Set up a symbolic electric current density boundary condition with a non-surface normal current density:

Specify a differential equation operator :

Specify a region:

On the left, a NeumannValue is set up. The default Neumann boundary integrand for this equation is . To model a boundary integrand of the form , a NeumannValue is set up:

Visualize the solution:

For nonconservative mass transport, boundary conditions like MassImpermeableBoundaryValue can produce NeumannBoundaryUnitNormal. Set up an impermeable boundary condition for a nonconservative model:

NeumannBoundaryUnitNormal can be used in NIntegrate to compute the flux through a boundary. Solve a Poisson equation on a unit Disk:

Visualize the solution:

Compute the total flux through the boundary of the region through the boundary region:

Compute the total flux through the boundary of a subregion:

Specify a time-dependent differential equation operator :

Specify a region:

On the left, a NeumannValue is set up. The default Neumann boundary integrand for this equation is . To model a boundary integrand of the form , a NeumannValue is set up:

Use a Neumann 0 boundary condition and solve the equation again:

Inspect how the solutions start to differ over time:

Create a tangential for a NeumannValue:

Make use of an Indexed component, the component, of a NeumannBoundaryUnitNormal to compute a NeumannValue:

The AcousticRadiationValue makes use of a NeumannBoundaryUnitNormal to automatically compute the sound direction vector. Define model variables vars for a frequency domain acoustic pressure field with model parameters pars:

Set up the equation with a radiation boundary at the left end and an acoustic absorbing boundary at the right end:

Set up the parametric solver:

Convert the solution to the time domain and visualize the solution in the frequency domain at various frequencies :

The boundary unit normal is computed by solving a Poisson equation over the region and specifying 0 Dirichlet conditions. Compute a Poisson equation over a unit Disk:

Compute the normalized gradient of the potential:

Visualize the unit normal: