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Linear steady transport on a translating sphere gives $B=2A{C}_{0}a/(k{a}^2+2D_s)$. The components of its unit normal vector are degree-one spherical harmonics. The condition $2|A|Ua/(k{a}^2+2D_s)\ll 1$ ensures the concentration perturbation is small relative to $C_0$.

For three non-collinear points with position vectors $\mathbf{a},\mathbf{b},\mathbf{c}$, a nonzero normal vector isThe plane is the set of $\mathbf{r}$ satisfying $(\mathbf{r}-\mathbf{a})\cdot \mathbf{N}=0$. Expanding the cross product and using the scalar triple product givesNon-collinearity is essential: otherwise $\mathbf{N}=0$ and the equation fails to select a unique plane.

A hypersurface in an $n$-dimensional smooth manifold is a smooth submanifold of dimension $n-1$. Locally, a regular level set $\{\varphi =0\}$ with $\nabla \varphi \ne 0$ is a hypersurface whose normal vector is $\nabla \varphi$.

A non-null hypersurface projection annihilates the unit normal vector and fixes tangent vectors to the hypersurface. It obeys $P^2=P$. Apply it to every tensor index to remove normal components, with the metric used to lower or raise the index pair for covariant slots. For a timelike unit normal in mostly-plus signature it is the spatial projection tensor. A null normal cannot be normalized this way, and a null hypersurface requires additional choices for a corresponding projection.

For a smooth plane at inclination $\theta$ rotating about a vertical axis at constant angular velocity $\omega$, choose rotating coordinates $\xi$ horizontally and $\eta$ uphill, with their cross product the upward normal vector. A sliding particle under gravity satisfiesThe normal reaction is $N=m(g\cos \theta -2\omega \dot{\xi }\sin \theta +{\omega }^2\eta \sin \theta \cos \theta )$ while contact is maintained. The Coriolis force does no work in the rotating frame, giving the conserved specific energyThe last term is the centrifugal potential; this quantity need not equal the inertial mechanical energy.