Chart Functions

In the context of a topological manifold \(M\) over a topological field \(K\), a chart function is a function from a chart codomain to \(K\). In other words, a chart function is a \(K\)-valued function of the coordinates associated to some chart. The internal coordinate expressions of chart functions and calculus on them are taken in charge by different calculus methods, at the choice of the user:

  • Sage’s default symbolic engine (Pynac + Maxima), implemented via the Symbolic Ring (SR)

  • SymPy engine, denoted sympy hereafter

See CalculusMethod for details.

AUTHORS:

  • Marco Mancini (2017) : initial version

  • Eric Gourgoulhon (2015) : for a previous class implementing only SR calculus (CoordFunctionSymb)

  • Florentin Jaffredo (2018) : series expansion with respect to a given parameter

class sage.manifolds.chart_func.ChartFunction(parent, expression=None, calc_method=None, expansion_symbol=None, order=None)[source]

Bases: AlgebraElement, ModuleElementWithMutability

Function of coordinates of a given chart.

If \((U, \varphi)\) is a chart on a topological manifold \(M\) of dimension \(n\) over a topological field \(K\), a chart function associated to \((U, \varphi)\) is a map

\[\begin{split}\begin{array}{llcl} f:& V \subset K^n & \longrightarrow & K \\ & (x^1, \ldots, x^n) & \longmapsto & f(x^1, \ldots, x^n), \end{array}\end{split}\]

where \(V\) is the codomain of \(\varphi\). In other words, \(f\) is a \(K\)-valued function of the coordinates associated to the chart \((U, \varphi)\).

The chart function \(f\) can be represented by expressions pertaining to different calculus methods; the currently implemented ones are

  • SR (Sage’s Symbolic Ring)

  • SymPy

See expr() for details.

INPUT:

  • parent – the algebra of chart functions on the chart \((U, \varphi)\)

  • expression – (default: None) a symbolic expression representing \(f(x^1, \ldots, x^n)\), where \((x^1, \ldots, x^n)\) are the coordinates of the chart \((U, \varphi)\)

  • calc_method – string (default: None); the calculus method with respect to which the internal expression of self must be initialized from expression. One of

    • 'SR': Sage’s default symbolic engine (Symbolic Ring)

    • 'sympy': SymPy

    • None: the chart current calculus method is assumed

  • expansion_symbol – (default: None) symbolic variable (the “small parameter”) with respect to which the coordinate expression is expanded in power series (around the zero value of this variable)

  • order – integer (default: None); the order of the expansion if expansion_symbol is not None; the order is defined as the degree of the polynomial representing the truncated power series in expansion_symbol

    Warning

    The value of order is \(n-1\), where \(n\) is the order of the big \(O\) in the power series expansion

EXAMPLES:

A symbolic chart function on a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x^2+3*y+1)
sage: type(f)
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>
sage: f.display()
(x, y) ↦ x^2 + 3*y + 1
sage: f(x,y)
x^2 + 3*y + 1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2)+Integer(3)*y+Integer(1))
>>> type(f)
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>
>>> f.display()
(x, y) ↦ x^2 + 3*y + 1
>>> f(x,y)
x^2 + 3*y + 1

The symbolic expression is returned when asking for the direct display of the function:

sage: f
x^2 + 3*y + 1
sage: latex(f)
x^{2} + 3 \, y + 1

>>> from sage.all import *
>>> f
x^2 + 3*y + 1
>>> latex(f)
x^{2} + 3 \, y + 1

A similar output is obtained by means of the method expr():

sage: f.expr()
x^2 + 3*y + 1

>>> from sage.all import *
>>> f.expr()
x^2 + 3*y + 1

The expression returned by expr() is by default a Sage symbolic expression:

sage: type(f.expr())
<class 'sage.symbolic.expression.Expression'>

>>> from sage.all import *
>>> type(f.expr())
<class 'sage.symbolic.expression.Expression'>

A SymPy expression can also be asked for:

sage: f.expr('sympy')
x**2 + 3*y + 1
sage: type(f.expr('sympy'))
<class 'sympy.core.add.Add'>

>>> from sage.all import *
>>> f.expr('sympy')
x**2 + 3*y + 1
>>> type(f.expr('sympy'))
<class 'sympy.core.add.Add'>

The value of the function at specified coordinates is obtained by means of the standard parentheses notation:

sage: f(2,-1)
2
sage: var('a b')
(a, b)
sage: f(a,b)
a^2 + 3*b + 1

>>> from sage.all import *
>>> f(Integer(2),-Integer(1))
2
>>> var('a b')
(a, b)
>>> f(a,b)
a^2 + 3*b + 1

An unspecified chart function:

sage: g = X.function(function('G')(x, y))
sage: g
G(x, y)
sage: g.display()
(x, y) ↦ G(x, y)
sage: g.expr()
G(x, y)
sage: g(2,3)
G(2, 3)

>>> from sage.all import *
>>> g = X.function(function('G')(x, y))
>>> g
G(x, y)
>>> g.display()
(x, y) ↦ G(x, y)
>>> g.expr()
G(x, y)
>>> g(Integer(2),Integer(3))
G(2, 3)

Coordinate functions can be compared to other values:

sage: f = X.function(x^2+3*y+1)
sage: f == 2
False
sage: f == x^2 + 3*y + 1
True
sage: g = X.function(x*y)
sage: f == g
False
sage: h = X.function(x^2+3*y+1)
sage: f == h
True

>>> from sage.all import *
>>> f = X.function(x**Integer(2)+Integer(3)*y+Integer(1))
>>> f == Integer(2)
False
>>> f == x**Integer(2) + Integer(3)*y + Integer(1)
True
>>> g = X.function(x*y)
>>> f == g
False
>>> h = X.function(x**Integer(2)+Integer(3)*y+Integer(1))
>>> f == h
True

A coercion by means of the restriction is implemented:

sage: D = M.open_subset('D')
sage: X_D = X.restrict(D, x^2+y^2<1)  # open disk
sage: c = X_D.function(x^2)
sage: c + f
2*x^2 + 3*y + 1

>>> from sage.all import *
>>> D = M.open_subset('D')
>>> X_D = X.restrict(D, x**Integer(2)+y**Integer(2)<Integer(1))  # open disk
>>> c = X_D.function(x**Integer(2))
>>> c + f
2*x^2 + 3*y + 1

Expansion to a given order with respect to a small parameter:

sage: t = var('t')  # the small parameter
sage: f = X.function(cos(t)*x*y, expansion_symbol=t, order=2)

>>> from sage.all import *
>>> t = var('t')  # the small parameter
>>> f = X.function(cos(t)*x*y, expansion_symbol=t, order=Integer(2))

The expansion is triggered by the call to simplify():

sage: f
x*y*cos(t)
sage: f.simplify()
-1/2*t^2*x*y + x*y

>>> from sage.all import *
>>> f
x*y*cos(t)
>>> f.simplify()
-1/2*t^2*x*y + x*y

Differences between ChartFunction and callable symbolic expressions

Callable symbolic expressions are defined directly from symbolic expressions of the coordinates:

sage: f0(x,y) = x^2 + 3*y + 1
sage: type(f0)
<class 'sage.symbolic.expression.Expression'>
sage: f0
(x, y) |--> x^2 + 3*y + 1
sage: f0(x,y)
x^2 + 3*y + 1

>>> from sage.all import *
>>> __tmp__=var("x,y"); f0 = symbolic_expression(x**Integer(2) + Integer(3)*y + Integer(1)).function(x,y)
>>> type(f0)
<class 'sage.symbolic.expression.Expression'>
>>> f0
(x, y) |--> x^2 + 3*y + 1
>>> f0(x,y)
x^2 + 3*y + 1

To get an output similar to that of f0 for a chart function, we must use the method display():

sage: f = X.function(x^2+3*y+1)
sage: f
x^2 + 3*y + 1
sage: f.display()
(x, y) ↦ x^2 + 3*y + 1
sage: f(x,y)
x^2 + 3*y + 1

>>> from sage.all import *
>>> f = X.function(x**Integer(2)+Integer(3)*y+Integer(1))
>>> f
x^2 + 3*y + 1
>>> f.display()
(x, y) ↦ x^2 + 3*y + 1
>>> f(x,y)
x^2 + 3*y + 1

More importantly, instances of ChartFunction differ from callable symbolic expression by the automatic simplifications in all operations. For instance, adding the two callable symbolic expressions:

sage: f0(x,y,z) = cos(x)^2 ; g0(x,y,z) = sin(x)^2

>>> from sage.all import *
>>> __tmp__=var("x,y,z"); f0 = symbolic_expression(cos(x)**Integer(2) ).function(x,y,z); __tmp__=var("x,y,z"); g0 = symbolic_expression(sin(x)**Integer(2)).function(x,y,z)

results in:

sage: f0 + g0
(x, y, z) |--> cos(x)^2 + sin(x)^2

>>> from sage.all import *
>>> f0 + g0
(x, y, z) |--> cos(x)^2 + sin(x)^2

To get \(1\), one has to call simplify_trig():

sage: (f0 + g0).simplify_trig()
(x, y, z) |--> 1

>>> from sage.all import *
>>> (f0 + g0).simplify_trig()
(x, y, z) |--> 1

On the contrary, the sum of the corresponding ChartFunction instances is automatically simplified (see simplify_chain_real() and simplify_chain_generic() for details):

sage: f = X.function(cos(x)^2) ; g = X.function(sin(x)^2)
sage: f + g
1

>>> from sage.all import *
>>> f = X.function(cos(x)**Integer(2)) ; g = X.function(sin(x)**Integer(2))
>>> f + g
1

Another difference regards the display of partial derivatives: for callable symbolic functions, it involves diff:

sage: g = function('g')(x, y)
sage: f0(x,y) = diff(g, x) + diff(g, y)
sage: f0
(x, y) |--> diff(g(x, y), x) + diff(g(x, y), y)

>>> from sage.all import *
>>> g = function('g')(x, y)
>>> __tmp__=var("x,y"); f0 = symbolic_expression(diff(g, x) + diff(g, y)).function(x,y)
>>> f0
(x, y) |--> diff(g(x, y), x) + diff(g(x, y), y)

while for chart functions, the display is more “textbook” like:

sage: f = X.function(diff(g, x) + diff(g, y))
sage: f
d(g)/dx + d(g)/dy

>>> from sage.all import *
>>> f = X.function(diff(g, x) + diff(g, y))
>>> f
d(g)/dx + d(g)/dy

The difference is even more dramatic on LaTeX outputs:

sage: latex(f0)
\left( x, y \right) \ {\mapsto} \ \frac{\partial}{\partial x}g\left(x, y\right) + \frac{\partial}{\partial y}g\left(x, y\right)
sage: latex(f)
\frac{\partial\,g}{\partial x} + \frac{\partial\,g}{\partial y}

>>> from sage.all import *
>>> latex(f0)
\left( x, y \right) \ {\mapsto} \ \frac{\partial}{\partial x}g\left(x, y\right) + \frac{\partial}{\partial y}g\left(x, y\right)
>>> latex(f)
\frac{\partial\,g}{\partial x} + \frac{\partial\,g}{\partial y}

Note that this regards only the display of coordinate functions: internally, the diff notation is still used, as we can check by asking for the symbolic expression stored in f:

sage: f.expr()
diff(g(x, y), x) + diff(g(x, y), y)

>>> from sage.all import *
>>> f.expr()
diff(g(x, y), x) + diff(g(x, y), y)

One can switch to Pynac notation by changing the options:

sage: Manifold.options.textbook_output=False
sage: latex(f)
\frac{\partial}{\partial x}g\left(x, y\right) + \frac{\partial}{\partial y}g\left(x, y\right)
sage: Manifold.options._reset()
sage: latex(f)
\frac{\partial\,g}{\partial x} + \frac{\partial\,g}{\partial y}

>>> from sage.all import *
>>> Manifold.options.textbook_output=False
>>> latex(f)
\frac{\partial}{\partial x}g\left(x, y\right) + \frac{\partial}{\partial y}g\left(x, y\right)
>>> Manifold.options._reset()
>>> latex(f)
\frac{\partial\,g}{\partial x} + \frac{\partial\,g}{\partial y}

Another difference between ChartFunction and callable symbolic expression is the possibility to switch off the display of the arguments of unspecified functions. Consider for instance:

sage: f = X.function(function('u')(x, y) * function('v')(x, y))
sage: f
u(x, y)*v(x, y)
sage: f0(x,y) = function('u')(x, y) * function('v')(x, y)
sage: f0
(x, y) |--> u(x, y)*v(x, y)

>>> from sage.all import *
>>> f = X.function(function('u')(x, y) * function('v')(x, y))
>>> f
u(x, y)*v(x, y)
>>> __tmp__=var("x,y"); f0 = symbolic_expression(function('u')(x, y) * function('v')(x, y)).function(x,y)
>>> f0
(x, y) |--> u(x, y)*v(x, y)

If there is a clear understanding that \(u\) and \(v\) are functions of \((x,y)\), the explicit mention of the latter can be cumbersome in lengthy tensor expressions. We can switch it off by:

sage: Manifold.options.omit_function_arguments=True
sage: f
u*v

>>> from sage.all import *
>>> Manifold.options.omit_function_arguments=True
>>> f
u*v

Note that neither the callable symbolic expression f0 nor the internal expression of f is affected by the above command:

sage: f0
(x, y) |--> u(x, y)*v(x, y)
sage: f.expr()
u(x, y)*v(x, y)

>>> from sage.all import *
>>> f0
(x, y) |--> u(x, y)*v(x, y)
>>> f.expr()
u(x, y)*v(x, y)

We revert to the default behavior by:

sage: Manifold.options._reset()
sage: f
u(x, y)*v(x, y)

>>> from sage.all import *
>>> Manifold.options._reset()
>>> f
u(x, y)*v(x, y)
__call__(*coords, **options)[source]

Compute the value of the function at specified coordinates.

INPUT:

  • *coords – list of coordinates \((x^1, \ldots, x^n)\), where the function \(f\) is to be evaluated

  • **options – allows to pass simplify=False to disable the call of the simplification chain on the result

OUTPUT:

  • the value \(f(x^1, \ldots, x^n)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(sin(x*y))
sage: f.__call__(-2, 3)
-sin(6)
sage: f(-2, 3)
-sin(6)
sage: var('a b')
(a, b)
sage: f.__call__(a, b)
sin(a*b)
sage: f(a,b)
sin(a*b)
sage: f.__call__(pi, 1)
0
sage: f.__call__(pi, 1/2)
1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(sin(x*y))
>>> f.__call__(-Integer(2), Integer(3))
-sin(6)
>>> f(-Integer(2), Integer(3))
-sin(6)
>>> var('a b')
(a, b)
>>> f.__call__(a, b)
sin(a*b)
>>> f(a,b)
sin(a*b)
>>> f.__call__(pi, Integer(1))
0
>>> f.__call__(pi, Integer(1)/Integer(2))
1

With SymPy:

sage: X.calculus_method().set('sympy')
sage: f(-2,3)
-sin(6)
sage: type(f(-2,3))
<class 'sympy.core.mul.Mul'>
sage: f(a,b)
sin(a*b)
sage: type(f(a,b))
sin
sage: type(f(pi,1))
<class 'sympy.core.numbers.Zero'>
sage: f(pi, 1/2)
1
sage: type(f(pi, 1/2))
<class 'sympy.core.numbers.One'>

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f(-Integer(2),Integer(3))
-sin(6)
>>> type(f(-Integer(2),Integer(3)))
<class 'sympy.core.mul.Mul'>
>>> f(a,b)
sin(a*b)
>>> type(f(a,b))
sin
>>> type(f(pi,Integer(1)))
<class 'sympy.core.numbers.Zero'>
>>> f(pi, Integer(1)/Integer(2))
1
>>> type(f(pi, Integer(1)/Integer(2)))
<class 'sympy.core.numbers.One'>
arccos()[source]

Arc cosine of self.

OUTPUT:

  • chart function \(\arccos(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.arccos()
arccos(x*y)
sage: arccos(f)  # equivalent to f.arccos()
arccos(x*y)
sage: acos(f)  # equivalent to f.arccos()
arccos(x*y)
sage: arccos(f).display()
(x, y) ↦ arccos(x*y)
sage: arccos(X.zero_function()).display()
(x, y) ↦ 1/2*pi

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.arccos()
arccos(x*y)
>>> arccos(f)  # equivalent to f.arccos()
arccos(x*y)
>>> acos(f)  # equivalent to f.arccos()
arccos(x*y)
>>> arccos(f).display()
(x, y) ↦ arccos(x*y)
>>> arccos(X.zero_function()).display()
(x, y) ↦ 1/2*pi

The same test with SymPy:

sage: M.set_calculus_method('sympy')
sage: f = X.function(x*y)
sage: f.arccos()
acos(x*y)
sage: arccos(f)  # equivalent to f.arccos()
acos(x*y)
sage: acos(f)  # equivalent to f.arccos()
acos(x*y)
sage: arccos(f).display()
(x, y) ↦ acos(x*y)

>>> from sage.all import *
>>> M.set_calculus_method('sympy')
>>> f = X.function(x*y)
>>> f.arccos()
acos(x*y)
>>> arccos(f)  # equivalent to f.arccos()
acos(x*y)
>>> acos(f)  # equivalent to f.arccos()
acos(x*y)
>>> arccos(f).display()
(x, y) ↦ acos(x*y)
arccosh()[source]

Inverse hyperbolic cosine of self.

OUTPUT:

  • chart function \(\mathrm{arccosh}(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.arccosh()
arccosh(x*y)
sage: arccosh(f)  # equivalent to f.arccosh()
arccosh(x*y)
sage: acosh(f)  # equivalent to f.arccosh()
arccosh(x*y)
sage: arccosh(f).display()
(x, y) ↦ arccosh(x*y)
sage: arccosh(X.function(1)) == X.zero_function()
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.arccosh()
arccosh(x*y)
>>> arccosh(f)  # equivalent to f.arccosh()
arccosh(x*y)
>>> acosh(f)  # equivalent to f.arccosh()
arccosh(x*y)
>>> arccosh(f).display()
(x, y) ↦ arccosh(x*y)
>>> arccosh(X.function(Integer(1))) == X.zero_function()
True

The same tests with SymPy:

sage: X.calculus_method().set('sympy')
sage: f.arccosh()
acosh(x*y)
sage: arccosh(f)  # equivalent to f.arccosh()
acosh(x*y)
sage: acosh(f)  # equivalent to f.arccosh()
acosh(x*y)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f.arccosh()
acosh(x*y)
>>> arccosh(f)  # equivalent to f.arccosh()
acosh(x*y)
>>> acosh(f)  # equivalent to f.arccosh()
acosh(x*y)
arcsin()[source]

Arc sine of self.

OUTPUT:

  • chart function \(\arcsin(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.arcsin()
arcsin(x*y)
sage: arcsin(f)  # equivalent to f.arcsin()
arcsin(x*y)
sage: asin(f)  # equivalent to f.arcsin()
arcsin(x*y)
sage: arcsin(f).display()
(x, y) ↦ arcsin(x*y)
sage: arcsin(X.zero_function()) == X.zero_function()
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.arcsin()
arcsin(x*y)
>>> arcsin(f)  # equivalent to f.arcsin()
arcsin(x*y)
>>> asin(f)  # equivalent to f.arcsin()
arcsin(x*y)
>>> arcsin(f).display()
(x, y) ↦ arcsin(x*y)
>>> arcsin(X.zero_function()) == X.zero_function()
True

The same tests with SymPy:

sage: X.calculus_method().set('sympy')
sage: f.arcsin()
asin(x*y)
sage: arcsin(f)  # equivalent to f.arcsin()
asin(x*y)
sage: asin(f)  # equivalent to f.arcsin()
asin(x*y)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f.arcsin()
asin(x*y)
>>> arcsin(f)  # equivalent to f.arcsin()
asin(x*y)
>>> asin(f)  # equivalent to f.arcsin()
asin(x*y)
arcsinh()[source]

Inverse hyperbolic sine of self.

OUTPUT:

  • chart function \(\mathrm{arcsinh}(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.arcsinh()
arcsinh(x*y)
sage: arcsinh(f)  # equivalent to f.arcsinh()
arcsinh(x*y)
sage: asinh(f)  # equivalent to f.arcsinh()
arcsinh(x*y)
sage: arcsinh(f).display()
(x, y) ↦ arcsinh(x*y)
sage: arcsinh(X.zero_function()) == X.zero_function()
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.arcsinh()
arcsinh(x*y)
>>> arcsinh(f)  # equivalent to f.arcsinh()
arcsinh(x*y)
>>> asinh(f)  # equivalent to f.arcsinh()
arcsinh(x*y)
>>> arcsinh(f).display()
(x, y) ↦ arcsinh(x*y)
>>> arcsinh(X.zero_function()) == X.zero_function()
True

The same tests with SymPy:

sage: X.calculus_method().set('sympy')
sage: f.arcsinh()
asinh(x*y)
sage: arcsinh(f)  # equivalent to f.arcsinh()
asinh(x*y)
sage: asinh(f)  # equivalent to f.arcsinh()
asinh(x*y)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f.arcsinh()
asinh(x*y)
>>> arcsinh(f)  # equivalent to f.arcsinh()
asinh(x*y)
>>> asinh(f)  # equivalent to f.arcsinh()
asinh(x*y)
arctan()[source]

Arc tangent of self.

OUTPUT:

  • chart function \(\arctan(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.arctan()
arctan(x*y)
sage: arctan(f)  # equivalent to f.arctan()
arctan(x*y)
sage: atan(f)  # equivalent to f.arctan()
arctan(x*y)
sage: arctan(f).display()
(x, y) ↦ arctan(x*y)
sage: arctan(X.zero_function()) == X.zero_function()
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.arctan()
arctan(x*y)
>>> arctan(f)  # equivalent to f.arctan()
arctan(x*y)
>>> atan(f)  # equivalent to f.arctan()
arctan(x*y)
>>> arctan(f).display()
(x, y) ↦ arctan(x*y)
>>> arctan(X.zero_function()) == X.zero_function()
True

The same tests with SymPy:

sage: X.calculus_method().set('sympy')
sage: f.arctan()
atan(x*y)
sage: arctan(f)  # equivalent to f.arctan()
atan(x*y)
sage: atan(f)  # equivalent to f.arctan()
atan(x*y)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f.arctan()
atan(x*y)
>>> arctan(f)  # equivalent to f.arctan()
atan(x*y)
>>> atan(f)  # equivalent to f.arctan()
atan(x*y)
arctanh()[source]

Inverse hyperbolic tangent of self.

OUTPUT:

  • chart function \(\mathrm{arctanh}(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.arctanh()
arctanh(x*y)
sage: arctanh(f)  # equivalent to f.arctanh()
arctanh(x*y)
sage: atanh(f)  # equivalent to f.arctanh()
arctanh(x*y)
sage: arctanh(f).display()
(x, y) ↦ arctanh(x*y)
sage: arctanh(X.zero_function()) == X.zero_function()
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.arctanh()
arctanh(x*y)
>>> arctanh(f)  # equivalent to f.arctanh()
arctanh(x*y)
>>> atanh(f)  # equivalent to f.arctanh()
arctanh(x*y)
>>> arctanh(f).display()
(x, y) ↦ arctanh(x*y)
>>> arctanh(X.zero_function()) == X.zero_function()
True

The same tests with SymPy:

sage: X.calculus_method().set('sympy')
sage: f.arctanh()
atanh(x*y)
sage: arctanh(f)  # equivalent to f.arctanh()
atanh(x*y)
sage: atanh(f)  # equivalent to f.arctanh()
atanh(x*y)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f.arctanh()
atanh(x*y)
>>> arctanh(f)  # equivalent to f.arctanh()
atanh(x*y)
>>> atanh(f)  # equivalent to f.arctanh()
atanh(x*y)
chart()[source]

Return the chart with respect to which self is defined.

OUTPUT: a Chart

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(1+x+y^2)
sage: f.chart()
Chart (M, (x, y))
sage: f.chart() is X
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(Integer(1)+x+y**Integer(2))
>>> f.chart()
Chart (M, (x, y))
>>> f.chart() is X
True
collect(s)[source]

Collect the coefficients of \(s\) in the expression of self into a group.

INPUT:

  • s – the symbol whose coefficients will be collected

OUTPUT:

  • self with the coefficients of s grouped in its expression

EXAMPLES:

Action on a 2-dimensional chart function:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x^2*y + x*y + (x*y)^2)
sage: f.display()
(x, y) ↦ x^2*y^2 + x^2*y + x*y
sage: f.collect(y)
x^2*y^2 + (x^2 + x)*y

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2)*y + x*y + (x*y)**Integer(2))
>>> f.display()
(x, y) ↦ x^2*y^2 + x^2*y + x*y
>>> f.collect(y)
x^2*y^2 + (x^2 + x)*y

The method collect() has changed the expression of f:

sage: f.display()
(x, y) ↦ x^2*y^2 + (x^2 + x)*y

>>> from sage.all import *
>>> f.display()
(x, y) ↦ x^2*y^2 + (x^2 + x)*y

The same test with SymPy

sage: X.calculus_method().set('sympy')
sage: f = X.function(x^2*y + x*y + (x*y)^2)
sage: f.display()
(x, y) ↦ x**2*y**2 + x**2*y + x*y
sage: f.collect(y)
x**2*y**2 + y*(x**2 + x)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f = X.function(x**Integer(2)*y + x*y + (x*y)**Integer(2))
>>> f.display()
(x, y) ↦ x**2*y**2 + x**2*y + x*y
>>> f.collect(y)
x**2*y**2 + y*(x**2 + x)
collect_common_factors()[source]

Collect common factors in the expression of self.

This method does not perform a full factorization but only looks for factors which are already explicitly present.

OUTPUT:

  • self with the common factors collected in its expression

EXAMPLES:

Action on a 2-dimensional chart function:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x/(x^2*y + x*y))
sage: f.display()
(x, y) ↦ x/(x^2*y + x*y)
sage: f.collect_common_factors()
1/((x + 1)*y)

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x/(x**Integer(2)*y + x*y))
>>> f.display()
(x, y) ↦ x/(x^2*y + x*y)
>>> f.collect_common_factors()
1/((x + 1)*y)

The method collect_common_factors() has changed the expression of f:

sage: f.display()
(x, y) ↦ 1/((x + 1)*y)

>>> from sage.all import *
>>> f.display()
(x, y) ↦ 1/((x + 1)*y)

The same test with SymPy:

sage: X.calculus_method().set('sympy')
sage: g = X.function(x/(x^2*y + x*y))
sage: g.display()
(x, y) ↦ x/(x**2*y + x*y)
sage: g.collect_common_factors()
1/(y*(x + 1))

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> g = X.function(x/(x**Integer(2)*y + x*y))
>>> g.display()
(x, y) ↦ x/(x**2*y + x*y)
>>> g.collect_common_factors()
1/(y*(x + 1))
copy()[source]

Return an exact copy of the object.

OUTPUT: a ChartFunctionSymb

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x+y^2)
sage: g = f.copy(); g
y^2 + x

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x+y**Integer(2))
>>> g = f.copy(); g
y^2 + x

By construction, g is identical to f:

sage: type(g) == type(f)
True
sage: g == f
True

>>> from sage.all import *
>>> type(g) == type(f)
True
>>> g == f
True

but it is not the same object:

sage: g is f
False

>>> from sage.all import *
>>> g is f
False
cos()[source]

Cosine of self.

OUTPUT:

  • chart function \(\cos(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.cos()
cos(x*y)
sage: cos(f)  # equivalent to f.cos()
cos(x*y)
sage: cos(f).display()
(x, y) ↦ cos(x*y)
sage: cos(X.zero_function()).display()
(x, y) ↦ 1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.cos()
cos(x*y)
>>> cos(f)  # equivalent to f.cos()
cos(x*y)
>>> cos(f).display()
(x, y) ↦ cos(x*y)
>>> cos(X.zero_function()).display()
(x, y) ↦ 1

The same tests with SymPy:

sage: X.calculus_method().set('sympy')
sage: f.cos()
cos(x*y)
sage: cos(f)  # equivalent to f.cos()
cos(x*y)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f.cos()
cos(x*y)
>>> cos(f)  # equivalent to f.cos()
cos(x*y)
cosh()[source]

Hyperbolic cosine of self.

OUTPUT:

  • chart function \(\cosh(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.cosh()
cosh(x*y)
sage: cosh(f)  # equivalent to f.cosh()
cosh(x*y)
sage: cosh(f).display()
(x, y) ↦ cosh(x*y)
sage: cosh(X.zero_function()).display()
(x, y) ↦ 1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.cosh()
cosh(x*y)
>>> cosh(f)  # equivalent to f.cosh()
cosh(x*y)
>>> cosh(f).display()
(x, y) ↦ cosh(x*y)
>>> cosh(X.zero_function()).display()
(x, y) ↦ 1

The same tests with SymPy:

sage: X.calculus_method().set('sympy')
sage: f.cosh()
cosh(x*y)
sage: cosh(f)  # equivalent to f.cosh()
cosh(x*y)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f.cosh()
cosh(x*y)
>>> cosh(f)  # equivalent to f.cosh()
cosh(x*y)
derivative(coord)[source]

Partial derivative with respect to a coordinate.

INPUT:

  • coord – either the coordinate \(x^i\) with respect to which the derivative of the chart function \(f\) is to be taken, or the index \(i\) labelling this coordinate (with the index convention defined on the chart domain via the parameter start_index)

OUTPUT:

  • a ChartFunction representing the partial derivative \(\frac{\partial f}{\partial x^i}\)

EXAMPLES:

Partial derivatives of a 2-dimensional chart function:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart(calc_method='SR')
sage: f = X.function(x^2+3*y+1); f
x^2 + 3*y + 1
sage: f.derivative(x)
2*x
sage: f.derivative(y)
3

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(calc_method='SR', names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2)+Integer(3)*y+Integer(1)); f
x^2 + 3*y + 1
>>> f.derivative(x)
2*x
>>> f.derivative(y)
3

An alias is diff:

sage: f.diff(x)
2*x

>>> from sage.all import *
>>> f.diff(x)
2*x

Each partial derivative is itself a chart function:

sage: type(f.diff(x))
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>

>>> from sage.all import *
>>> type(f.diff(x))
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>

The same result is returned by the function diff:

sage: diff(f, x)
2*x

>>> from sage.all import *
>>> diff(f, x)
2*x

An index can be used instead of the coordinate symbol:

sage: f.diff(0)
2*x
sage: diff(f, 1)
3

>>> from sage.all import *
>>> f.diff(Integer(0))
2*x
>>> diff(f, Integer(1))
3

The index range depends on the convention used on the chart’s domain:

sage: M = Manifold(2, 'M', structure='topological', start_index=1)
sage: X.<x,y> = M.chart()
sage: f = X.function(x^2+3*y+1)
sage: f.diff(0)
Traceback (most recent call last):
...
ValueError: coordinate index out of range
sage: f.diff(1)
2*x
sage: f.diff(2)
3

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological', start_index=Integer(1))
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2)+Integer(3)*y+Integer(1))
>>> f.diff(Integer(0))
Traceback (most recent call last):
...
ValueError: coordinate index out of range
>>> f.diff(Integer(1))
2*x
>>> f.diff(Integer(2))
3

The same test with SymPy:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart(calc_method='sympy')
sage: f = X.function(x^2+3*y+1); f
x**2 + 3*y + 1
sage: f.diff(x)
2*x
sage: f.diff(y)
3

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(calc_method='sympy', names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2)+Integer(3)*y+Integer(1)); f
x**2 + 3*y + 1
>>> f.diff(x)
2*x
>>> f.diff(y)
3
diff(coord)[source]

Partial derivative with respect to a coordinate.

INPUT:

  • coord – either the coordinate \(x^i\) with respect to which the derivative of the chart function \(f\) is to be taken, or the index \(i\) labelling this coordinate (with the index convention defined on the chart domain via the parameter start_index)

OUTPUT:

  • a ChartFunction representing the partial derivative \(\frac{\partial f}{\partial x^i}\)

EXAMPLES:

Partial derivatives of a 2-dimensional chart function:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart(calc_method='SR')
sage: f = X.function(x^2+3*y+1); f
x^2 + 3*y + 1
sage: f.derivative(x)
2*x
sage: f.derivative(y)
3

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(calc_method='SR', names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2)+Integer(3)*y+Integer(1)); f
x^2 + 3*y + 1
>>> f.derivative(x)
2*x
>>> f.derivative(y)
3

An alias is diff:

sage: f.diff(x)
2*x

>>> from sage.all import *
>>> f.diff(x)
2*x

Each partial derivative is itself a chart function:

sage: type(f.diff(x))
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>

>>> from sage.all import *
>>> type(f.diff(x))
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>

The same result is returned by the function diff:

sage: diff(f, x)
2*x

>>> from sage.all import *
>>> diff(f, x)
2*x

An index can be used instead of the coordinate symbol:

sage: f.diff(0)
2*x
sage: diff(f, 1)
3

>>> from sage.all import *
>>> f.diff(Integer(0))
2*x
>>> diff(f, Integer(1))
3

The index range depends on the convention used on the chart’s domain:

sage: M = Manifold(2, 'M', structure='topological', start_index=1)
sage: X.<x,y> = M.chart()
sage: f = X.function(x^2+3*y+1)
sage: f.diff(0)
Traceback (most recent call last):
...
ValueError: coordinate index out of range
sage: f.diff(1)
2*x
sage: f.diff(2)
3

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological', start_index=Integer(1))
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2)+Integer(3)*y+Integer(1))
>>> f.diff(Integer(0))
Traceback (most recent call last):
...
ValueError: coordinate index out of range
>>> f.diff(Integer(1))
2*x
>>> f.diff(Integer(2))
3

The same test with SymPy:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart(calc_method='sympy')
sage: f = X.function(x^2+3*y+1); f
x**2 + 3*y + 1
sage: f.diff(x)
2*x
sage: f.diff(y)
3

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(calc_method='sympy', names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2)+Integer(3)*y+Integer(1)); f
x**2 + 3*y + 1
>>> f.diff(x)
2*x
>>> f.diff(y)
3
disp()[source]

Display self in arrow notation. For display the standard SR representation is used.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

EXAMPLES:

Coordinate function on a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(cos(x*y/2))
sage: f.display()
(x, y) ↦ cos(1/2*x*y)
sage: latex(f.display())
\left(x, y\right) \mapsto \cos\left(\frac{1}{2} \, x y\right)

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(cos(x*y/Integer(2)))
>>> f.display()
(x, y) ↦ cos(1/2*x*y)
>>> latex(f.display())
\left(x, y\right) \mapsto \cos\left(\frac{1}{2} \, x y\right)

A shortcut is disp():

sage: f.disp()
(x, y) ↦ cos(1/2*x*y)

>>> from sage.all import *
>>> f.disp()
(x, y) ↦ cos(1/2*x*y)

Display of the zero function:

sage: X.zero_function().display()
(x, y) ↦ 0

>>> from sage.all import *
>>> X.zero_function().display()
(x, y) ↦ 0
display()[source]

Display self in arrow notation. For display the standard SR representation is used.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

EXAMPLES:

Coordinate function on a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(cos(x*y/2))
sage: f.display()
(x, y) ↦ cos(1/2*x*y)
sage: latex(f.display())
\left(x, y\right) \mapsto \cos\left(\frac{1}{2} \, x y\right)

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(cos(x*y/Integer(2)))
>>> f.display()
(x, y) ↦ cos(1/2*x*y)
>>> latex(f.display())
\left(x, y\right) \mapsto \cos\left(\frac{1}{2} \, x y\right)

A shortcut is disp():

sage: f.disp()
(x, y) ↦ cos(1/2*x*y)

>>> from sage.all import *
>>> f.disp()
(x, y) ↦ cos(1/2*x*y)

Display of the zero function:

sage: X.zero_function().display()
(x, y) ↦ 0

>>> from sage.all import *
>>> X.zero_function().display()
(x, y) ↦ 0
exp()[source]

Exponential of self.

OUTPUT:

  • chart function \(\exp(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x+y)
sage: f.exp()
e^(x + y)
sage: exp(f) # equivalent to f.exp()
e^(x + y)
sage: exp(f).display()
(x, y) ↦ e^(x + y)
sage: exp(X.zero_function())
1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x+y)
>>> f.exp()
e^(x + y)
>>> exp(f) # equivalent to f.exp()
e^(x + y)
>>> exp(f).display()
(x, y) ↦ e^(x + y)
>>> exp(X.zero_function())
1

The same test with SymPy:

sage: X.calculus_method().set('sympy')
sage: f = X.function(x+y)
sage: f.exp()
exp(x + y)
sage: exp(f) # equivalent to f.exp()
exp(x + y)
sage: exp(f).display()
(x, y) ↦ exp(x + y)
sage: exp(X.zero_function())
1

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f = X.function(x+y)
>>> f.exp()
exp(x + y)
>>> exp(f) # equivalent to f.exp()
exp(x + y)
>>> exp(f).display()
(x, y) ↦ exp(x + y)
>>> exp(X.zero_function())
1
expand()[source]

Expand the coordinate expression of self.

OUTPUT: self with its expression expanded

EXAMPLES:

Expanding a 2-dimensional chart function:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function((x - y)^2)
sage: f.display()
(x, y) ↦ (x - y)^2
sage: f.expand()
x^2 - 2*x*y + y^2

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function((x - y)**Integer(2))
>>> f.display()
(x, y) ↦ (x - y)^2
>>> f.expand()
x^2 - 2*x*y + y^2

The method expand() has changed the expression of f:

sage: f.display()
(x, y) ↦ x^2 - 2*x*y + y^2

>>> from sage.all import *
>>> f.display()
(x, y) ↦ x^2 - 2*x*y + y^2

The same test with SymPy

sage: X.calculus_method().set('sympy')
sage: g = X.function((x - y)^2)
sage: g.expand()
x**2 - 2*x*y + y**2

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> g = X.function((x - y)**Integer(2))
>>> g.expand()
x**2 - 2*x*y + y**2
expr(method=None)[source]

Return the symbolic expression of self in terms of the chart coordinates, as an object of a specified calculus method.

INPUT:

  • method – string (default: None); the calculus method which the returned expression belongs to. One of

    • 'SR': Sage’s default symbolic engine (Symbolic Ring)

    • 'sympy': SymPy

    • None: the chart current calculus method is assumed

OUTPUT:

EXAMPLES:

Chart function on a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x^2+y)
sage: f.expr()
x^2 + y
sage: type(f.expr())
<class 'sage.symbolic.expression.Expression'>

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2)+y)
>>> f.expr()
x^2 + y
>>> type(f.expr())
<class 'sage.symbolic.expression.Expression'>

Asking for the SymPy expression:

sage: f.expr('sympy')
x**2 + y
sage: type(f.expr('sympy'))
<class 'sympy.core.add.Add'>

>>> from sage.all import *
>>> f.expr('sympy')
x**2 + y
>>> type(f.expr('sympy'))
<class 'sympy.core.add.Add'>

The default corresponds to the current calculus method, here the one based on the Symbolic Ring SR:

sage: f.expr() is f.expr('SR')
True

>>> from sage.all import *
>>> f.expr() is f.expr('SR')
True

If we change the current calculus method on chart X, we change the default:

sage: X.calculus_method().set('sympy')
sage: f.expr()
x**2 + y
sage: f.expr() is f.expr('sympy')
True
sage: X.calculus_method().set('SR')  # revert back to SR

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f.expr()
x**2 + y
>>> f.expr() is f.expr('sympy')
True
>>> X.calculus_method().set('SR')  # revert back to SR

Internally, the expressions corresponding to various calculus methods are stored in the dictionary _express:

sage: for method in sorted(f._express):
....:     print("'{}': {}".format(method, f._express[method]))
....:
'SR': x^2 + y
'sympy': x**2 + y

>>> from sage.all import *
>>> for method in sorted(f._express):
...     print("'{}': {}".format(method, f._express[method]))
....:
'SR': x^2 + y
'sympy': x**2 + y

The method expr() is useful for accessing to all the symbolic expression functionalities in Sage; for instance:

sage: var('a')
a
sage: f = X.function(a*x*y); f.display()
(x, y) ↦ a*x*y
sage: f.expr()
a*x*y
sage: f.expr().subs(a=2)
2*x*y

>>> from sage.all import *
>>> var('a')
a
>>> f = X.function(a*x*y); f.display()
(x, y) ↦ a*x*y
>>> f.expr()
a*x*y
>>> f.expr().subs(a=Integer(2))
2*x*y

Note that for substituting the value of a coordinate, the function call can be used as well:

sage: f(x,3)
3*a*x
sage: bool( f(x,3) == f.expr().subs(y=3) )
True

>>> from sage.all import *
>>> f(x,Integer(3))
3*a*x
>>> bool( f(x,Integer(3)) == f.expr().subs(y=Integer(3)) )
True
factor()[source]

Factorize the coordinate expression of self.

OUTPUT: self with its expression factorized

EXAMPLES:

Factorization of a 2-dimensional chart function:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x^2 + 2*x*y + y^2)
sage: f.display()
(x, y) ↦ x^2 + 2*x*y + y^2
sage: f.factor()
(x + y)^2

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2) + Integer(2)*x*y + y**Integer(2))
>>> f.display()
(x, y) ↦ x^2 + 2*x*y + y^2
>>> f.factor()
(x + y)^2

The method factor() has changed the expression of f:

sage: f.display()
(x, y) ↦ (x + y)^2

>>> from sage.all import *
>>> f.display()
(x, y) ↦ (x + y)^2

The same test with SymPy

sage: X.calculus_method().set('sympy')
sage: g = X.function(x^2 + 2*x*y + y^2)
sage: g.display()
(x, y) ↦ x**2 + 2*x*y + y**2
sage: g.factor()
(x + y)**2

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> g = X.function(x**Integer(2) + Integer(2)*x*y + y**Integer(2))
>>> g.display()
(x, y) ↦ x**2 + 2*x*y + y**2
>>> g.factor()
(x + y)**2
is_trivial_one()[source]

Check if self is trivially equal to one without any simplification.

This method is supposed to be fast as compared with self == 1 and is intended to be used in library code where trying to obtain a mathematically correct result by applying potentially expensive rewrite rules is not desirable.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(1)
sage: f.is_trivial_one()
True
sage: f = X.function(float(1.0))
sage: f.is_trivial_one()
True
sage: f = X.function(x-x+1)
sage: f.is_trivial_one()
True
sage: X.one_function().is_trivial_one()
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(Integer(1))
>>> f.is_trivial_one()
True
>>> f = X.function(float(RealNumber('1.0')))
>>> f.is_trivial_one()
True
>>> f = X.function(x-x+Integer(1))
>>> f.is_trivial_one()
True
>>> X.one_function().is_trivial_one()
True

No simplification is attempted, so that False is returned for non-trivial cases:

sage: f = X.function(cos(x)^2 + sin(x)^2)
sage: f.is_trivial_one()
False

>>> from sage.all import *
>>> f = X.function(cos(x)**Integer(2) + sin(x)**Integer(2))
>>> f.is_trivial_one()
False

On the contrary, the method is_zero() and the direct comparison to one involve some simplification algorithms and return True:

sage: (f - 1).is_zero()
True
sage: f == 1
True

>>> from sage.all import *
>>> (f - Integer(1)).is_zero()
True
>>> f == Integer(1)
True
is_trivial_zero()[source]

Check if self is trivially equal to zero without any simplification.

This method is supposed to be fast as compared with self.is_zero() or self == 0 and is intended to be used in library code where trying to obtain a mathematically correct result by applying potentially expensive rewrite rules is not desirable.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(0)
sage: f.is_trivial_zero()
True
sage: f = X.function(float(0.0))
sage: f.is_trivial_zero()
True
sage: f = X.function(x-x)
sage: f.is_trivial_zero()
True
sage: X.zero_function().is_trivial_zero()
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(Integer(0))
>>> f.is_trivial_zero()
True
>>> f = X.function(float(RealNumber('0.0')))
>>> f.is_trivial_zero()
True
>>> f = X.function(x-x)
>>> f.is_trivial_zero()
True
>>> X.zero_function().is_trivial_zero()
True

No simplification is attempted, so that False is returned for non-trivial cases:

sage: f = X.function(cos(x)^2 + sin(x)^2 - 1)
sage: f.is_trivial_zero()
False

>>> from sage.all import *
>>> f = X.function(cos(x)**Integer(2) + sin(x)**Integer(2) - Integer(1))
>>> f.is_trivial_zero()
False

On the contrary, the method is_zero() and the direct comparison to zero involve some simplification algorithms and return True:

sage: f.is_zero()
True
sage: f == 0
True

>>> from sage.all import *
>>> f.is_zero()
True
>>> f == Integer(0)
True
is_unit()[source]

Return True iff self is not trivially zero since most chart functions are invertible and an actual computation would take too much time.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x^2+3*y+1)
sage: f.is_unit()
True
sage: zero = X.function(0)
sage: zero.is_unit()
False

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x**Integer(2)+Integer(3)*y+Integer(1))
>>> f.is_unit()
True
>>> zero = X.function(Integer(0))
>>> zero.is_unit()
False
log(base=None)[source]

Logarithm of self.

INPUT:

  • base – (default: None) base of the logarithm; if None, the natural logarithm (i.e. logarithm to base \(e\)) is returned

OUTPUT:

  • chart function \(\log_a(f)\), where \(f\) is the current chart function and \(a\) is the base

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x+y)
sage: f.log()
log(x + y)
sage: log(f) # equivalent to f.log()
log(x + y)
sage: log(f).display()
(x, y) ↦ log(x + y)
sage: f.log(2)
log(x + y)/log(2)
sage: log(f, 2)
log(x + y)/log(2)

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x+y)
>>> f.log()
log(x + y)
>>> log(f) # equivalent to f.log()
log(x + y)
>>> log(f).display()
(x, y) ↦ log(x + y)
>>> f.log(Integer(2))
log(x + y)/log(2)
>>> log(f, Integer(2))
log(x + y)/log(2)

The same test with SymPy:

sage: X.calculus_method().set('sympy')
sage: f = X.function(x+y)
sage: f.log()
log(x + y)
sage: log(f) # equivalent to f.log()
log(x + y)
sage: log(f).display()
(x, y) ↦ log(x + y)
sage: f.log(2)
log(x + y)/log(2)
sage: log(f, 2)
log(x + y)/log(2)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f = X.function(x+y)
>>> f.log()
log(x + y)
>>> log(f) # equivalent to f.log()
log(x + y)
>>> log(f).display()
(x, y) ↦ log(x + y)
>>> f.log(Integer(2))
log(x + y)/log(2)
>>> log(f, Integer(2))
log(x + y)/log(2)
scalar_field(name=None, latex_name=None)[source]

Construct the scalar field that has self as coordinate expression.

The domain of the scalar field is the open subset covered by the chart on which self is defined.

INPUT:

  • name – (default: None) name given to the scalar field

  • latex_name – (default: None) LaTeX symbol to denote the scalar field; if None, the LaTeX symbol is set to name

OUTPUT: a ScalarField

EXAMPLES:

Construction of a scalar field on a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: c_xy.<x,y> = M.chart()
sage: fc = c_xy.function(x+2*y^3)
sage: f = fc.scalar_field() ; f
Scalar field on the 2-dimensional topological manifold M
sage: f.display()
M → ℝ
(x, y) ↦ 2*y^3 + x
sage: f.coord_function(c_xy) is fc
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2)
>>> fc = c_xy.function(x+Integer(2)*y**Integer(3))
>>> f = fc.scalar_field() ; f
Scalar field on the 2-dimensional topological manifold M
>>> f.display()
M → ℝ
(x, y) ↦ 2*y^3 + x
>>> f.coord_function(c_xy) is fc
True
set_expr(calc_method, expression)[source]

Add an expression in a particular calculus method self. Some control is done to verify the consistency between the different representations of the same expression.

INPUT:

  • calc_method – calculus method

  • expression – symbolic expression

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(1+x^2)
sage: f._repr_()
'x^2 + 1'
sage: f.set_expr('sympy','x**2+1')
sage: f  # indirect doctest
x^2 + 1

sage: g = X.function(1+x^3)
sage: g._repr_()
'x^3 + 1'
sage: g.set_expr('sympy','x**2+y')
Traceback (most recent call last):
...
ValueError: Expressions are not equal

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(Integer(1)+x**Integer(2))
>>> f._repr_()
'x^2 + 1'
>>> f.set_expr('sympy','x**2+1')
>>> f  # indirect doctest
x^2 + 1

>>> g = X.function(Integer(1)+x**Integer(3))
>>> g._repr_()
'x^3 + 1'
>>> g.set_expr('sympy','x**2+y')
Traceback (most recent call last):
...
ValueError: Expressions are not equal
simplify()[source]

Simplify the coordinate expression of self.

For details about the employed chain of simplifications for the SR calculus method, see simplify_chain_real() for chart functions on real manifolds and simplify_chain_generic() for the generic case.

If self has been defined with the small parameter expansion_symbol and some truncation order, the coordinate expression of self will be expanded in power series of that parameter and truncated to the given order.

OUTPUT: self with its coordinate expression simplified

EXAMPLES:

Simplification of a chart function on a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(cos(x)^2 + sin(x)^2 + sqrt(x^2))
sage: f.display()
(x, y) ↦ cos(x)^2 + sin(x)^2 + abs(x)
sage: f.simplify()
abs(x) + 1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(cos(x)**Integer(2) + sin(x)**Integer(2) + sqrt(x**Integer(2)))
>>> f.display()
(x, y) ↦ cos(x)^2 + sin(x)^2 + abs(x)
>>> f.simplify()
abs(x) + 1

The method simplify() has changed the expression of f:

sage: f.display()
(x, y) ↦ abs(x) + 1

>>> from sage.all import *
>>> f.display()
(x, y) ↦ abs(x) + 1

Another example:

sage: f = X.function((x^2-1)/(x+1)); f
(x^2 - 1)/(x + 1)
sage: f.simplify()
x - 1

>>> from sage.all import *
>>> f = X.function((x**Integer(2)-Integer(1))/(x+Integer(1))); f
(x^2 - 1)/(x + 1)
>>> f.simplify()
x - 1

Examples taking into account the declared range of a coordinate:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart('x:(1,+oo) y')
sage: f = X.function(sqrt(x^2-2*x+1)); f
sqrt(x^2 - 2*x + 1)
sage: f.simplify()
x - 1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart('x:(1,+oo) y', names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(sqrt(x**Integer(2)-Integer(2)*x+Integer(1))); f
sqrt(x^2 - 2*x + 1)
>>> f.simplify()
x - 1


sage: forget()  # to clear the previous assumption on x
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart('x:(-oo,0) y')
sage: f = X.function(sqrt(x^2-2*x+1)); f
sqrt(x^2 - 2*x + 1)
sage: f.simplify()
-x + 1

>>> from sage.all import *
>>> forget()  # to clear the previous assumption on x
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart('x:(-oo,0) y', names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(sqrt(x**Integer(2)-Integer(2)*x+Integer(1))); f
sqrt(x^2 - 2*x + 1)
>>> f.simplify()
-x + 1

The same tests with SymPy:

sage: forget()  # to clear the previous assumption on x
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart(calc_method='sympy')
sage: f = X.function(cos(x)^2 + sin(x)^2 + sqrt(x^2)); f
sin(x)**2 + cos(x)**2 + Abs(x)
sage: f.simplify()
Abs(x) + 1

>>> from sage.all import *
>>> forget()  # to clear the previous assumption on x
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(calc_method='sympy', names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(cos(x)**Integer(2) + sin(x)**Integer(2) + sqrt(x**Integer(2))); f
sin(x)**2 + cos(x)**2 + Abs(x)
>>> f.simplify()
Abs(x) + 1


sage: f = X.function((x^2-1)/(x+1)); f
(x**2 - 1)/(x + 1)
sage: f.simplify()
x - 1

>>> from sage.all import *
>>> f = X.function((x**Integer(2)-Integer(1))/(x+Integer(1))); f
(x**2 - 1)/(x + 1)
>>> f.simplify()
x - 1


sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart('x:(1,+oo) y', calc_method='sympy')
sage: f = X.function(sqrt(x^2-2*x+1)); f
sqrt(x**2 - 2*x + 1)
sage: f.simplify()
x - 1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart('x:(1,+oo) y', calc_method='sympy', names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(sqrt(x**Integer(2)-Integer(2)*x+Integer(1))); f
sqrt(x**2 - 2*x + 1)
>>> f.simplify()
x - 1


sage: forget()  # to clear the previous assumption on x
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart('x:(-oo,0) y', calc_method='sympy')
sage: f = X.function(sqrt(x^2-2*x+1)); f
sqrt(x**2 - 2*x + 1)
sage: f.simplify()
1 - x

>>> from sage.all import *
>>> forget()  # to clear the previous assumption on x
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart('x:(-oo,0) y', calc_method='sympy', names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(sqrt(x**Integer(2)-Integer(2)*x+Integer(1))); f
sqrt(x**2 - 2*x + 1)
>>> f.simplify()
1 - x

Power series expansion with respect to a small parameter \(t\) (at the moment, this is implemented only for the SR calculus backend, hence the first line below):

sage: X.calculus_method().set('SR')
sage: t = var('t')
sage: f = X.function(exp(t*x), expansion_symbol=t, order=3)

>>> from sage.all import *
>>> X.calculus_method().set('SR')
>>> t = var('t')
>>> f = X.function(exp(t*x), expansion_symbol=t, order=Integer(3))

At this stage, \(f\) is not expanded in power series:

sage: f
e^(t*x)

>>> from sage.all import *
>>> f
e^(t*x)

Invoking simplify() triggers the expansion to the given order:

sage: f.simplify()
1/6*t^3*x^3 + 1/2*t^2*x^2 + t*x + 1
sage: f.display()
(x, y) ↦ 1/6*t^3*x^3 + 1/2*t^2*x^2 + t*x + 1

>>> from sage.all import *
>>> f.simplify()
1/6*t^3*x^3 + 1/2*t^2*x^2 + t*x + 1
>>> f.display()
(x, y) ↦ 1/6*t^3*x^3 + 1/2*t^2*x^2 + t*x + 1
sin()[source]

Sine of self.

OUTPUT:

  • chart function \(\sin(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.sin()
sin(x*y)
sage: sin(f)  # equivalent to f.sin()
sin(x*y)
sage: sin(f).display()
(x, y) ↦ sin(x*y)
sage: sin(X.zero_function()) == X.zero_function()
True
sage: f = X.function(2-cos(x)^2+y)
sage: g = X.function(-sin(x)^2+y)
sage: (f+g).simplify()
2*y + 1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.sin()
sin(x*y)
>>> sin(f)  # equivalent to f.sin()
sin(x*y)
>>> sin(f).display()
(x, y) ↦ sin(x*y)
>>> sin(X.zero_function()) == X.zero_function()
True
>>> f = X.function(Integer(2)-cos(x)**Integer(2)+y)
>>> g = X.function(-sin(x)**Integer(2)+y)
>>> (f+g).simplify()
2*y + 1

The same tests with SymPy:

sage: X.calculus_method().set('sympy')
sage: f = X.function(x*y)
sage: f.sin()
sin(x*y)
sage: sin(f)  # equivalent to f.sin()
sin(x*y)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f = X.function(x*y)
>>> f.sin()
sin(x*y)
>>> sin(f)  # equivalent to f.sin()
sin(x*y)
sinh()[source]

Hyperbolic sine of self.

OUTPUT:

  • chart function \(\sinh(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.sinh()
sinh(x*y)
sage: sinh(f)  # equivalent to f.sinh()
sinh(x*y)
sage: sinh(f).display()
(x, y) ↦ sinh(x*y)
sage: sinh(X.zero_function()) == X.zero_function()
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.sinh()
sinh(x*y)
>>> sinh(f)  # equivalent to f.sinh()
sinh(x*y)
>>> sinh(f).display()
(x, y) ↦ sinh(x*y)
>>> sinh(X.zero_function()) == X.zero_function()
True

The same tests with SymPy:

sage: X.calculus_method().set('sympy')
sage: f.sinh()
sinh(x*y)
sage: sinh(f)  # equivalent to f.sinh()
sinh(x*y)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f.sinh()
sinh(x*y)
>>> sinh(f)  # equivalent to f.sinh()
sinh(x*y)
sqrt()[source]

Square root of self.

OUTPUT:

  • chart function \(\sqrt{f}\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x+y)
sage: f.sqrt()
sqrt(x + y)
sage: sqrt(f)  # equivalent to f.sqrt()
sqrt(x + y)
sage: sqrt(f).display()
(x, y) ↦ sqrt(x + y)
sage: sqrt(X.zero_function()).display()
(x, y) ↦ 0

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x+y)
>>> f.sqrt()
sqrt(x + y)
>>> sqrt(f)  # equivalent to f.sqrt()
sqrt(x + y)
>>> sqrt(f).display()
(x, y) ↦ sqrt(x + y)
>>> sqrt(X.zero_function()).display()
(x, y) ↦ 0
tan()[source]

Tangent of self.

OUTPUT:

  • chart function \(\tan(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.tan()
sin(x*y)/cos(x*y)
sage: tan(f)  # equivalent to f.tan()
sin(x*y)/cos(x*y)
sage: tan(f).display()
(x, y) ↦ sin(x*y)/cos(x*y)
sage: tan(X.zero_function()) == X.zero_function()
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.tan()
sin(x*y)/cos(x*y)
>>> tan(f)  # equivalent to f.tan()
sin(x*y)/cos(x*y)
>>> tan(f).display()
(x, y) ↦ sin(x*y)/cos(x*y)
>>> tan(X.zero_function()) == X.zero_function()
True

The same test with SymPy:

sage: M.set_calculus_method('sympy')
sage: g = X.function(x*y)
sage: g.tan()
tan(x*y)
sage: tan(g)  # equivalent to g.tan()
tan(x*y)
sage: tan(g).display()
(x, y) ↦ tan(x*y)

>>> from sage.all import *
>>> M.set_calculus_method('sympy')
>>> g = X.function(x*y)
>>> g.tan()
tan(x*y)
>>> tan(g)  # equivalent to g.tan()
tan(x*y)
>>> tan(g).display()
(x, y) ↦ tan(x*y)
tanh()[source]

Hyperbolic tangent of self.

OUTPUT:

  • chart function \(\tanh(f)\), where \(f\) is the current chart function

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(x*y)
sage: f.tanh()
sinh(x*y)/cosh(x*y)
sage: tanh(f)  # equivalent to f.tanh()
sinh(x*y)/cosh(x*y)
sage: tanh(f).display()
(x, y) ↦ sinh(x*y)/cosh(x*y)
sage: tanh(X.zero_function()) == X.zero_function()
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.function(x*y)
>>> f.tanh()
sinh(x*y)/cosh(x*y)
>>> tanh(f)  # equivalent to f.tanh()
sinh(x*y)/cosh(x*y)
>>> tanh(f).display()
(x, y) ↦ sinh(x*y)/cosh(x*y)
>>> tanh(X.zero_function()) == X.zero_function()
True

The same tests with SymPy:

sage: X.calculus_method().set('sympy')
sage: f.tanh()
tanh(x*y)
sage: tanh(f)  # equivalent to f.tanh()
tanh(x*y)

>>> from sage.all import *
>>> X.calculus_method().set('sympy')
>>> f.tanh()
tanh(x*y)
>>> tanh(f)  # equivalent to f.tanh()
tanh(x*y)
class sage.manifolds.chart_func.ChartFunctionRing(chart)[source]

Bases: Parent, UniqueRepresentation

Ring of all chart functions on a chart.

INPUT:

  • chart – a coordinate chart, as an instance of class Chart

EXAMPLES:

The ring of all chart functions w.r.t. to a chart:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: FR = X.function_ring(); FR
Ring of chart functions on Chart (M, (x, y))
sage: type(FR)
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category'>
sage: FR.category()
Category of commutative algebras over Symbolic Ring

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> FR = X.function_ring(); FR
Ring of chart functions on Chart (M, (x, y))
>>> type(FR)
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category'>
>>> FR.category()
Category of commutative algebras over Symbolic Ring

Coercions by means of restrictions are implemented:

sage: FR_X = X.function_ring()
sage: D = M.open_subset('D')
sage: X_D = X.restrict(D, x^2+y^2<1)  # open disk
sage: FR_X_D = X_D.function_ring()
sage: FR_X_D.has_coerce_map_from(FR_X)
True

>>> from sage.all import *
>>> FR_X = X.function_ring()
>>> D = M.open_subset('D')
>>> X_D = X.restrict(D, x**Integer(2)+y**Integer(2)<Integer(1))  # open disk
>>> FR_X_D = X_D.function_ring()
>>> FR_X_D.has_coerce_map_from(FR_X)
True

But only if the charts are compatible:

sage: Y.<t,z> = D.chart()
sage: FR_Y = Y.function_ring()
sage: FR_Y.has_coerce_map_from(FR_X)
False

>>> from sage.all import *
>>> Y = D.chart(names=('t', 'z',)); (t, z,) = Y._first_ngens(2)
>>> FR_Y = Y.function_ring()
>>> FR_Y.has_coerce_map_from(FR_X)
False
Element[source]

alias of ChartFunction

is_field(proof=True)[source]

Return False as self is not an integral domain.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: FR = X.function_ring()
sage: FR.is_integral_domain()
False
sage: FR.is_field()
False

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> FR = X.function_ring()
>>> FR.is_integral_domain()
False
>>> FR.is_field()
False
is_integral_domain(proof=True)[source]

Return False as self is not an integral domain.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: FR = X.function_ring()
sage: FR.is_integral_domain()
False
sage: FR.is_field()
False

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> FR = X.function_ring()
>>> FR.is_integral_domain()
False
>>> FR.is_field()
False
one()[source]

Return the constant function \(1\) in self.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: FR = X.function_ring()
sage: FR.one()
1

sage: M = Manifold(2, 'M', structure='topological', field=Qp(5))
sage: X.<x,y> = M.chart()
sage: X.function_ring().one()
1 + O(5^20)

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> FR = X.function_ring()
>>> FR.one()
1

>>> M = Manifold(Integer(2), 'M', structure='topological', field=Qp(Integer(5)))
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> X.function_ring().one()
1 + O(5^20)
zero()[source]

Return the constant function \(0\) in self.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: FR = X.function_ring()
sage: FR.zero()
0

sage: M = Manifold(2, 'M', structure='topological', field=Qp(5))
sage: X.<x,y> = M.chart()
sage: X.function_ring().zero()
0

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> FR = X.function_ring()
>>> FR.zero()
0

>>> M = Manifold(Integer(2), 'M', structure='topological', field=Qp(Integer(5)))
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> X.function_ring().zero()
0
class sage.manifolds.chart_func.MultiCoordFunction(chart, expressions)[source]

Bases: SageObject, Mutability

Coordinate function to some Cartesian power of the base field.

If \(n\) and \(m\) are two positive integers and \((U, \varphi)\) is a chart on a topological manifold \(M\) of dimension \(n\) over a topological field \(K\), a multi-coordinate function associated to \((U, \varphi)\) is a map

\[\begin{split}\begin{array}{llcl} f:& V \subset K^n & \longrightarrow & K^m \\ & (x^1, \ldots, x^n) & \longmapsto & (f_1(x^1, \ldots, x^n), \ldots, f_m(x^1, \ldots, x^n)), \end{array}\end{split}\]

where \(V\) is the codomain of \(\varphi\). In other words, \(f\) is a \(K^m\)-valued function of the coordinates associated to the chart \((U, \varphi)\). Each component \(f_i\) (\(1 \leq i \leq m\)) is a coordinate function and is therefore stored as a ChartFunction.

INPUT:

  • chart – the chart \((U, \varphi)\)

  • expressions – list (or tuple) of length \(m\) of elements to construct the coordinate functions \(f_i\) (\(1 \leq i \leq m\)); for symbolic coordinate functions, this must be symbolic expressions involving the chart coordinates, while for numerical coordinate functions, this must be data file names

EXAMPLES:

A function \(f: V \subset \RR^2 \longrightarrow \RR^3\):

sage: forget()  # to clear the previous assumption on x
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.multifunction(x-y, x*y, cos(x)*exp(y)); f
Coordinate functions (x - y, x*y, cos(x)*e^y) on the Chart (M, (x, y))
sage: type(f)
<class 'sage.manifolds.chart_func.MultiCoordFunction'>
sage: f(x,y)
(x - y, x*y, cos(x)*e^y)
sage: latex(f)
\left(x - y, x y, \cos\left(x\right) e^{y}\right)

>>> from sage.all import *
>>> forget()  # to clear the previous assumption on x
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.multifunction(x-y, x*y, cos(x)*exp(y)); f
Coordinate functions (x - y, x*y, cos(x)*e^y) on the Chart (M, (x, y))
>>> type(f)
<class 'sage.manifolds.chart_func.MultiCoordFunction'>
>>> f(x,y)
(x - y, x*y, cos(x)*e^y)
>>> latex(f)
\left(x - y, x y, \cos\left(x\right) e^{y}\right)

Each real-valued function \(f_i\) (\(1 \leq i \leq m\)) composing \(f\) can be accessed via the square-bracket operator, by providing \(i-1\) as an argument:

sage: f[0]
x - y
sage: f[1]
x*y
sage: f[2]
cos(x)*e^y

>>> from sage.all import *
>>> f[Integer(0)]
x - y
>>> f[Integer(1)]
x*y
>>> f[Integer(2)]
cos(x)*e^y

We can give a more verbose explanation of each function:

sage: f[0].display()
(x, y) ↦ x - y

>>> from sage.all import *
>>> f[Integer(0)].display()
(x, y) ↦ x - y

Each f[i-1] is an instance of ChartFunction:

sage: isinstance(f[0], sage.manifolds.chart_func.ChartFunction)
True

>>> from sage.all import *
>>> isinstance(f[Integer(0)], sage.manifolds.chart_func.ChartFunction)
True

A class MultiCoordFunction can represent a real-valued function (case \(m = 1\)), although one should rather employ the class ChartFunction for this purpose:

sage: g = X.multifunction(x*y^2)
sage: g(x,y)
(x*y^2,)

>>> from sage.all import *
>>> g = X.multifunction(x*y**Integer(2))
>>> g(x,y)
(x*y^2,)

Evaluating the functions at specified coordinates:

sage: f(1,2)
(-1, 2, cos(1)*e^2)
sage: var('a b')
(a, b)
sage: f(a,b)
(a - b, a*b, cos(a)*e^b)
sage: g(1,2)
(4,)

>>> from sage.all import *
>>> f(Integer(1),Integer(2))
(-1, 2, cos(1)*e^2)
>>> var('a b')
(a, b)
>>> f(a,b)
(a - b, a*b, cos(a)*e^b)
>>> g(Integer(1),Integer(2))
(4,)
chart()[source]

Return the chart with respect to which self is defined.

OUTPUT: a Chart

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.multifunction(x-y, x*y, cos(x)*exp(y))
sage: f.chart()
Chart (M, (x, y))
sage: f.chart() is X
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.multifunction(x-y, x*y, cos(x)*exp(y))
>>> f.chart()
Chart (M, (x, y))
>>> f.chart() is X
True
expr(method=None)[source]

Return a tuple of data, the item no. \(i\) being sufficient to reconstruct the coordinate function no. \(i\).

In other words, if f is a multi-coordinate function, then f.chart().multifunction(*(f.expr())) results in a multi-coordinate function identical to f.

INPUT:

  • method – string (default: None); the calculus method which the returned expressions belong to. One of

    • 'SR': Sage’s default symbolic engine (Symbolic Ring)

    • 'sympy': SymPy

    • None: the chart current calculus method is assumed

OUTPUT:

  • a tuple of the symbolic expressions of the chart functions composing self

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.multifunction(x-y, x*y, cos(x)*exp(y))
sage: f.expr()
(x - y, x*y, cos(x)*e^y)
sage: type(f.expr()[0])
<class 'sage.symbolic.expression.Expression'>

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.multifunction(x-y, x*y, cos(x)*exp(y))
>>> f.expr()
(x - y, x*y, cos(x)*e^y)
>>> type(f.expr()[Integer(0)])
<class 'sage.symbolic.expression.Expression'>

A SymPy output:

sage: f.expr('sympy')
(x - y, x*y, exp(y)*cos(x))
sage: type(f.expr('sympy')[0])
<class 'sympy.core.add.Add'>

>>> from sage.all import *
>>> f.expr('sympy')
(x - y, x*y, exp(y)*cos(x))
>>> type(f.expr('sympy')[Integer(0)])
<class 'sympy.core.add.Add'>

One shall always have:

sage: f.chart().multifunction(*(f.expr())) == f
True

>>> from sage.all import *
>>> f.chart().multifunction(*(f.expr())) == f
True
jacobian()[source]

Return the Jacobian matrix of the system of coordinate functions.

jacobian() is a 2-dimensional array of size \(m \times n\), where \(m\) is the number of functions and \(n\) the number of coordinates, the generic element being \(J_{ij} = \frac{\partial f_i}{\partial x^j}\) with \(1 \leq i \leq m\) (row index) and \(1 \leq j \leq n\) (column index).

OUTPUT:

  • Jacobian matrix as a 2-dimensional array J of coordinate functions with J[i-1][j-1] being \(J_{ij} = \frac{\partial f_i}{\partial x^j}\) for \(1 \leq i \leq m\) and \(1 \leq j \leq n\)

EXAMPLES:

Jacobian of a set of 3 functions of 2 coordinates:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.multifunction(x-y, x*y, y^3*cos(x))
sage: f.jacobian()
[           1           -1]
[           y            x]
[ -y^3*sin(x) 3*y^2*cos(x)]

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.multifunction(x-y, x*y, y**Integer(3)*cos(x))
>>> f.jacobian()
[           1           -1]
[           y            x]
[ -y^3*sin(x) 3*y^2*cos(x)]

Each element of the result is a chart function:

sage: type(f.jacobian()[2,0])
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>
sage: f.jacobian()[2,0].display()
(x, y) ↦ -y^3*sin(x)

>>> from sage.all import *
>>> type(f.jacobian()[Integer(2),Integer(0)])
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>
>>> f.jacobian()[Integer(2),Integer(0)].display()
(x, y) ↦ -y^3*sin(x)

Test of the computation:

sage: [[f.jacobian()[i,j] == f[i].diff(j) for j in range(2)] for i in range(3)]
[[True, True], [True, True], [True, True]]

>>> from sage.all import *
>>> [[f.jacobian()[i,j] == f[i].diff(j) for j in range(Integer(2))] for i in range(Integer(3))]
[[True, True], [True, True], [True, True]]

Test with start_index = 1:

sage: M = Manifold(2, 'M', structure='topological', start_index=1)
sage: X.<x,y> = M.chart()
sage: f = X.multifunction(x-y, x*y, y^3*cos(x))
sage: f.jacobian()
[           1           -1]
[           y            x]
[ -y^3*sin(x) 3*y^2*cos(x)]
sage: [[f.jacobian()[i,j] == f[i].diff(j+1) for j in range(2)]  # note the j+1
....:                                         for i in range(3)]
[[True, True], [True, True], [True, True]]

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological', start_index=Integer(1))
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.multifunction(x-y, x*y, y**Integer(3)*cos(x))
>>> f.jacobian()
[           1           -1]
[           y            x]
[ -y^3*sin(x) 3*y^2*cos(x)]
>>> [[f.jacobian()[i,j] == f[i].diff(j+Integer(1)) for j in range(Integer(2))]  # note the j+1
...                                         for i in range(Integer(3))]
[[True, True], [True, True], [True, True]]
jacobian_det()[source]

Return the Jacobian determinant of the system of functions.

The number \(m\) of coordinate functions must equal the number \(n\) of coordinates.

OUTPUT: a ChartFunction representing the determinant

EXAMPLES:

Jacobian determinant of a set of 2 functions of 2 coordinates:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.multifunction(x-y, x*y)
sage: f.jacobian_det()
x + y

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> f = X.multifunction(x-y, x*y)
>>> f.jacobian_det()
x + y

The output of jacobian_det() is an instance of ChartFunction and can therefore be called on specific values of the coordinates, e.g. \((x,y) = (1,2)\):

sage: type(f.jacobian_det())
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>
sage: f.jacobian_det().display()
(x, y) ↦ x + y
sage: f.jacobian_det()(1,2)
3

>>> from sage.all import *
>>> type(f.jacobian_det())
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>
>>> f.jacobian_det().display()
(x, y) ↦ x + y
>>> f.jacobian_det()(Integer(1),Integer(2))
3

The result is cached:

sage: f.jacobian_det() is f.jacobian_det()
True

>>> from sage.all import *
>>> f.jacobian_det() is f.jacobian_det()
True

We verify the determinant of the Jacobian:

sage: f.jacobian_det() == det(matrix([[f[i].diff(j).expr() for j in range(2)]
....:                                 for i in range(2)]))
True

>>> from sage.all import *
>>> f.jacobian_det() == det(matrix([[f[i].diff(j).expr() for j in range(Integer(2))]
...                                 for i in range(Integer(2))]))
True

An example using SymPy:

sage: M.set_calculus_method('sympy')
sage: g = X.multifunction(x*y^3, e^x)
sage: g.jacobian_det()
-3*x*y**2*exp(x)
sage: type(g.jacobian_det().expr())
<class 'sympy.core.mul.Mul'>

>>> from sage.all import *
>>> M.set_calculus_method('sympy')
>>> g = X.multifunction(x*y**Integer(3), e**x)
>>> g.jacobian_det()
-3*x*y**2*exp(x)
>>> type(g.jacobian_det().expr())
<class 'sympy.core.mul.Mul'>

Jacobian determinant of a set of 3 functions of 3 coordinates:

sage: M = Manifold(3, 'M', structure='topological')
sage: X.<x,y,z> = M.chart()
sage: f = X.multifunction(x*y+z^2, z^2*x+y^2*z, (x*y*z)^3)
sage: f.jacobian_det().display()
(x, y, z) ↦ 6*x^3*y^5*z^3 - 3*x^4*y^3*z^4 - 12*x^2*y^4*z^5 + 6*x^3*y^2*z^6

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = X._first_ngens(3)
>>> f = X.multifunction(x*y+z**Integer(2), z**Integer(2)*x+y**Integer(2)*z, (x*y*z)**Integer(3))
>>> f.jacobian_det().display()
(x, y, z) ↦ 6*x^3*y^5*z^3 - 3*x^4*y^3*z^4 - 12*x^2*y^4*z^5 + 6*x^3*y^2*z^6

We verify the determinant of the Jacobian:

sage: f.jacobian_det() == det(matrix([[f[i].diff(j).expr() for j in range(3)]
....:                                 for i in range(3)]))
True

>>> from sage.all import *
>>> f.jacobian_det() == det(matrix([[f[i].diff(j).expr() for j in range(Integer(3))]
...                                 for i in range(Integer(3))]))
True
set_immutable()[source]

Set self and all chart functions of self immutable.

EXAMPLES:

Declare a coordinate function immutable:

sage: M = Manifold(3, 'M', structure='topological')
sage: X.<x,y,z> = M.chart()
sage: f = X.multifunction(x+y+z, x*y*z)
sage: f.is_immutable()
False
sage: f.set_immutable()
sage: f.is_immutable()
True

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = X._first_ngens(3)
>>> f = X.multifunction(x+y+z, x*y*z)
>>> f.is_immutable()
False
>>> f.set_immutable()
>>> f.is_immutable()
True

The chart functions are now immutable, too:

sage: f[0].parent()
Ring of chart functions on Chart (M, (x, y, z))
sage: f[0].is_immutable()
True

>>> from sage.all import *
>>> f[Integer(0)].parent()
Ring of chart functions on Chart (M, (x, y, z))
>>> f[Integer(0)].is_immutable()
True