cdb.core.manip

Manipulating equations, inequalities and similar expressions

This package contains various helper functions to manipulate expressions in elementary ways, e.g. moving terms around or multiplying sides of an expression by a factor. In order to use this package, import it with the standard Python import statement: if you do import cdb.core.manip as manip then e.g. the isolate function is available as manip.isolate.

eq_to_subrule(ex: Ex) -> Ex

Create substitution rule from an equation.

Creates a substitution rule from an equation Ex.

eq_to_subrule($a = b + c + d$);

$\displaystyle{}b+c+d \rightarrow a$

b + c + d -> a

multiply_through(ex: Ex, factor: Ex, auto_distribute=False) -> Ex

Multiply all terms by the same factor.

Multiplies all terms in ex by factor. If auto_distribute is True, distribute is called on the resulting expression.

multiply_through($a+b < c$, $k$);

$\displaystyle{}k \left(a+b\right) < k c$

k (a + b) < k c

add_through(ex: Ex, factor: Ex) -> Ex

Add a factor to both sides of an equation.

add_through($A_{\mu}B_{\nu} = C_{\mu \nu}$, $D_{\mu \nu}$);

$\displaystyle{}A_{\mu} B_{\nu}+D_{\mu \nu} = C_{\mu \nu}+D_{\mu \nu}$

A_{\mu} B_{\nu} + D_{\mu \nu} = C_{\mu \nu} + D_{\mu \nu}

apply_through(ex: Ex, op: Ex) -> Ex

Apply an operator through both sides of an equation.

apply_through($\partial{x_{\mu}} = 0$, $\partial{#}$);

$\displaystyle{}\partial\left(\partial\left(x_{\mu}\right)\right) = \partial\left(0\right)$

\partial(\partial(x_{\mu})) = \partial(0)

get_lhs(ex: Ex) -> Ex

Return left-hand side of an expression

Returns the left hand side of an equation, inequality or expression whose top node contains two children.

get_lhs($a=b$);

$\displaystyle{}a$

get_rhs(ex: Ex) -> Ex

Return right-hand side of an expression

Returns the right-hand side of an equation, inequality or expression whose top node contains two children.

get_rhs($a=b$);

$\displaystyle{}b$

swap_sides(ex: Ex) -> Ex

Swap the left and right-hand sides of an expression.

Swaps the two sides of an equation, inequality or expression whose top node contains two children. Inequalities will flip as appropriate.

get_factor(node: ExNode, term: Ex) -> Ex

Get all multiplicative factors of term in node

get_factor($3/2 R$.top(), $R$);

$\displaystyle{}\frac{3}{2}$

3/2

get_basis_component(ex: Ex, basis: Ex) -> Ex

Returns the multiplicative coefficients of \texttt{basis

in \texttt{ex}}

test := A_{\mu\nu} \exp(i*k_\mu*x^{\mu})+B_{\mu\nu} \exp(-i*k_\mu*x^{\mu}); get_basis_component(test, $\exp(-i*k_\mu*x^{\mu})$); get_basis_component(test, $A_{\mu\nu}$); get_basis_component(test, $k_{\mu}$); get_basis_component(test, $C_{\mu\nu}$);

$\displaystyle{}A_{\mu \nu} \exp\left(i k_{\mu} x^{\mu}\right)+B_{\mu \nu} \exp\left(-i k_{\mu} x^{\mu}\right)$

A_{\mu \nu} \exp(i k_{\mu} x^{\mu}) + B_{\mu \nu} \exp(-i k_{\mu} x^{\mu})

$\displaystyle{}B_{\mu \nu}$

B_{\mu \nu}

$\displaystyle{}\exp\left(i k_{\mu} x^{\mu}\right)$

\exp(i k_{\mu} x^{\mu})

$\displaystyle{}A_{\mu \nu} \exp\left(i x^{\mu}\right)+B_{\mu \nu} \exp\left(-i x^{\mu}\right)$

A_{\mu \nu} \exp(i x^{\mu}) + B_{\mu \nu} \exp(-i x^{\mu})

$\displaystyle{}0$

to_rhs(ex: Ex, *parts: Ex..., match_type: str = "all") -> Ex

Move the indicated term of the expression to the right-hand side.

If ex is an equation or inequality which contains part on its left hand side, then moves all parts to the right hand side. match_type can be either all which will move all terms which contain each part as a factor, numeric which will move terms up to a numeric prefactor or exact which will only move the term if the subtree matches exactly.

to_rhs($2a+b+j*b+k*b = c+d$);

$\displaystyle{}0 = c+d-2a-b-j b-k b$

0 = c + d-2a-b-j b-k b

\partial{#}::Derivative. to_rhs($-\partial_{\alpha}{b_{\beta}} - \partial_{\beta}{a_{\alpha}} = 0$, $\partial_{\alpha}{b_{\beta}}$, match_type="exact");

$\displaystyle{}-\partial_{\alpha}{b_{\beta}}-\partial_{\beta}{a_{\alpha}} = 0$

-\partial_{\alpha}(b_{\beta})-\partial_{\beta}(a_{\alpha}) = 0

to_lhs(ex: Ex, *parts: Ex..., match_type: str = "all") -> Ex

Move the indicated term of the expression to the left-hand side.

Complement of to_rhs.

isolate()

Isolates a term on the left hand side of an equation or inequality

ex := 2a + 1/2 a * k - b = c * a + b * k + a - 4; isolate(ex, $a$, True); ex2 := -3/2 R + 5\Lambda = T\kappa; isolate(ex2, $R$);

$\displaystyle{}2a+\frac{1}{2}a k-b = c a+b k+a-4$

2a + 1/2 a k-b = c a + b k + a-4

$\displaystyle{}a = \left(b k-4+b\right) {\left(1-c\right)}^{-1}$

a = (b k-4 + b) (1-c)**(-1)

$\displaystyle{} - \frac{3}{2}R+5\Lambda = T \kappa$

- 3/2 R + 5\Lambda = T \kappa

$\displaystyle{}R = - \frac{2}{3}T \kappa+\frac{10}{3}\Lambda$

R = - 2/3 T \kappa + 10/3 \Lambda