# Source code for ufl.utils.derivativetuples

```# -*- coding: utf-8 -*-
"This module contains a collection of utilities for representing partial derivatives as integer tuples."

# Copyright (C) 2013-2016 Martin Sandve Alnæs and Anders Logg
#
# This file is part of UFL (https://www.fenicsproject.org)
#

import itertools

[docs]def derivative_counts_to_listing(derivative_counts):
"""Convert a derivative count tuple to a derivative listing tuple.

The derivative d^3 / dy^2 dz is represented
in counting form as (0, 2, 1) meaning (dx^0, dy^2, dz^1)
and in listing form as (1, 1, 2) meaning (dy, dy, dz).
"""
derivatives = []  # = 1
for i, d in enumerate(derivative_counts):
derivatives.extend((i,) * d)  # *= d/dx_i^d
return tuple(derivatives)

[docs]def derivative_listing_to_counts(derivatives, gdim):
"""Convert a derivative listing tuple to a derivative count tuple.

The derivative d^3 / dy^2 dz is represented
in counting form as (0, 2, 1) meaning (dx^0, dy^2, dz^1)
and in listing form as (1, 1, 2) meaning (dy, dy, dz).
"""
derivative_counts = [0] * gdim
for d in derivatives:
derivative_counts[d] += 1
return tuple(derivative_counts)

[docs]def compute_derivative_tuples(n, gdim):
"""Compute the list of all derivative tuples for derivatives of
given total order n and given geometric dimension gdim. This
function returns two lists. The first is a list of tuples, where
each tuple of length n specifies the coordinate directions of the
n derivatives. The second is a corresponding list of tuples, where
each tuple of length gdim specifies the number of derivatives in
each direction. Both lists have length gdim^n and are ordered as
expected by the UFC function tabulate_basis_derivatives.

Example: If n = 2 and gdim = 3, then the nice tuples are

(0, 0)  <-->  (2, 0, 0)  <-->  d^2/dxdx
(0, 1)  <-->  (1, 1, 0)  <-->  d^2/dxdy
(0, 2)  <-->  (1, 0, 1)  <-->  d^2/dxdz
(1, 0)  <-->  (1, 1, 0)  <-->  d^2/dydx
(1, 1)  <-->  (0, 2, 0)  <-->  d^2/dydy
(1, 2)  <-->  (0, 1, 1)  <-->  d^2/dydz
(2, 0)  <-->  (1, 0, 1)  <-->  d^2/dzdx
(2, 1)  <-->  (0, 1, 1)  <-->  d^2/dzdy
(2, 2)  <-->  (0, 0, 2)  <-->  d^2/dzdz
"""

# Create list of derivatives (note that we have d^n derivatives)
deriv_tuples = [d for d in itertools.product(*(n * [range(0, gdim)]))]

# Translate from list of derivative tuples to list of tuples
# expressing the number of derivatives in each dimension...
_deriv_tuples = [tuple(len([_d for _d in d if _d == i]) for i in range(gdim))
for d in deriv_tuples]

return deriv_tuples, _deriv_tuples
```