314 lines
8.4 KiB
Python
314 lines
8.4 KiB
Python
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from sympy.core.function import diff
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from sympy.core.singleton import S
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from sympy.integrals.integrals import integrate
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from sympy.physics.vector import Vector, express
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from sympy.physics.vector.frame import _check_frame
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from sympy.physics.vector.vector import _check_vector
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__all__ = ['curl', 'divergence', 'gradient', 'is_conservative',
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'is_solenoidal', 'scalar_potential',
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'scalar_potential_difference']
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def curl(vect, frame):
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"""
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Returns the curl of a vector field computed wrt the coordinate
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symbols of the given frame.
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Parameters
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==========
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vect : Vector
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The vector operand
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frame : ReferenceFrame
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The reference frame to calculate the curl in
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame
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>>> from sympy.physics.vector import curl
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>>> R = ReferenceFrame('R')
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>>> v1 = R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z
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>>> curl(v1, R)
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0
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>>> v2 = R[0]*R[1]*R[2]*R.x
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>>> curl(v2, R)
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R_x*R_y*R.y - R_x*R_z*R.z
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"""
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_check_vector(vect)
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if vect == 0:
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return Vector(0)
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vect = express(vect, frame, variables=True)
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# A mechanical approach to avoid looping overheads
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vectx = vect.dot(frame.x)
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vecty = vect.dot(frame.y)
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vectz = vect.dot(frame.z)
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outvec = Vector(0)
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outvec += (diff(vectz, frame[1]) - diff(vecty, frame[2])) * frame.x
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outvec += (diff(vectx, frame[2]) - diff(vectz, frame[0])) * frame.y
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outvec += (diff(vecty, frame[0]) - diff(vectx, frame[1])) * frame.z
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return outvec
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def divergence(vect, frame):
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"""
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Returns the divergence of a vector field computed wrt the coordinate
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symbols of the given frame.
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Parameters
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==========
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vect : Vector
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The vector operand
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frame : ReferenceFrame
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The reference frame to calculate the divergence in
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame
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>>> from sympy.physics.vector import divergence
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>>> R = ReferenceFrame('R')
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>>> v1 = R[0]*R[1]*R[2] * (R.x+R.y+R.z)
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>>> divergence(v1, R)
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R_x*R_y + R_x*R_z + R_y*R_z
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>>> v2 = 2*R[1]*R[2]*R.y
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>>> divergence(v2, R)
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2*R_z
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"""
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_check_vector(vect)
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if vect == 0:
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return S.Zero
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vect = express(vect, frame, variables=True)
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vectx = vect.dot(frame.x)
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vecty = vect.dot(frame.y)
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vectz = vect.dot(frame.z)
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out = S.Zero
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out += diff(vectx, frame[0])
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out += diff(vecty, frame[1])
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out += diff(vectz, frame[2])
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return out
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def gradient(scalar, frame):
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"""
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Returns the vector gradient of a scalar field computed wrt the
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coordinate symbols of the given frame.
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Parameters
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==========
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scalar : sympifiable
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The scalar field to take the gradient of
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frame : ReferenceFrame
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The frame to calculate the gradient in
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame
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>>> from sympy.physics.vector import gradient
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>>> R = ReferenceFrame('R')
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>>> s1 = R[0]*R[1]*R[2]
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>>> gradient(s1, R)
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R_y*R_z*R.x + R_x*R_z*R.y + R_x*R_y*R.z
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>>> s2 = 5*R[0]**2*R[2]
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>>> gradient(s2, R)
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10*R_x*R_z*R.x + 5*R_x**2*R.z
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"""
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_check_frame(frame)
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outvec = Vector(0)
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scalar = express(scalar, frame, variables=True)
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for i, x in enumerate(frame):
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outvec += diff(scalar, frame[i]) * x
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return outvec
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def is_conservative(field):
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"""
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Checks if a field is conservative.
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Parameters
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==========
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field : Vector
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The field to check for conservative property
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame
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>>> from sympy.physics.vector import is_conservative
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>>> R = ReferenceFrame('R')
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>>> is_conservative(R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z)
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True
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>>> is_conservative(R[2] * R.y)
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False
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"""
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# Field is conservative irrespective of frame
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# Take the first frame in the result of the separate method of Vector
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if field == Vector(0):
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return True
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frame = list(field.separate())[0]
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return curl(field, frame).simplify() == Vector(0)
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def is_solenoidal(field):
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"""
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Checks if a field is solenoidal.
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Parameters
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==========
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field : Vector
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The field to check for solenoidal property
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame
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>>> from sympy.physics.vector import is_solenoidal
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>>> R = ReferenceFrame('R')
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>>> is_solenoidal(R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z)
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True
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>>> is_solenoidal(R[1] * R.y)
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False
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"""
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# Field is solenoidal irrespective of frame
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# Take the first frame in the result of the separate method in Vector
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if field == Vector(0):
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return True
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frame = list(field.separate())[0]
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return divergence(field, frame).simplify() is S.Zero
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def scalar_potential(field, frame):
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"""
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Returns the scalar potential function of a field in a given frame
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(without the added integration constant).
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Parameters
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==========
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field : Vector
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The vector field whose scalar potential function is to be
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calculated
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frame : ReferenceFrame
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The frame to do the calculation in
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame
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>>> from sympy.physics.vector import scalar_potential, gradient
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>>> R = ReferenceFrame('R')
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>>> scalar_potential(R.z, R) == R[2]
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True
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>>> scalar_field = 2*R[0]**2*R[1]*R[2]
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>>> grad_field = gradient(scalar_field, R)
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>>> scalar_potential(grad_field, R)
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2*R_x**2*R_y*R_z
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"""
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# Check whether field is conservative
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if not is_conservative(field):
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raise ValueError("Field is not conservative")
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if field == Vector(0):
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return S.Zero
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# Express the field exntirely in frame
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# Substitute coordinate variables also
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_check_frame(frame)
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field = express(field, frame, variables=True)
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# Make a list of dimensions of the frame
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dimensions = list(frame)
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# Calculate scalar potential function
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temp_function = integrate(field.dot(dimensions[0]), frame[0])
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for i, dim in enumerate(dimensions[1:]):
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partial_diff = diff(temp_function, frame[i + 1])
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partial_diff = field.dot(dim) - partial_diff
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temp_function += integrate(partial_diff, frame[i + 1])
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return temp_function
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def scalar_potential_difference(field, frame, point1, point2, origin):
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"""
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Returns the scalar potential difference between two points in a
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certain frame, wrt a given field.
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If a scalar field is provided, its values at the two points are
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considered. If a conservative vector field is provided, the values
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of its scalar potential function at the two points are used.
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Returns (potential at position 2) - (potential at position 1)
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Parameters
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==========
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field : Vector/sympyfiable
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The field to calculate wrt
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frame : ReferenceFrame
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The frame to do the calculations in
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point1 : Point
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The initial Point in given frame
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position2 : Point
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The second Point in the given frame
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origin : Point
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The Point to use as reference point for position vector
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calculation
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame, Point
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>>> from sympy.physics.vector import scalar_potential_difference
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>>> R = ReferenceFrame('R')
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>>> O = Point('O')
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>>> P = O.locatenew('P', R[0]*R.x + R[1]*R.y + R[2]*R.z)
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>>> vectfield = 4*R[0]*R[1]*R.x + 2*R[0]**2*R.y
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>>> scalar_potential_difference(vectfield, R, O, P, O)
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2*R_x**2*R_y
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>>> Q = O.locatenew('O', 3*R.x + R.y + 2*R.z)
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>>> scalar_potential_difference(vectfield, R, P, Q, O)
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-2*R_x**2*R_y + 18
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"""
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_check_frame(frame)
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if isinstance(field, Vector):
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# Get the scalar potential function
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scalar_fn = scalar_potential(field, frame)
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else:
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# Field is a scalar
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scalar_fn = field
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# Express positions in required frame
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position1 = express(point1.pos_from(origin), frame, variables=True)
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position2 = express(point2.pos_from(origin), frame, variables=True)
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# Get the two positions as substitution dicts for coordinate variables
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subs_dict1 = {}
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subs_dict2 = {}
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for i, x in enumerate(frame):
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subs_dict1[frame[i]] = x.dot(position1)
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subs_dict2[frame[i]] = x.dot(position2)
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return scalar_fn.subs(subs_dict2) - scalar_fn.subs(subs_dict1)
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