  Active Brownian Particles (ABPs) transmit a swim pressure $\Pi^{swim}=n\zeta
D^{swim}$ to the container boundaries, where $\zeta$ is the drag coefficient,
$D^{swim}$ is the swim diffusivity and $n$ is the uniform bulk number density
far from the container walls. In this work we extend the notion of the
isotropic swim pressure to the anisotropic tensorial swim stress
$\mathbf{\sigma}^{swim} = - n \zeta \mathbf{D}^{swim}$, which is related to the
anisotropic swim diffusivity $\mathbf{D}^{swim}$. We demonstrate this
relationship with ABPs that achieve nematic orientational order via a bulk
external field. The anisotropic swim stress is obtained analytically for dilute
ABPs in both 2D and 3D systems, and the anisotropy is shown to grow
exponentially with the strength of the external field. We verify that the
normal component of the anisotropic swim stress applies a pressure
$\Pi^{swim}=-(\mathbf{\sigma}^{swim}\cdot\mathbf{n})\cdot\mathbf{n}$ on a wall
with normal vector $\mathbf{n}$, and, through Brownian dynamics simulations,
this pressure is shown to be the force per unit area transmitted by the active
particles. Since ABPs have no friction with a wall, the difference between the
normal and tangential stress components -- the normal stress difference --
generates a net flow of ABPs along the wall, which is a generic property of
active matter systems.
