Initialisation du repository de Beta

venv/lib/python3.12/site-packages/sympy/vector/operators.py

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+import collections
+from sympy.core.expr import Expr
+from sympy.core import sympify, S, preorder_traversal
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.vector import Vector, VectorMul, VectorAdd, Cross, Dot
+from sympy.core.function import Derivative
+from sympy.core.add import Add
+from sympy.core.mul import Mul
+
+
+def _get_coord_systems(expr):
+    g = preorder_traversal(expr)
+    ret = set()
+    for i in g:
+        if isinstance(i, CoordSys3D):
+            ret.add(i)
+            g.skip()
+    return frozenset(ret)
+
+
+def _split_mul_args_wrt_coordsys(expr):
+    d = collections.defaultdict(lambda: S.One)
+    for i in expr.args:
+        d[_get_coord_systems(i)] *= i
+    return list(d.values())
+
+
+class Gradient(Expr):
+    """
+    Represents unevaluated Gradient.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, Gradient
+    >>> R = CoordSys3D('R')
+    >>> s = R.x*R.y*R.z
+    >>> Gradient(s)
+    Gradient(R.x*R.y*R.z)
+
+    """
+
+    def __new__(cls, expr):
+        expr = sympify(expr)
+        obj = Expr.__new__(cls, expr)
+        obj._expr = expr
+        return obj
+
+    def doit(self, **hints):
+        return gradient(self._expr, doit=True)
+
+
+class Divergence(Expr):
+    """
+    Represents unevaluated Divergence.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, Divergence
+    >>> R = CoordSys3D('R')
+    >>> v = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
+    >>> Divergence(v)
+    Divergence(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k)
+
+    """
+
+    def __new__(cls, expr):
+        expr = sympify(expr)
+        obj = Expr.__new__(cls, expr)
+        obj._expr = expr
+        return obj
+
+    def doit(self, **hints):
+        return divergence(self._expr, doit=True)
+
+
+class Curl(Expr):
+    """
+    Represents unevaluated Curl.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, Curl
+    >>> R = CoordSys3D('R')
+    >>> v = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
+    >>> Curl(v)
+    Curl(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k)
+
+    """
+
+    def __new__(cls, expr):
+        expr = sympify(expr)
+        obj = Expr.__new__(cls, expr)
+        obj._expr = expr
+        return obj
+
+    def doit(self, **hints):
+        return curl(self._expr, doit=True)
+
+
+def curl(vect, doit=True):
+    """
+    Returns the curl of a vector field computed wrt the base scalars
+    of the given coordinate system.
+
+    Parameters
+    ==========
+
+    vect : Vector
+        The vector operand
+
+    doit : bool
+        If True, the result is returned after calling .doit() on
+        each component. Else, the returned expression contains
+        Derivative instances
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, curl
+    >>> R = CoordSys3D('R')
+    >>> v1 = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
+    >>> curl(v1)
+    0
+    >>> v2 = R.x*R.y*R.z*R.i
+    >>> curl(v2)
+    R.x*R.y*R.j + (-R.x*R.z)*R.k
+
+    """
+
+    coord_sys = _get_coord_systems(vect)
+
+    if len(coord_sys) == 0:
+        return Vector.zero
+    elif len(coord_sys) == 1:
+        coord_sys = next(iter(coord_sys))
+        i, j, k = coord_sys.base_vectors()
+        x, y, z = coord_sys.base_scalars()
+        h1, h2, h3 = coord_sys.lame_coefficients()
+        vectx = vect.dot(i)
+        vecty = vect.dot(j)
+        vectz = vect.dot(k)
+        outvec = Vector.zero
+        outvec += (Derivative(vectz * h3, y) -
+                   Derivative(vecty * h2, z)) * i / (h2 * h3)
+        outvec += (Derivative(vectx * h1, z) -
+                   Derivative(vectz * h3, x)) * j / (h1 * h3)
+        outvec += (Derivative(vecty * h2, x) -
+                   Derivative(vectx * h1, y)) * k / (h2 * h1)
+
+        if doit:
+            return outvec.doit()
+        return outvec
+    else:
+        if isinstance(vect, (Add, VectorAdd)):
+            from sympy.vector import express
+            try:
+                cs = next(iter(coord_sys))
+                args = [express(i, cs, variables=True) for i in vect.args]
+            except ValueError:
+                args = vect.args
+            return VectorAdd.fromiter(curl(i, doit=doit) for i in args)
+        elif isinstance(vect, (Mul, VectorMul)):
+            vector = [i for i in vect.args if isinstance(i, (Vector, Cross, Gradient))][0]
+            scalar = Mul.fromiter(i for i in vect.args if not isinstance(i, (Vector, Cross, Gradient)))
+            res = Cross(gradient(scalar), vector).doit() + scalar*curl(vector, doit=doit)
+            if doit:
+                return res.doit()
+            return res
+        elif isinstance(vect, (Cross, Curl, Gradient)):
+            return Curl(vect)
+        else:
+            raise ValueError("Invalid argument for curl")
+
+
+def divergence(vect, doit=True):
+    """
+    Returns the divergence of a vector field computed wrt the base
+    scalars of the given coordinate system.
+
+    Parameters
+    ==========
+
+    vector : Vector
+        The vector operand
+
+    doit : bool
+        If True, the result is returned after calling .doit() on
+        each component. Else, the returned expression contains
+        Derivative instances
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, divergence
+    >>> R = CoordSys3D('R')
+    >>> v1 = R.x*R.y*R.z * (R.i+R.j+R.k)
+
+    >>> divergence(v1)
+    R.x*R.y + R.x*R.z + R.y*R.z
+    >>> v2 = 2*R.y*R.z*R.j
+    >>> divergence(v2)
+    2*R.z
+
+    """
+    coord_sys = _get_coord_systems(vect)
+    if len(coord_sys) == 0:
+        return S.Zero
+    elif len(coord_sys) == 1:
+        if isinstance(vect, (Cross, Curl, Gradient)):
+            return Divergence(vect)
+        # TODO: is case of many coord systems, this gets a random one:
+        coord_sys = next(iter(coord_sys))
+        i, j, k = coord_sys.base_vectors()
+        x, y, z = coord_sys.base_scalars()
+        h1, h2, h3 = coord_sys.lame_coefficients()
+        vx = _diff_conditional(vect.dot(i), x, h2, h3) \
+             / (h1 * h2 * h3)
+        vy = _diff_conditional(vect.dot(j), y, h3, h1) \
+             / (h1 * h2 * h3)
+        vz = _diff_conditional(vect.dot(k), z, h1, h2) \
+             / (h1 * h2 * h3)
+        res = vx + vy + vz
+        if doit:
+            return res.doit()
+        return res
+    else:
+        if isinstance(vect, (Add, VectorAdd)):
+            return Add.fromiter(divergence(i, doit=doit) for i in vect.args)
+        elif isinstance(vect, (Mul, VectorMul)):
+            vector = [i for i in vect.args if isinstance(i, (Vector, Cross, Gradient))][0]
+            scalar = Mul.fromiter(i for i in vect.args if not isinstance(i, (Vector, Cross, Gradient)))
+            res = Dot(vector, gradient(scalar)) + scalar*divergence(vector, doit=doit)
+            if doit:
+                return res.doit()
+            return res
+        elif isinstance(vect, (Cross, Curl, Gradient)):
+            return Divergence(vect)
+        else:
+            raise ValueError("Invalid argument for divergence")
+
+
+def gradient(scalar_field, doit=True):
+    """
+    Returns the vector gradient of a scalar field computed wrt the
+    base scalars of the given coordinate system.
+
+    Parameters
+    ==========
+
+    scalar_field : SymPy Expr
+        The scalar field to compute the gradient of
+
+    doit : bool
+        If True, the result is returned after calling .doit() on
+        each component. Else, the returned expression contains
+        Derivative instances
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, gradient
+    >>> R = CoordSys3D('R')
+    >>> s1 = R.x*R.y*R.z
+    >>> gradient(s1)
+    R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
+    >>> s2 = 5*R.x**2*R.z
+    >>> gradient(s2)
+    10*R.x*R.z*R.i + 5*R.x**2*R.k
+
+    """
+    coord_sys = _get_coord_systems(scalar_field)
+
+    if len(coord_sys) == 0:
+        return Vector.zero
+    elif len(coord_sys) == 1:
+        coord_sys = next(iter(coord_sys))
+        h1, h2, h3 = coord_sys.lame_coefficients()
+        i, j, k = coord_sys.base_vectors()
+        x, y, z = coord_sys.base_scalars()
+        vx = Derivative(scalar_field, x) / h1
+        vy = Derivative(scalar_field, y) / h2
+        vz = Derivative(scalar_field, z) / h3
+
+        if doit:
+            return (vx * i + vy * j + vz * k).doit()
+        return vx * i + vy * j + vz * k
+    else:
+        if isinstance(scalar_field, (Add, VectorAdd)):
+            return VectorAdd.fromiter(gradient(i) for i in scalar_field.args)
+        if isinstance(scalar_field, (Mul, VectorMul)):
+            s = _split_mul_args_wrt_coordsys(scalar_field)
+            return VectorAdd.fromiter(scalar_field / i * gradient(i) for i in s)
+        return Gradient(scalar_field)
+
+
+class Laplacian(Expr):
+    """
+    Represents unevaluated Laplacian.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, Laplacian
+    >>> R = CoordSys3D('R')
+    >>> v = 3*R.x**3*R.y**2*R.z**3
+    >>> Laplacian(v)
+    Laplacian(3*R.x**3*R.y**2*R.z**3)
+
+    """
+
+    def __new__(cls, expr):
+        expr = sympify(expr)
+        obj = Expr.__new__(cls, expr)
+        obj._expr = expr
+        return obj
+
+    def doit(self, **hints):
+        from sympy.vector.functions import laplacian
+        return laplacian(self._expr)
+
+
+def _diff_conditional(expr, base_scalar, coeff_1, coeff_2):
+    """
+    First re-expresses expr in the system that base_scalar belongs to.
+    If base_scalar appears in the re-expressed form, differentiates
+    it wrt base_scalar.
+    Else, returns 0
+    """
+    from sympy.vector.functions import express
+    new_expr = express(expr, base_scalar.system, variables=True)
+    arg = coeff_1 * coeff_2 * new_expr
+    return Derivative(arg, base_scalar) if arg else S.Zero