Utilities for Calculus¶
This module defines helper functions which are used for simplifications and display of symbolic expressions.
AUTHORS:
- Michal Bejger (2015) : class
ExpressionNice
- Eric Gourgoulhon (2015) : simplification functions
- Travis Scrimshaw (2016): review tweaks
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class
sage.manifolds.utilities.
ExpressionNice
(ex)¶ Bases:
sage.symbolic.expression.Expression
Subclass of
Expression
for a “human-friendly” display of partial derivatives and the possibility to shorten the display by skipping the arguments of symbolic functions.INPUT:
ex
– symbolic expression
EXAMPLES:
An expression formed with callable symbolic expressions:
sage: var('x y z') (x, y, z) sage: f = function('f')(x, y) sage: g = f.diff(y).diff(x) sage: h = function('h')(y, z) sage: k = h.diff(z) sage: fun = x*g + y*(k-z)^2
The standard Pynac display of partial derivatives:
sage: fun y*(z - diff(h(y, z), z))^2 + x*diff(f(x, y), x, y) sage: latex(fun) y {\left(z - \frac{\partial}{\partial z}h\left(y, z\right)\right)}^{2} + x \frac{\partial^{2}}{\partial x\partial y}f\left(x, y\right)
With
ExpressionNice
, the Pynac notationD[...]
is replaced by textbook-like notation:sage: from sage.manifolds.utilities import ExpressionNice sage: ExpressionNice(fun) y*(z - d(h)/dz)^2 + x*d^2(f)/dxdy sage: latex(ExpressionNice(fun)) y {\left(z - \frac{\partial\,h}{\partial z}\right)}^{2} + x \frac{\partial^2\,f}{\partial x\partial y}
An example when function variables are themselves functions:
sage: f = function('f')(x, y) sage: g = function('g')(x, f) # the second variable is the function f sage: fun = (g.diff(x))*x - x^2*f.diff(x,y) sage: fun -x^2*diff(f(x, y), x, y) + (diff(f(x, y), x)*D[1](g)(x, f(x, y)) + D[0](g)(x, f(x, y)))*x sage: ExpressionNice(fun) -x^2*d^2(f)/dxdy + (d(f)/dx*d(g)/d(f(x, y)) + d(g)/dx)*x sage: latex(ExpressionNice(fun)) -x^{2} \frac{\partial^2\,f}{\partial x\partial y} + {\left(\frac{\partial\,f}{\partial x} \frac{\partial\,g}{\partial \left( f\left(x, y\right) \right)} + \frac{\partial\,g}{\partial x}\right)} x
Note that
D[1](g)(x, f(x,y))
is rendered asd(g)/d(f(x, y))
.An example with multiple differentiations:
sage: fun = f.diff(x,x,y,y,x)*x sage: fun x*diff(f(x, y), x, x, x, y, y) sage: ExpressionNice(fun) x*d^5(f)/dx^3dy^2 sage: latex(ExpressionNice(fun)) x \frac{\partial^5\,f}{\partial x ^ 3\partial y ^ 2}
Parentheses are added around powers of partial derivatives to avoid any confusion:
sage: fun = f.diff(y)^2 sage: fun diff(f(x, y), y)^2 sage: ExpressionNice(fun) (d(f)/dy)^2 sage: latex(ExpressionNice(fun)) \left(\frac{\partial\,f}{\partial y}\right)^{2}
The explicit mention of function arguments can be omitted for the sake of brevity:
sage: fun = fun*f sage: ExpressionNice(fun) f(x, y)*(d(f)/dy)^2 sage: Manifold.options.omit_function_arguments=True sage: ExpressionNice(fun) f*(d(f)/dy)^2 sage: latex(ExpressionNice(fun)) f \left(\frac{\partial\,f}{\partial y}\right)^{2} sage: Manifold.options._reset() sage: ExpressionNice(fun) f(x, y)*(d(f)/dy)^2 sage: latex(ExpressionNice(fun)) f\left(x, y\right) \left(\frac{\partial\,f}{\partial y}\right)^{2}
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sage.manifolds.utilities.
set_axes_labels
(graph, xlabel, ylabel, zlabel, **kwds)¶ Set axes labels for a 3D graphics object
graph
.This is a workaround for the lack of axes labels in 3D plots. This sets the labels as
text3d()
objects at locations determined from the bounding box of the graphic objectgraph
.INPUT:
graph
–Graphics3d
; a 3D graphic objectxlabel
– string for the x-axis labelylabel
– string for the y-axis labelzlabel
– string for the z-axis label**kwds
– options (e.g. color) for text3d
OUTPUT:
- the 3D graphic object with text3d labels added
EXAMPLES:
sage: g = sphere() sage: g.all [Graphics3d Object] sage: from sage.manifolds.utilities import set_axes_labels sage: ga = set_axes_labels(g, 'X', 'Y', 'Z', color='red') sage: ga.all # the 3D frame has now axes labels [Graphics3d Object, Graphics3d Object, Graphics3d Object, Graphics3d Object]
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sage.manifolds.utilities.
simplify_abs_trig
(expr)¶ Simplify
abs(sin(...))
in symbolic expressions.EXAMPLES:
sage: forget() # for doctests only sage: M = Manifold(3, 'M', structure='topological') sage: X.<x,y,z> = M.chart(r'x y:(0,pi) z:(-pi/3,0)') sage: X.coord_range() x: (-oo, +oo); y: (0, pi); z: (-1/3*pi, 0)
Since
x
spans all \(\RR\), no simplification ofabs(sin(x))
occurs, whileabs(sin(y))
andabs(sin(3*z))
are correctly simplified, given that \(y \in (0,\pi)\) and \(z \in (-\pi/3,0)\):sage: from sage.manifolds.utilities import simplify_abs_trig sage: simplify_abs_trig( abs(sin(x)) + abs(sin(y)) + abs(sin(3*z)) ) abs(sin(x)) + sin(y) - sin(3*z)
Note that neither Sage’s function
simplify_trig()
norsimplify_full()
works in this case:sage: s = abs(sin(x)) + abs(sin(y)) + abs(sin(3*z)) sage: s.simplify_trig() abs(4*cos(z)^2 - 1)*abs(sin(z)) + abs(sin(x)) + abs(sin(y)) sage: s.simplify_full() abs(4*cos(z)^2 - 1)*abs(sin(z)) + abs(sin(x)) + abs(sin(y))
despite the following assumptions hold:
sage: assumptions() [x is real, y is real, y > 0, y < pi, z is real, z > -1/3*pi, z < 0]
Additional checks are:
sage: simplify_abs_trig( abs(sin(y/2)) ) # shall simplify sin(1/2*y) sage: simplify_abs_trig( abs(sin(2*y)) ) # must not simplify abs(sin(2*y)) sage: simplify_abs_trig( abs(sin(z/2)) ) # shall simplify -sin(1/2*z) sage: simplify_abs_trig( abs(sin(4*z)) ) # must not simplify abs(sin(4*z))
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sage.manifolds.utilities.
simplify_chain_generic
(expr)¶ Apply a chain of simplifications to a symbolic expression.
This is the simplification chain used in calculus involving coordinate functions on manifolds over fields different from \(\RR\), as implemented in
CoordFunctionSymb
.The chain is formed by the following functions, called successively:
NB: for the time being, this is identical to
simplify_full()
.EXAMPLES:
We consider variables that are coordinates of a chart on a complex manifold:
sage: forget() # for doctest only sage: M = Manifold(2, 'M', structure='topological', field='complex') sage: X.<x,y> = M.chart()
Then neither
x
nory
is assumed to be real:sage: assumptions() []
Accordingly,
simplify_chain_generic
does not simplifysqrt(x^2)
toabs(x)
:sage: from sage.manifolds.utilities import simplify_chain_generic sage: s = sqrt(x^2) sage: simplify_chain_generic(s) sqrt(x^2)
This contrasts with the behavior of
simplify_chain_real()
.Other simplifications:
sage: s = (x+y)^2 - x^2 -2*x*y - y^2 sage: simplify_chain_generic(s) 0 sage: s = (x^2 - 2*x + 1) / (x^2 -1) sage: simplify_chain_generic(s) (x - 1)/(x + 1) sage: s = cos(2*x) - 2*cos(x)^2 + 1 sage: simplify_chain_generic(s) 0
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sage.manifolds.utilities.
simplify_chain_real
(expr)¶ Apply a chain of simplifications to a symbolic expression, assuming the real domain.
This is the simplification chain used in calculus involving coordinate functions on real manifolds, as implemented in
CoordFunctionSymb
.The chain is formed by the following functions, called successively:
simplify_factorial()
simplify_trig()
simplify_rational()
simplify_sqrt_real()
simplify_abs_trig()
canonicalize_radical()
simplify_log()
simplify_rational()
simplify_trig()
EXAMPLES:
We consider variables that are coordinates of a chart on a real manifold:
sage: forget() # for doctest only sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart('x:(0,1) y')
The following assumptions then hold:
sage: assumptions() [x is real, x > 0, x < 1, y is real]
and we have:
sage: from sage.manifolds.utilities import simplify_chain_real sage: s = sqrt(y^2) sage: simplify_chain_real(s) abs(y)
The above result is correct since
y
is real. It is obtained bysimplify_real()
as well, but not bysimplify_full()
:sage: s.simplify_real() abs(y) sage: s.simplify_full() sqrt(y^2)
Furthermore, we have:
sage: s = sqrt(x^2-2*x+1) sage: simplify_chain_real(s) -x + 1
which is correct since \(x \in (0,1)\). On this example, neither
simplify_real()
norsimplify_full()
, norcanonicalize_radical()
give satisfactory results:sage: s.simplify_real() # unsimplified output sqrt(x^2 - 2*x + 1) sage: s.simplify_full() # unsimplified output sqrt(x^2 - 2*x + 1) sage: s.canonicalize_radical() # wrong output since x in (0,1) x - 1
Other simplifications:
sage: s = abs(sin(pi*x)) sage: simplify_chain_real(s) # correct output since x in (0,1) sin(pi*x) sage: s.simplify_real() # unsimplified output abs(sin(pi*x)) sage: s.simplify_full() # unsimplified output abs(sin(pi*x))
sage: s = cos(y)^2 + sin(y)^2 sage: simplify_chain_real(s) 1 sage: s.simplify_real() # unsimplified output cos(y)^2 + sin(y)^2 sage: s.simplify_full() # OK 1
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sage.manifolds.utilities.
simplify_sqrt_real
(expr)¶ Simplify
sqrt
in symbolic expressions in the real domain.EXAMPLES:
Simplifications of basic expressions:
sage: from sage.manifolds.utilities import simplify_sqrt_real sage: simplify_sqrt_real( sqrt(x^2) ) abs(x) sage: assume(x<0) sage: simplify_sqrt_real( sqrt(x^2) ) -x sage: simplify_sqrt_real( sqrt(x^2-2*x+1) ) -x + 1 sage: simplify_sqrt_real( sqrt(x^2) + sqrt(x^2-2*x+1) ) -2*x + 1
This improves over Sage’s
canonicalize_radical()
, which yields incorrect results whenx < 0
:sage: forget() # removes the assumption x<0 sage: sqrt(x^2).canonicalize_radical() x sage: assume(x<0) sage: sqrt(x^2).canonicalize_radical() # wrong output x sage: sqrt(x^2-2*x+1).canonicalize_radical() # wrong output x - 1 sage: ( sqrt(x^2) + sqrt(x^2-2*x+1) ).canonicalize_radical() # wrong output 2*x - 1
Simplification of nested
sqrt
‘s:sage: forget() # removes the assumption x<0 sage: simplify_sqrt_real( sqrt(1 + sqrt(x^2)) ) sqrt(abs(x) + 1) sage: assume(x<0) sage: simplify_sqrt_real( sqrt(1 + sqrt(x^2)) ) sqrt(-x + 1) sage: simplify_sqrt_real( sqrt(x^2 + sqrt(4*x^2) + 1) ) -x + 1
Again,
canonicalize_radical()
fails on the last one:sage: (sqrt(x^2 + sqrt(4*x^2) + 1)).canonicalize_radical() # wrong output x + 1