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Take the upward-oriented graph $z=c\sqrt{1-x^2/a^2-y^2/b^2}$ over $0\le x\le L_x$, $0\le y\le L_y$, where $a,b,c,L_x,L_y>0$, $A=L_x^2/a^2$, $B=L_y^2/b^2$ and $A+B<1$. The surface flux of the vector field $F=(-y,x,0)$ is the displayed formula. Indeed the vector surface element is $(-z_x,-z_y,1)dxdy$, soThe change of variables $u=x^2/a^2$, $v=y^2/b^2$ changes $xydxdy$ to $a^2{b}^{2}dudv/4$. Integrating $(1-u-v{)}^{-1/2}$ over $[0,A]\times [0,B]$ gives $\frac{4}{3}[1-(1-A{)}^{3/2}-(1-B{)}^{3/2}+(1-A-B{)}^{3/2}]$, proving the result. When $a=b$, the value of the vector field is a tangent vector to the ellipsoid at every point, and the surface flux vanishes pointwise.

Half the length of one principal diameter of an ellipsoid. For a centred quadratic form $x^{T}Sx=C$ with $S$ symmetric positive definite and $C>0$, its semi-axis lengths are $\sqrt{C/{\lambda }_i}$, where ${\lambda }_i$ are the eigenvalues of $S$. The shortest centre-to-surface distance corresponds to the largest eigenvalue.

After an orthogonal diagonalization, the sign of the third coefficient classifies the displayed quadric surface. If $C>0$, it is an ellipsoid with semiaxes $A^{-1/2},B^{-1/2},C^{-1/2}$. If $C=0$, it is an elliptic cylinder parallel to the $w$ axis, with elliptical transverse section. If $C<0$, it is a one-sheet hyperboloid with axis $w$ and waist ellipse $A{u}^2+B{v}^2=1$. The continuous limit $C\to 0$ sends the associated axial length scale to infinity. Repeated positive eigenvalues make the corresponding principal-axis choices nonunique within their eigenspace, without changing the surface.

The interior bounded by an ellipsoid is a solid ellipsoid. Positive coordinate dilations multiply its symplectic capacity by the square of the dilation factor.

A spheroid is an ellipsoid of revolution obtained by rotating an ellipse about a principal axis. Rotation about the shorter axis gives an oblate spheroid; rotation about the longer axis gives a prolate spheroid. Thus the oblate form is a member of the general family.

Let $S$ be a real symmetric positive definite matrix, $C>0$ and $q\ne 0$. The Cauchy-Schwarz inequality in the $S$-weighted inner product gives $q^{T}x=(S^{-1}q{)}^{T}Sx\le \sqrt{q^T{S}^{-1}q}\sqrt{x^{T}Sx}$. Equality is attained by a positive multiple of $S^{-1}q$. Thus the displayed maximum holds. For a translated ellipsoid, add $q^{T}a$ for its centre $a$. This computes a coordinate extremum without constructing a rotation matrix.

For a symmetric positive-definite matrix $A$, solving $A{x}_{\ast }=b$ is equivalent to minimizing the quadratic form with linear termWriting the error vector $e=x-x_{\ast }$ and expanding gives the exact energy norm identityThe cross term vanishes because $A{x}_{\ast }=b$. With $\eta =Se$ and $B=S^{-1}A{S}^{-1}$, the same quantity is ${\eta }^{T}B\eta /2$. Symmetric positive-definite preconditioning changes coordinates without changing the energy assigned to a corresponding error vector.

In an orthonormal eigenbasis of $A$, the level set $e^{T}Ae=t>0$ has semiaxis lengths $\sqrt{t/{\lambda }_i}$. Its longest-to-shortest axis ratio is $\sqrt{{\kappa }_2(A)}$. A large spectral condition number of a positive-definite matrix therefore appears geometrically as a thin ellipsoid. Good scaling makes the transformed ellipsoid rounder, allowing a simple search direction to improve all coordinates together.

The negative gradient of $F$ is the linear-system residual $r$. Relative to the norm $\Vert v{\Vert }_M=\sqrt{v^{T}Mv}$, a steepest direction is $z=M^{-1}r$: the Cauchy-Schwarz inequality applied to $r^{T}v=(S^{-1}r{)}^T(Sv)$ shows that the greatest decrease per unit $M$-length occurs along this direction. Equivalently, ordinary gradient descent in $y=Sx$ becomes preconditioned steepest descent in $x$.^[ref]

If $D=\mathrm{diag}(a_1,\dots ,a_{n+1})$ is invertible, $x\mapsto Dx$ sends the ellipsoid ${\sum }_i{a}_i^2{x}_i^2=1$ to the unit sphere. On $u=(v,s)\in S^n$, the two stereographic projections $v/(1-s)$ and $v/(1+s)$ omit opposite poles and have inverses $(2w/(1+|w{|}^2),\pm (|w{|}^2-1)/(1+|w{|}^2))$, with the sign chosen for the appropriate pole. Their overlap transition is $w\mapsto w/|w{|}^2$, smooth and involutive on ${\mathbb{R}}^n\setminus \{0\}$. Composing their inverses with $D^{-1}$ gives a two-chart smooth atlas of the ellipsoid. Its subspace topology is Hausdorff and has a second-countable space topology, so it is a smooth manifold. For $n=0$ the overlap is empty and each chart is a point.

Let $A$ be a real positive-definite matrix and minimize the displayed quadratic function over all of ${\mathbb{R}}^n$. Its gradient is $Ax-b$, so its only stationary point solves the linear system $A{x}_{\ast }=b$. The matrix is an invertible matrix, making this solution exist and be unique. Completing the square around that solution proves the stronger global statement:The quadratic function is strictly convex, and its level sets above the minimum are translated ellipsoids. Positivity therefore turns solving a linear system into locating the bottom of a bowl.^[ref]

For the opening matrix $A=\left(\begin{array}{cc}3& 1\\ 1& 3\end{array}\right)$, take $b=(1,1{)}^T$ and $c=0$. Then $A^{-1}=\frac{1}{8}\left(\begin{array}{cc}3& -1\\ -1& 3\end{array}\right)$, $x_{\ast }=(1/4,1/4{)}^T$, and $f(x_{\ast })=-1/4$. Every displacement $h$ raises the value by $\frac{1}{2}{h}^{T}Ah$; along the unit eigenvectors $e_{+}$ and $e_{-}$ the increases are respectively $2s^2$ and $s^2$ for displacement $s{e}_{+}$ or $s{e}_{-}$.

For a singular positive semidefinite matrix, a solution of $Ax=b$ is still a global minimizer, but adding any kernel vector gives another. If $b$ is outside the image of a linear map defined by $A$, some kernel vector $v$ has $b^{T}v\ne 0$, and $f(sv)=c-s{b}^{T}v$ is unbounded below. For an indefinite quadratic form, a negative eigenvalue gives a direction along which the quadratic term tends to $-\infty$ regardless of $b$. Thus a stationary point alone does not establish a minimum. The same energy-difference identity extends to the quadratic variational principle for a symmetric positive operator, with additional existence hypotheses in infinite dimension.

The velocity ellipsoid represents the positive velocity covariance tensor by its orthogonal principal axes and eigenvalue square roots. The uncentered second-moment tensor differs from covariance when a population streams. Vanishing mixed second moments align the second-moment ellipsoid with the chosen frame; for pure azimuthal streaming in a phase-mixed axisymmetric model, the off-diagonal covariance also vanishes.