# Scalar and Vector Field Functionality¶

## Implementation in sympy.vector¶

### Scalar and vector fields¶

In `sympy.vector`

, every `CoordSysCartesian`

instance is assigned basis
vectors corresponding to the \(X\), \(Y\) and
\(Z\) axes. These can be accessed using the properties
named `i`

, `j`

and `k`

respectively. Hence, to define a vector
\(\mathbf{v}\) of the form
\(3\mathbf{\hat{i}} + 4\mathbf{\hat{j}} + 5\mathbf{\hat{k}}\) with
respect to a given frame \(\mathbf{R}\), you would do

```
>>> from sympy.vector import CoordSys3D
>>> R = CoordSys3D('R')
>>> v = 3*R.i + 4*R.j + 5*R.k
```

Vector math and basic calculus operations with respect to vectors have already been elaborated upon in the earlier section of this module’s documentation.

On the other hand, base scalars (or coordinate variables) are implemented
in a special class called `BaseScalar`

, and are assigned to every
coordinate system, one for each axis from \(X\), \(Y\) and
\(Z\). These coordinate variables are used to form the expressions of
vector or scalar fields in 3D space.
For a system `R`

, the \(X\), \(Y\) and \(Z\)
`BaseScalars`

instances can be accessed using the `R.x`

, `R.y`

and `R.z`

expressions respectively.

Therefore, to generate the expression for the aforementioned electric potential field \(2{x}^{2}y\), you would have to do

```
>>> from sympy.vector import CoordSys3D
>>> R = CoordSys3D('R')
>>> electric_potential = 2*R.x**2*R.y
>>> electric_potential
2*R.x**2*R.y
```

It is to be noted that `BaseScalar`

instances can be used just
like any other SymPy `Symbol`

, except that they store the information
about the coordinate system and axis they correspond to.

Scalar fields can be treated just as any other SymPy expression,
for any math/calculus functionality. Hence, to differentiate the above
electric potential with respect to \(x\) (i.e. `R.x`

), you would
use the `diff`

method.

```
>>> from sympy.vector import CoordSys3D
>>> R = CoordSys3D('R')
>>> electric_potential = 2*R.x**2*R.y
>>> from sympy import diff
>>> diff(electric_potential, R.x)
4*R.x*R.y
```

It is worth noting that having a `BaseScalar`

in the expression implies
that a ‘field’ changes with position, in 3D space. Technically speaking, a
simple `Expr`

with no `BaseScalar`

s is still a field, though
constant.

Like scalar fields, vector fields that vary with position can also be
constructed using `BaseScalar`

s in the measure-number expressions.

```
>>> from sympy.vector import CoordSys3D
>>> R = CoordSys3D('R')
>>> v = R.x**2*R.i + 2*R.x*R.z*R.k
```

### The Del operator¶

The Del, or ‘Nabla’ operator - written as \(\mathbf{\nabla}\) is commonly known as the vector differential operator. Depending on its usage in a mathematical expression, it may denote the gradient of a scalar field, the divergence of a vector field, or the curl of a vector field.

Essentially, \(\mathbf{\nabla}\) is not technically an ‘operator’, but a convenient mathematical notation to denote any one of the aforementioned field operations.

In `sympy.vector`

, \(\mathbf{\nabla}\) has been implemented
as the `Del()`

class. The instance of this class is independent of
coordinate system. Hence, the \(\mathbf{\nabla}\) operator would
be accessible as `Del()`

.

Given below is an example of usage of the `Del()`

class.

```
>>> from sympy.vector import CoordSys3D, Del
>>> C = CoordSys3D('C')
>>> delop = Del()
>>> gradient_field = delop(C.x*C.y*C.z)
>>> gradient_field
(Derivative(C.x*C.y*C.z, C.x))*C.i + (Derivative(C.x*C.y*C.z, C.y))*C.j
+ (Derivative(C.x*C.y*C.z, C.z))*C.k
```

The above expression can be evaluated using the SymPy `doit()`

routine.

```
>>> gradient_field.doit()
C.y*C.z*C.i + C.x*C.z*C.j + C.x*C.y*C.k
```

Usage of the \(\mathbf{\nabla}\) notation in `sympy.vector`

has been described in greater detail in the subsequent subsections.

## Conservative and Solenoidal fields¶

In vector calculus, a conservative field is a field that is the gradient of some scalar field. Conservative fields have the property that their line integral over any path depends only on the end-points, and is independent of the path travelled. A conservative vector field is also said to be ‘irrotational’, since the curl of a conservative field is always zero.

In physics, conservative fields represent forces in physical systems where energy is conserved.

To check if a vector field is conservative in `sympy.vector`

, the
`is_conservative`

function can be used.

```
>>> from sympy.vector import CoordSys3D, is_conservative
>>> R = CoordSys3D('R')
>>> field = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
>>> is_conservative(field)
True
>>> curl(field)
0
```

A solenoidal field, on the other hand, is a vector field whose divergence is zero at all points in space.

To check if a vector field is solenoidal in `sympy.vector`

, the
`is_solenoidal`

function can be used.

```
>>> from sympy.vector import CoordSys3D, is_solenoidal
>>> R = CoordSys3D('R')
>>> field = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
>>> is_solenoidal(field)
True
>>> divergence(field)
0
```

## Scalar potential functions¶

We have previously mentioned that every conservative field can be defined as the gradient of some scalar field. This scalar field is also called the ‘scalar potential field’ corresponding to the aforementioned conservative field.

The `scalar_potential`

function in `sympy.vector`

calculates the
scalar potential field corresponding to a given conservative vector field in
3D space - minus the extra constant of integration, of course.

Example of usage -

```
>>> from sympy.vector import CoordSys3D, scalar_potential
>>> R = CoordSys3D('R')
>>> conservative_field = 4*R.x*R.y*R.z*R.i + 2*R.x**2*R.z*R.j + 2*R.x**2*R.y*R.k
>>> scalar_potential(conservative_field, R)
2*R.x**2*R.y*R.z
```

Providing a non-conservative vector field as an argument to
`scalar_potential`

raises a `ValueError`

.

The scalar potential difference, or simply ‘potential difference’, corresponding to a conservative vector field can be defined as the difference between the values of its scalar potential function at two points in space. This is useful in calculating a line integral with respect to a conservative function, since it depends only on the endpoints of the path.

This computation is performed as follows in `sympy.vector`

.

```
>>> from sympy.vector import CoordSys3D, Point
>>> from sympy.vector import scalar_potential_difference
>>> R = CoordSys3D('R')
>>> P = R.origin.locate_new('P', 1*R.i + 2*R.j + 3*R.k)
>>> vectfield = 4*R.x*R.y*R.i + 2*R.x**2*R.j
>>> scalar_potential_difference(vectfield, R, R.origin, P)
4
```

If provided with a scalar expression instead of a vector field,
`scalar_potential_difference`

returns the difference between the values
of that scalar field at the two given points in space.