqrt (1 - sin(x)^2))` redefines the derivative of `sin`.

  `gradef` cannot define partial derivatives for a subscripted function.

  `printprops ([f_1, ..., f_n], gradef)` displays the partial derivatives of the functions f_1, ..., f_n, as defined by `gradef`.

  `printprops ([a_n, ..., a_n], atomgrad)` displays the partial derivatives of the variables a_n, ..., a_n, as defined by `gradef`.

  `gradefs` is the list of the functions for which partial derivatives have been defined by `gradef`. `gradefs` does not include any variables for which partial derivatives have been defined by `gradef`.

  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.

System variable: gradefs

  Default value: `[]`

  `gradefs` is the list of the functions for which partial derivatives have been defined by `gradef`. `gradefs` does not include any variables for which partial derivatives have been defined by `gradef`.

Function: laplace (expr, t, s)

  Attempts to compute the Laplace transform of expr with respect to the variable t and transform parameter s. If `laplace` cannot find a solution, a noun `'laplace` is returned.

  `laplace` recognizes in expr the functions `delta`, `exp`, `log`, `sin`, `cos`, `sinh`, `cosh`, and `erf`, as well as `derivative`, `integrate`, `sum`, and `ilt`. If some other functions are present, `laplace` may not be able to compute the transform.

  expr may also be a linear, constant coefficient differential equation in which case `atvalue` of the dependent variable is used. The required atvalue may be supplied either before or after the transform is computed. 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.

  `laplace` recognizes convolution integrals of the form `integrate (f(x) * g(t - x), x, 0, t)`; other kinds of convolutions are not recognized.

  Functional relations must be explicitly represented in expr; implicit relations, established by `depends`, are not recognized. That is, if f depends on x and y, `f (x, y)` must appear in expr.

  See also `ilt`, the inverse Laplace transform.

  Examples:

      (%i1) laplace (exp (2*t + a) * sin(t) * t, t, s);
                                  a
                                %e  (2 s - 4)
      (%o1)                    ---------------
                                 2           2
                               (s  - 4 s + 5)
      (%i2) laplace ('diff (f (x), x), x, s);
      (%o2)             s laplace(f(x), x, s) - f(0)
      (%i3) diff (diff (delta (t), t), t);
                                2
                               d
      (%o3)                    --- (delta(t))
                                 2
                               dt
      (%i4) laplace (%, t, s);
                                  !
                     d            !         2
      (%o4)        - -- (delta(t))!      + s  - delta(0) s
                     dt           !
                                  !t = 0

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