# Docstrings for basic field functions¶

## Field operation functions¶

These functions implement some basic operations pertaining to fields in general.

### curl¶

sympy.physics.vector.fieldfunctions.curl(vect, frame)[source]

Returns the curl of a vector field computed wrt the coordinate symbols of the given frame.

Parameters : vect : Vector The vector operand frame : ReferenceFrame The reference frame to calculate the curl in

Examples

>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import curl
>>> R = ReferenceFrame('R')
>>> v1 = R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z
>>> curl(v1, R)
0
>>> v2 = R[0]*R[1]*R[2]*R.x
>>> curl(v2, R)
R_x*R_y*R.y - R_x*R_z*R.z


### divergence¶

sympy.physics.vector.fieldfunctions.divergence(vect, frame)[source]

Returns the divergence of a vector field computed wrt the coordinate symbols of the given frame.

Parameters : vect : Vector The vector operand frame : ReferenceFrame The reference frame to calculate the divergence in

Examples

>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import divergence
>>> R = ReferenceFrame('R')
>>> v1 = R[0]*R[1]*R[2] * (R.x+R.y+R.z)
>>> divergence(v1, R)
R_x*R_y + R_x*R_z + R_y*R_z
>>> v2 = 2*R[1]*R[2]*R.y
>>> divergence(v2, R)
2*R_z


Returns the vector gradient of a scalar field computed wrt the coordinate symbols of the given frame.

Parameters : scalar : sympifiable The scalar field to take the gradient of frame : ReferenceFrame The frame to calculate the gradient in

Examples

>>> from sympy.physics.vector import ReferenceFrame
>>> R = ReferenceFrame('R')
>>> s1 = R[0]*R[1]*R[2]
R_y*R_z*R.x + R_x*R_z*R.y + R_x*R_y*R.z
>>> s2 = 5*R[0]**2*R[2]
10*R_x*R_z*R.x + 5*R_x**2*R.z


### scalar_potential¶

sympy.physics.vector.fieldfunctions.scalar_potential(field, frame)[source]

Returns the scalar potential function of a field in a given frame (without the added integration constant).

Parameters : field : Vector The vector field whose scalar potential function is to be calculated frame : ReferenceFrame The frame to do the calculation in

Examples

>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import scalar_potential, gradient
>>> R = ReferenceFrame('R')
>>> scalar_potential(R.z, R) == R[2]
True
>>> scalar_field = 2*R[0]**2*R[1]*R[2]
2*R_x**2*R_y*R_z


### scalar_potential_difference¶

sympy.physics.vector.fieldfunctions.scalar_potential_difference(field, frame, point1, point2, origin)[source]

Returns the scalar potential difference between two points in a certain frame, wrt a given field.

If a scalar field is provided, its values at the two points are considered. If a conservative vector field is provided, the values of its scalar potential function at the two points are used.

Returns (potential at position 2) - (potential at position 1)

Parameters : field : Vector/sympyfiable The field to calculate wrt frame : ReferenceFrame The frame to do the calculations in point1 : Point The initial Point in given frame position2 : Point The second Point in the given frame origin : Point The Point to use as reference point for position vector calculation

Examples

>>> from sympy.physics.vector import ReferenceFrame, Point
>>> from sympy.physics.vector import scalar_potential_difference
>>> R = ReferenceFrame('R')
>>> O = Point('O')
>>> P = O.locatenew('P', R[0]*R.x + R[1]*R.y + R[2]*R.z)
>>> vectfield = 4*R[0]*R[1]*R.x + 2*R[0]**2*R.y
>>> scalar_potential_difference(vectfield, R, O, P, O)
2*R_x**2*R_y
>>> Q = O.locatenew('O', 3*R.x + R.y + 2*R.z)
>>> scalar_potential_difference(vectfield, R, P, Q, O)
-2*R_x**2*R_y + 18


## Checking the type of vector field¶

### is_conservative¶

sympy.physics.vector.fieldfunctions.is_conservative(field)[source]

Checks if a field is conservative.

Examples

>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import is_conservative
>>> R = ReferenceFrame('R')
>>> is_conservative(R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z)
True
>>> is_conservative(R[2] * R.y)
False


Paramaters

field : Vector
The field to check for conservative property

### is_solenoidal¶

sympy.physics.vector.fieldfunctions.is_solenoidal(field)[source]

Checks if a field is solenoidal.

Examples

>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import is_solenoidal
>>> R = ReferenceFrame('R')
>>> is_solenoidal(R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z)
True
>>> is_solenoidal(R[1] * R.y)
False


Paramaters

field : Vector
The field to check for solenoidal property

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