def gradient(scalar_field, coord_sys=None, 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 coord_sys : CoordSys3D The coordinate system to calculate the gradient in Deprecated since version 1.1 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_sys_from_expr(scalar_field, coord_sys) 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)
def gradient(scalar_field, coord_sys=None, 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 coord_sys : CoordSys3D The coordinate system to calculate the gradient in Deprecated since version 1.1 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_sys_from_expr(scalar_field, coord_sys) 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)
def curl(vect, coord_sys=None, 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 coord_sys : CoordSys3D The coordinate system to calculate the gradient in. Deprecated since version 1.1 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_sys_from_expr(vect, coord_sys) 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 Curl(vect)
def curl(vect, coord_sys=None, 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 coord_sys : CoordSys3D The coordinate system to calculate the gradient in. Deprecated since version 1.1 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_sys_from_expr(vect, coord_sys) 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 Curl(vect)