Math::Symbolic::VectorCalculus(3pm)

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NAME

6       Math::Symbolic::VectorCalculus - Symbolically comp. grad, Jacobi matri‐
7       ces etc.
8

SYNOPSIS

10         use Math::Symbolic qw/:all/;
11         use Math::Symbolic::VectorCalculus; # not loaded by Math::Symbolic
12
13         @gradient = grad 'x+y*z';
14         # or:
15         $function = parse_from_string('a*b^c');
16         @gradient = grad $function;
17         # or:
18         @signature = qw(x y z);
19         @gradient = grad 'a*x+b*y+c*z', @signature; # Gradient only for x, y, z
20         # or:
21         @gradient = grad $function, @signature;
22
23         # Similar syntax variations as with the gradient:
24         $divergence = div @functions;
25         $divergence = div @functions, @signature;
26
27         # Again, similar DWIM syntax variations as with grad:
28         @rotation = rot @functions;
29         @rotation = rot @functions, @signature;
30
31         # Signatures always inferred from the functions here:
32         @matrix = Jacobi @functions;
33         # $matrix is now array of array references. These hold
34         # Math::Symbolic trees. Or:
35         @matrix = Jacobi @functions, @signature;
36
37         # Similar to Jacobi:
38         @matrix = Hesse $function;
39         # or:
40         @matrix = Hesse $function, @signature;
41
42         $wronsky_determinant = WronskyDet @functions, @vars;
43         # or:
44         $wronsky_determinant = WronskyDet @functions; # functions of 1 variable
45
46         $differential = TotalDifferential $function;
47         $differential = TotalDifferential $function, @signature;
48         $differential = TotalDifferential $function, @signature, @point;
49
50         $dir_deriv = DirectionalDerivative $function, @vector;
51         $dir_deriv = DirectionalDerivative $function, @vector, @signature;
52
53         $taylor = TaylorPolyTwoDim $function, $var1, $var2, $degree;
54         $taylor = TaylorPolyTwoDim $function, $var1, $var2,
55                                    $degree, $var1_0, $var2_0;
56         # example:
57         $taylor = TaylorPolyTwoDim 'sin(x)*cos(y)', 'x', 'y', 2;
58

DESCRIPTION

60       This module provides several subroutines related to vector calculus
61       such as computing gradients, divergence, rotation, and Jacobi/Hesse
62       Matrices of Math::Symbolic trees.  Furthermore it provides means of
63       computing directional derivatives and the total differential of a
64       scalar function and the Wronsky Determinant of a set of n scalar func‐
65       tions.
66
67       Please note that the code herein may or may not be refactored into the
68       OO-interface of the Math::Symbolic module in the future.
69
70       EXPORT
71
72       None by default.
73
74       You may choose to have any of the following routines exported to the
75       calling namespace. ':all' tag exports all of the following:
76
77         grad
78         div
79         rot
80         Jacobi
81         Hesse
82         WronskyDet
83         TotalDifferential
84         DirectionalDerivative
85         TaylorPolyTwoDim
86

SUBROUTINES

88       grad
89
90       This subroutine computes the gradient of a Math::Symbolic tree repre‐
91       senting a function.
92
93       The gradient of a function f(x1, x2, ..., xn) is defined as the vector:
94
95         ( df(x1, x2, ..., xn) / d(x1),
96           df(x1, x2, ..., xn) / d(x2),
97           ...,
98           df(x1, x2, ..., xn) / d(xn) )
99
100       (These are all partial derivatives.) Any good book on calculus will
101       have more details on this.
102
103       grad uses prototypes to allow for a variety of usages. In its most
104       basic form, it accepts only one argument which may either be a
105       Math::Symbolic tree or a string both of which will be interpreted as
106       the function to compute the gradient for. Optionally, you may specify a
107       second argument which must be a (literal) array of Math::Sym‐
108       bolic::Variable objects or valid Math::Symbolic variable names
109       (strings). These variables will the be used for the gradient instead of
110       the x1, ..., xn inferred from the function signature.
111
112       div
113
114       This subroutine computes the divergence of a set of Math::Symbolic
115       trees representing a vectorial function.
116
117       The divergence of a vectorial function F = (f1(x1, ..., xn), ...,
118       fn(x1, ..., xn)) is defined like follows:
119
120         sum_from_i=1_to_n( dfi(x1, ..., xn) / dxi )
121
122       That is, the sum of all partial derivatives of the i-th component func‐
123       tion to the i-th coordinate. See your favourite book on calculus for
124       details.  Obviously, it is important to keep in mind that the number of
125       function components must be equal to the number of variables/coordi‐
126       nates.
127
128       Similar to grad, div uses prototypes to offer a comfortable interface.
129       First argument must be a (literal) array of strings and Math::Symbolic
130       trees which represent the vectorial function's components. If no second
131       argument is passed, the variables used for computing the divergence
132       will be inferred from the functions. That means the function signatures
133       will be joined to form a signature for the vectorial function.
134
135       If the optional second argument is specified, it has to be a (literal)
136       array of Math::Symbolic::Variable objects and valid variable names
137       (strings).  These will then be interpreted as the list of variables for
138       computing the divergence.
139
140       rot
141
142       This subroutine computes the rotation of a set of three Math::Symbolic
143       trees representing a vectorial function.
144
145       The rotation of a vectorial function F = (f1(x1, x2, x3), f2(x1, x2,
146       x3), f3(x1, x2, x3)) is defined as the following vector:
147
148         ( ( df3/dx2 - df2/dx3 ),
149           ( df1/dx3 - df3/dx1 ),
150           ( df2/dx1 - df1/dx2 ) )
151
152       Or "nabla x F" for short. Again, I have to refer to the literature for
153       the details on what rotation is. Please note that there have to be
154       exactly three function components and three coordinates because the
155       cross product and hence rotation is only defined in three dimensions.
156
157       As with the previously introduced subroutines div and grad, rot offers
158       a prototyped interface.  First argument must be a (literal) array of
159       strings and Math::Symbolic trees which represent the vectorial func‐
160       tion's components. If no second argument is passed, the variables used
161       for computing the rotation will be inferred from the functions. That
162       means the function signatures will be joined to form a signature for
163       the vectorial function.
164
165       If the optional second argument is specified, it has to be a (literal)
166       array of Math::Symbolic::Variable objects and valid variable names
167       (strings).  These will then be interpreted as the list of variables for
168       computing the rotation. (And please excuse my copying the last two
169       paragraphs from above.)
170
171       Jacobi
172
173       Jacobi() returns the Jacobi matrix of a given vectorial function.  It
174       expects any number of arguments (strings and/or Math::Symbolic trees)
175       which will be interpreted as the vectorial function's components.
176       Variables used for computing the matrix are, by default, inferred from
177       the combined signature of the components. By specifying a second lit‐
178       eral array of variable names as (second) argument, you may override
179       this behaviour.
180
181       The Jacobi matrix is the vector of gradient vectors of the vectorial
182       function's components.
183
184       Hesse
185
186       Hesse() returns the Hesse matrix of a given scalar function. First
187       argument must be a string (to be parsed as a Math::Symbolic tree) or a
188       Math::Symbolic tree. As with Jacobi(), Hesse() optionally accepts an
189       array of signature variables as second argument.
190
191       The Hesse matrix is the Jacobi matrix of the gradient of a scalar func‐
192       tion.
193
194       TotalDifferential
195
196       This function computes the total differential of a scalar function of
197       multiple variables in a certain point.
198
199       First argument must be the function to derive. The second argument is
200       an optional (literal) array of variable names (strings) and Math::Sym‐
201       bolic::Variable objects to be used for deriving. If the argument is not
202       specified, the functions signature will be used. The third argument is
203       also an optional array and denotes the set of variable (names) to use
204       for indicating the point for which to evaluate the differential. It
205       must have the same number of elements as the second argument.  If not
206       specified the variable names used as coordinated (the second argument)
207       with an appended '_0' will be used as the point's components.
208
209       DirectionalDerivative
210
211       DirectionalDerivative computes the directional derivative of a scalar
212       function in the direction of a specified vector. With f being the func‐
213       tion and X, A being vectors, it looks like this: (this is a partial de‐
214       rivative)
215
216         df(X)/dA = grad(f(X)) * (A / ⎪A⎪)
217
218       First argument must be the function to derive (either a string or a
219       valid Math::Symbolic tree). Second argument must be vector into whose
220       direction to derive. It is to be specified as an array of variable
221       names and objects.  Third argument is the optional signature to be used
222       for computing the gradient.  Please see the documentation of the grad
223       function for details. It's dimension must match that of the directional
224       vector.
225
226       TaylorPolyTwoDim
227
228       This subroutine computes the Taylor Polynomial for functions of two
229       variables. Please refer to the documentation of the TaylorPolynomial
230       function in the Math::Symbolic::MiscCalculus package for an explanation
231       of single dimensional Taylor Polynomials. This is the counterpart in
232       two dimensions.
233
234       First argument must be the function to approximate with the Taylor
235       Polynomial either as a string or a Math::Symbolic tree. Second and
236       third argument must be the names of the two coordinates. (These may
237       alternatively be Math::Symbolic::Variable objects.) Fourth argument
238       must be the degree of the Taylor Polynomial. Fifth and Sixth arguments
239       are optional and specify the names of the variables to introduce as the
240       point of approximation. These default to the names of the coordinates
241       with '_0' appended.
242
243       WronskyDet
244
245       WronskyDet() computes the Wronsky Determinant of a set of n functions.
246
247       First argument is required and a (literal) array of n functions. Second
248       argument is optional and a (literal) array of n variables or variable
249       names.  If the second argument is omitted, the variables used for
250       deriving are inferred from function signatures. This requires, however,
251       that the function signatures have exactly one element. (And the func‐
252       tion this exactly one variable.)
253

AUTHOR

255       Please send feedback, bug reports, and support requests to the
256       Math::Symbolic support mailing list: math-symbolic-support at lists dot
257       sourceforge dot net. Please consider letting us know how you use
258       Math::Symbolic. Thank you.
259
260       If you're interested in helping with the development or extending the
261       module's functionality, please contact the developers' mailing list:
262       math-symbolic-develop at lists dot sourceforge dot net.
263
264       List of contributors:
265
266         Steffen Müller, symbolic-module at steffen-mueller dot net
267         Stray Toaster, mwk at users dot sourceforge dot net
268         Oliver Ebenhöh
269

NAME

SYNOPSIS

DESCRIPTION

SUBROUTINES

AUTHOR

SEE ALSO