# Documentation

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# `vectorPotential`

Vector potential of a three-dimensional vector field

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## Syntax

```vectorPotential(`j`, `[x1, x2, x3]`, <Test>)
```

## Description

`vectorPotential(j, x)` returns the vector potential of the vector field with respect to . This is a vector field with .

The vector potential of a vector function `j` exists if and only if the divergence of `j` is zero. It is uniquely determined.

If the vector potential of `j` does not exist, then `vectorPotential` returns `FALSE`.

If `j` is a vector then the component ring of `j` must be a field (i.e., a domain of category `Cat::Field`) for which definite integration can be performed.

If `j` is given as a list of three arithmetical expressions, then `vectorPotential` returns a vector of the domain `Dom::Matrix``()`.

## Examples

### Example 1

We check if the vector function has a vector potential:

```delete x, y, z: vectorPotential( [x^2*y, -1/2*y^2*x, -x*y*z], [x, y, z], Test )```

The answer is yes, so let us compute the vector potential of :

```vectorPotential( [x^2*y, -1/2*y^2*x, -x*y*z], [x, y, z] )```

We check the result:

`curl(%, [x, y, z])`

### Example 2

The vector function does not have a vector potential:

`vectorPotential([x^2, 2*y, z], [x, y, z])`

## Parameters

 `j` A list of three arithmetical expressions, or a 3-dimensional vector (i.e., a 3×1 or 1 ×3 matrix of a domain of category `Cat::Matrix`) `x1`, `x2`, `x3` (indexed) identifiers

## Options

 `Test` Check whether the vector field `j` has a vector potential and return `TRUE` or `FALSE`, respectively.

## Return Values

Vector with three components, i.e., an 3 ×1 or n matrix of a domain of category `Cat::Matrix`, or a boolean value.