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Sympy Support

generalised_chain_rule_for and generalised_chain_rule

  • This code defines two functions in Python, generalised_chain_rule_for and generalised_chain_rule, both of which are related to the concept of the chain rule in calculus.

  • generalised_chain_rule_for is a function that takes in four arguments: j, f, variables, and args. j is an integer representing the index of the variable with respect to which the differentiation is performed. f is a symbolic function, variables is a dictionary containing the independent variables and their corresponding expressions, and args is a tuple of the independent variables.

  • The function returns the differential of the composite function f with respect to the j-th component of args, calculated using the chain rule. The chain rule is implemented using a sum of the product of the partial derivative of f with respect to each independent variable and the derivative of the corresponding expression with respect to the j-th component of args.

  • The function also includes two examples to illustrate its usage.

    • The first example calculates the differential of the composite function f(x, y) with respect to r using the chain rule, where x and y are expressed in terms of r and theta.
    • The second example calculates the differential of the same composite function with respect to theta using the chain rule.

  • generalised_chain_rule is a function that takes in three arguments: w, variables, and args. w is a symbolic function, variables is a dictionary containing the independent variables and their corresponding expressions, and args is a tuple of the independent variables.

  • The function returns the differential of the composite function w with respect to the independent variables args, calculated using the chain rule. The chain rule is implemented using a sum of the product of the partial derivative of w with respect to each independent variable and the derivative of the corresponding expression with respect to the independent variables.

>>> import sympy as sp
>>> from  sympy_support.generalised_chain_rule import generalised_chain_rule_for, generalised_chain_rule

>>> x, y, z, r, theta = sp.symbols("x, y, z, r, theta", real=True)
>>> f = sp.symbols("f", cls=sp.Function)

>>> generalised_chain_rule_for(0, f, {x: r * sp.cos(theta), y: r * sp.sin(theta)}, r, theta)
sin(theta)*Derivative(f(x, y), y) + cos(theta)*Derivative(f(x, y), x)

>>> generalised_chain_rule_for(1, f, {x: r * sp.cos(theta), y: r * sp.sin(theta)}, r, theta)
-r*sin(theta)*Derivative(f(x, y), x) + r*cos(theta)*Derivative(f(x, y), y)

# w = f(x,y), x = sin(t), y = cos(t) calculate df/dt
>>> x, y, t = sp.symbols("x, y, t", real=True)
>>> dependents = {x: sp.sin(t), y: sp.cos(t)}
>>> f = sp.Function("f")
>>> generalised_chain_rule(f, dependents, t)
(Eq(Derivative(f(t), t), -sin(t)*Derivative(f(x, y), y) + cos(t)*Derivative(f(x, y), x)),)

>>> generalised_chain_rule(f, dependents, t)[0].lhs
Derivative(f(t), t)

>>> generalised_chain_rule(f, dependents, t)[0].rhs
-sin(t)*Derivative(f(x, y), y) + cos(t)*Derivative(f(x, y), x)

find_dependent_variable_differentials

  • find_dependent_variable_differentials that calculates the partial derivatives of a given function w (represented as a SymPy Function symbol) with respect to a set of independent variables args. The function takes into account the dependencies of the variables on other variables, which are provided in the variables dictionary.
  • The function defines the find_dependent_variable_differentials function, which takes in the function symbol w, the dictionary of dependent variables, and the independent variables args.
  • The function then calculates the partial derivatives of the function w with respect to each of the independent variables in args. It does this by using the chain rule to express the partial derivatives in terms of the partial derivatives of the dependent variables in variables.
  • The function returns a list of equations, where each equation gives the partial derivative of the function w with respect to one of the independent variables in args.
  • The example provided in the docstring demonstrates how to use the function.
    • It defines two dependent variables x and y in terms of the independent variables r and theta
    • It then uses find_dependent_variable_differentials to calculate the partial derivatives of a function f with respect to x and y.
>>> import sympy as sp
>>> from  sympy_support.generalised_chain_rule import find_dependent_variable_differentials
>>> x, y, r, theta = sp.symbols("x, y, r, theta", real=True)
>>> dependents = {x: r * sp.cos(theta), y: r * sp.sin(theta)}
>>> f = sp.Function("f")

>>> result = find_dependent_variable_differentials(f, dependents, r, theta)
>>> result[sp.Derivative(f(x, y), x)]
cos(theta)*Derivative(f(r, theta), r) - sin(theta)*Derivative(f(r, theta), theta)/r

>>> result[sp.Derivative(f(x, y), y)]
sin(theta)*Derivative(f(r, theta), r) + cos(theta)*Derivative(f(r, theta), theta)/r