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# vectorPotential

Vector potential of vector field

## Syntax

``vectorPotential(V,X)``
``vectorPotential(V)``

## Description

````vectorPotential(V,X)` computes the vector potential of the vector field `V` with respect to the vector `X` in Cartesian coordinates. The vector field `V` and the vector `X` are both three-dimensional.```

example

````vectorPotential(V)` returns the vector potential `V` with respect to a vector constructed from the first three symbolic variables found in `V` by `symvar`.```

## Examples

### Compute Vector Potential of Field

Compute the vector potential of this row vector field with respect to the vector `[x, y, z]`:

```syms x y z vectorPotential([x^2*y, -1/2*y^2*x, -x*y*z], [x y z])```
```ans = -(x*y^2*z)/2 -x^2*y*z 0```

Compute the vector potential of this column vector field with respect to the vector `[x, y, z]`:

```syms x y z f(x,y,z) = 2*y^3 - 4*x*y; g(x,y,z) = 2*y^2 - 16*z^2+18; h(x,y,z) = -32*x^2 - 16*x*y^2; A = vectorPotential([f; g; h], [x y z])```
```A(x, y, z) = z*(2*y^2 + 18) - (16*z^3)/3 + (16*x*y*(y^2 + 6*x))/3 2*y*z*(- y^2 + 2*x) 0```

### Test if Vector Potential Exists for Field

To check whether the vector potential exists for a particular vector field, compute the divergence of that vector field:

```syms x y z V = [x^2 2*y z]; divergence(V, [x y z])```
```ans = 2*x + 3```

If the divergence is not equal to 0, the vector potential does not exist. In this case, `vectorPotential` returns the vector with all three components equal to `NaN`:

`vectorPotential(V, [x y z])`
```ans = NaN NaN NaN```

## Input Arguments

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Vector field, specified as a 3-D vector of symbolic expressions or functions.

Input, specified as a vector of three symbolic variables with respect to which you compute the vector potential.

## More About

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### Vector Potential of a Vector Field

The vector potential of a vector field `V` is a vector field `A`, such that:

`$V=\nabla ×A=curl\left(A\right)$`

## Tips

• The vector potential exists if and only if the divergence of a vector field `V` with respect to `X` equals 0. If `vectorPotential` cannot verify that `V` has a vector potential, it returns the vector with all three components equal to `NaN`.

## See Also

#### Mathematical Modeling with Symbolic Math Toolbox

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