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A Cartesian basis is an ordered basis of unit vectors along the mutually perpendicular axes of a Cartesian coordinate system. It is an orthonormal basis of a real Euclidean space: $e_i\cdot e_j={\delta }_{ij}$, with ${\delta }_{ij}$ the Kronecker delta. In the plane, the familiar standard basis is $e_1=(1,0{)}^T$, $e_2=(0,1{)}^T$; the vector $p=(2,1{)}^T$ is $2e_1+e_2$. A rotated pair of perpendicular unit vectors is also a Cartesian basis, even though its basis vectors have different standard basis entries.

The advantage is that coordinates can be measured separately by dot products, and the squared norm is the sum of their squares:These are the real Euclidean space instances of the orthonormal basis expansion.^[ref]

An orthogonal basis only requires perpendicular basis vectors; their lengths need not be one. For example, $(2,0{)}^T,(0,3{)}^T$ are orthogonal, but $p=(2,1{)}^T$ has coordinates $(1,1/3{)}^T$, while its dot products with them are $4,3$. For an orthogonal basis $(a_i)$, the coordinate along $a_i$ is $(p\cdot a_i)/(a_i\cdot a_i)$. Dividing each nonzero basis vector by its norm makes it a unit vector without changing its direction. Unit length alone does not imply orthogonality: $(1,0{)}^T$ and $(1,1{)}^T/\sqrt{2}$ are both unit vectors but have dot product $1/\sqrt{2}$.

The examples below show why these distinctions matter: rotating a Cartesian basis changes the coordinate vector while keeping lengths easy to compute; using nonperpendicular basis vectors still gives unique coordinates, but requires the dual basis or a matrix inverse to extract them.

For real harmonic fields with complex amplitudes $\hat{p}$ and $\hat{u}$, their period-averaged product is $\mathrm{Re}(\hat{p}{\hat{u}}^{\ast })/2$.

A surface wave with phase speed below the bulk sound speed decays normally as $e^{-\alpha z}$, where $\alpha =k\sqrt{1-c^2/c_0^2}$.

Linear acoustic fields obeywhere

For irrotational linear acoustic flow, $\mathbf{u}=\nabla \varphi$, $p^{\prime }=-{\rho }_0{\varphi }_t$, and $\varphi$ obeys the wave equation.

For a homentropic reference state, first-order pressure and density perturbations satisfy $p^{\prime }=c_0^2{\rho }^{\prime }$, where $c_0$ is the sound speed.

For $L=mc|\dot{x}|-V(x)$,The momentum loses the speed, so the velocity cannot be recovered from $(x,p)$ and the ordinary inverse Legendre transform does not exist.

For $u_t=i{u}_{xx}$ on $x>0$, the half-line Fourier transform satisfiesSubtracting the relation at $-k$ cancels the unknown Neumann datum. Fourier inversion of the resulting odd combination gives the solution in terms of the initial transform and the prescribed Dirichlet datum, with boundary kernel $G(k,t)=kh(t)$.

For ${\Phi }_x+U\Phi =0$ and ${\Phi }_y+V\Phi =0$, compatibility is $V_x-U_y+[U,V]=0$.

For the auxiliary equations ${\psi }_x=U\psi$ and ${\psi }_t=V\psi$, equality of mixed derivatives gives $U_t-V_x+[U,V]=0$. This is the compatibility condition for a Lax pair.

For $\dot{L}=[L,A]$, the operator ${\partial }_t+A$ maps each eigenspace of $L$ into itself, so the spectrum is time-independent.

For complex fields $q,r$, the standard two-by-two AKNS pair has zero-curvature equationsThe reductions $q=\overline{r}$ and $q=-\overline{r}$ give the focusing and defocusing cubic nonlinear Schrödinger equations, respectively.

The Airy equation is the linear dispersive partial differential equation $q_t+q_{xxx}=0$. Under the spatial Fourier transform, each mode evolves by multiplication by $e^{i{k}^{3}t}$.

For positive time its fundamental solution is $(3t{)}^{-1/3}\mathrm{Ai}(x/(3t{)}^{1/3})$, connecting this evolution equation to the Airy function without identifying the two concepts.

An analytic function on an annulus $r<|z-a|<R$ has a unique locally uniformly convergent expansionwhere $C$ is any positively oriented circle around $a$ in the annulus.

A Dyck path starts at $(0,0)$, ends at $(2n,0)$, uses steps $U=(1,1)$ and $D=(1,-1)$, and never passes below the horizontal axis. Its semilength of a Dyck path is $n$: there are $n$ up steps and $n$ down steps. The nonnegative height records a constraint throughout the journey, not just the endpoint. The Catalan number $C_n$ counts the Dyck paths of this size.

Soft thresholding maps $u$ to $\mathrm{sgn}(u)(|u|-\lambda {)}_{+}$ and is the proximal operator of the absolute-value penalty.

The square-root Lasso uses the unsquared residual norm plus an ${\ell }^1$ penalty.

The Dantzig selector minimizes the coefficient ${\ell }^1$ norm subject to an ${\ell }^{\infty }$ bound on residual correlations.

A dark-matter halo is a gravitationally bound concentration of dark matter that hosts galaxies and larger structures.

Standard perturbation theory expands the density contrast and velocity divergence of a pressureless irrotational fluid in powers of the linear density field.

An Einstein-de Sitter universe is spatially flat and contains only pressureless matter. In conformal time, $a\propto {\tau }^2$ and $\mathcal{H}=a^{\prime }/a=2/\tau$.