## 18 Differentiation ¶

### 18.1 Functions and Variables 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
                        /

  Categories: Integral calculus ·

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
                        /

  Categories: Integral calculus ·

Function: at
at (expr, [eqn_1, …, eqn_n])
at (expr, eqn) ¶

  Evaluates the expression expr with the variables assuming the values as specified for them in the list of equations `[eqn_1, ..., eqn_n]` or the single equation eqn.

  If a subexpression depends on any of the variables for which a value is specified but there is no `atvalue` specified and it can’t be otherwise evaluated, then a noun form of the `at` is returned which displays in a two-dimensional form.

  `at` carries out multiple substitutions in parallel.

  See also `atvalue`. For other functions which carry out substitutions, see also `subst` and `ev`.

  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, 1))!     )
                                      dx          !
                                                  !x = 0

  Note that in the last line `y` is treated differently to `x` as `y` isn’t used as a differentiation variable.

  The difference between `subst`, `at` and `ev` can be seen in the following example:

      (%i1) e1:I(t)=C*diff(U(t),t)$
      (%i2) e2:U(t)=L*diff(I(t),t)$

      (%i3) at(e1,e2);
                                     !
                            d        !
      (%o3)       I(t) = C (-- (U(t))!                    )
                            dt       !          d
                                     !U(t) = L (-- (I(t)))
                                                dt

      (%i4) subst(e2,e1);
                                  d      d
      (%o4)             I(t) = C (-- (L (-- (I(t)))))
                                  dt     dt

      (%i5) ev(e1,e2,diff);
                                        2
                                       d
      (%o5)                I(t) = C L (--- (I(t)))
                                         2
                                       dt

  Categories: Evaluation · Differential equations ·

Property: atomgrad ¶

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

  Categories: Differential calculus ·

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

  See also `at`.

  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, 1))!     )
                                      dx          !
                                                  !x = 0

  Categories: Differential equations · Declarations and inferences ·

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".

  See also `diff`, `del` and `derivdegree`.

  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)

  Categories: Differential calculus ·

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

  Categories: Mathematical functions · Laplace transform ·

System variable: dependencies ¶
Function: dependencies (f_1, …, f_n) ¶

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

  The function `dependencies(f_1, …, f_n)` appends f_1, …, f_n, to the `dependencies` list, where f_1, …, f_n are expressions of the form `f(x_1, …, x_m)`, and x_1, …, x_m are any number of arguments.

  `dependencies(f(x_1, …, x_m))` is equivalent to `depends(f, [x_1, …, x_m])`.

  See also `depends` and `gradef`.

      (%i1) dependencies;
      (%o1)                          []

      (%i2) depends (foo, [bar, baz]);
      (%o2)                    [foo(bar, baz)]

      (%i3) depends ([g, h], [a, b, c]);
      (%o3)               [g(a, b, c), h(a, b, c)]

      (%i4) dependencies;
      (%o4)        [foo(bar, baz), g(a, b, c), h(a, b, c)]

      (%i5) dependencies (quux (x, y), mumble (u));
      (%o5)                [quux(x, y), mumble(u)]

      (%i6) dependencies;
      (%o6) [foo(bar, baz), g(a, b, c), h(a, b, c), quux(x, y),
                                                             mumble(u)]

      (%i7) remove (quux, dependency);
      (%o7)                         done

      (%i8) dependencies;
      (%o8)  [foo(bar, baz), g(a, b, c), h(a, b, c), mumble(u)]

  Categories: Declarations and inferences · Global variables ·

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)`.

  `depends(f, [x_1, …, x_n])` is equivalent to `dependencies(f(x_1, …, x_n))`.

  See also `diff`, `del`, `derivdegree` and `derivabbrev`.

      (%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

  Categories: Differential calculus · Declarations and inferences ·

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`.

  Categories: Differential calculus · Global flags ·

Function: derivdegree (expr, y, x) ¶

  Returns the highest degree of the derivative of the dependent variable y with respect to the independent variable x occurring 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

  Categories: Differential calculus · Expressions ·

Function: derivlist (var_1, …, var_k) ¶

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

  Categories: Differential calculus · Evaluation ·

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)`.

  Categories: Differential calculus · Expressions ·

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

  See also `depends`, `del`, `derivdegree`, `derivabbrev`, and `gradef`.

  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.

  Categories: Differential calculus ·

Special symbol: diff ¶

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

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]

  Categories: Differential calculus · Vectors · Operators ·

Function: gradef
gradef (f(x_1, …, x_n), g_1, …, g_m)
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.

  Categories: Differential calculus · Declarations and inferences ·

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`.

  Categories: Differential calculus · Declarations and inferences ·

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