In this demo, we look at how Basix can be used to compute the integral of the normal derivative of a basis function over a triangular facet of a tetrahedral cell.

As an example, we integrate the normal derivative of the fifth basis function (note: counting starts at 0) of a degree 3 Lagrange space over the zeroth facet of a tetrahedral cell. This facet will have vertices at (1,0,0), (0,1,0) and (0,0,1).

We start by importing Basis and Numpy.

```
import basix
import numpy as np
from basix import ElementFamily, CellType, LagrangeVariant
```

We define a degree 3 Lagrange space on a tetrahedron.

```
lagrange = basix.create_element(
ElementFamily.P, CellType.tetrahedron, 3, LagrangeVariant.equispaced)
```

The facets of a tetrahedron are triangular, so we create a quadrature rule on a triangle. We use an order 3 rule so that we can integrate the basis functions in our space exactly.

```
points, weights = basix.make_quadrature(CellType.triangle, 3)
```

Next, we must map the quadrature points to our facet. We use the function geometry to get the coordinates of the vertices of the tetrahedron, and we use sub_entity_connectivity to see which vertices are adjacent to our facet. We get the sub-entity connectivity using the indices 2 (facets of 3D cells have dimension 2), 0 (vertices have dimension 0), and 0 (the index of the facet we chose to use).

Using this information, we can map the quadrature points to the facet.

```
vertices = basix.geometry(CellType.tetrahedron)
facet = basix.cell.sub_entity_connectivity(CellType.tetrahedron)[2][0][0]
mapped_points = np.array([
vertices[facet[0]] * (1 - x - y) + vertices[facet[1]] * x + vertices[facet[2]] * y
for x, y in points
])
```

We now compute the normal derivative of the fifth basis function at the quadrature points. First, we use facet_outward_normals to get the normal vector to the facet.

We then tabulate the basis functions of our space at the quadrature points. We pass 1 in as the first argument, as we want the derivatives of the basis functions. The result of tabulation will be an array of size 4 by number of quadrature points by number of degrees of freedom. To get the data that we want, we use the indices 1: (to get the derivatives and not also the function values), : (to include every point), 5 (to get the fifth basis function), and 0 (to get the only entry as the value size is 1).

We then multiply the three derivatives of the basis function by the three componenets of the normal.

```
normal = basix.cell.facet_outward_normals(CellType.tetrahedron)[0]
tab = lagrange.tabulate(1, mapped_points)[1:, :, 5, 0]
normal_deriv = tab[0] * normal[0] + tab[1] * normal[1] + tab[2] * normal[2]
```

As our facet is not the reference triangle, we must multiply the integrand by the norm of the Jacobian. We compute this by taking the cross product of the two columns of the Jacobian, and then compute the integral.

```
jacobian = basix.cell.facet_jacobians(CellType.tetrahedron)[0]
size_jacobian = np.linalg.norm(np.cross(jacobian[:, 0], jacobian[:, 1]))
print(np.sum(normal_deriv * weights) * size_jacobian)
```