For the Falkner-Skan equation $f^{(3)}+\beta f{f}^{\prime \prime }+m(1-(f^{\prime }{)}^2)=0$, an attached branch has $f^{\prime }>0$ away from the wall. It is then possible to replace the similarity coordinate by $f$ itself. Define $w(f)=f^{\prime }$. The chain rule gives $f^{\prime \prime }=w{w}_f$ and $f^{(3)}=w(w_f^2+w{w}_{ff})$, proving the displayed second-order equation. The wall and outer conditions become $w(0)=0$ and $w(\infty )=1$ when the stream function is set to zero at the wall.

The reduction is singular at the wall: if $s=f^{\prime \prime }(0)>0$, then $f\sim s{\eta }^2/2$ and $w\sim \sqrt{2sf}$. Hence $w_f$ diverges while $w{w}_f\to s$ remains finite. A finite-slope condition on $w$ at zero would incorrectly exclude the ordinary positive-wall-shear branch. At flow reversal or separation, the invertibility of the coordinate change must be checked again.

A historical name for a crystalline material isolated from human faeces. Early descriptions and formulas are not sufficient to establish that every preparation was one pure modern compound.

A tarsal plate is a dense connective-tissue support within an eyelid. It contains specialized glands and helps maintain lid shape. It is not a plate of bone, despite the stiffness implied by its name.

The medial rectus muscle adducts the eye and is supplied by the oculomotor nerve. Coordinated activity with the opposite eye supports convergence. A local muscle action and the binocular control problem are therefore different explanatory levels.

The lateral rectus muscle abducts the eye. It is supplied by the abducens nerve, unlike most other muscles rotating the eyeball. Its relatively direct horizontal action provides a useful anatomical example connecting one muscle, one nerve and one principal movement.

The superior rectus muscle primarily elevates the eye and also contributes to intorsion and adduction. It is supplied by the oculomotor nerve. Its line of pull explains why an extraocular muscle need not produce only the direction suggested by its name.

The inferior rectus muscle primarily depresses the eye and also contributes to extorsion and adduction. It is supplied by the oculomotor nerve. Its vertical action is tested differently from depression supplied by the superior oblique muscle because the pulling geometry differs.

The superior oblique muscle passes through the trochlea of superior oblique before attaching to the eyeball. It contributes to intorsion, depression and abduction, depending on gaze position. It is supplied by the trochlear nerve.

The inferior oblique muscle is an extraocular muscle contributing to extorsion, elevation and abduction of the eye. Its effect depends on gaze position; elevation is especially informative when the eye is adducted. Its course differs from the rectus muscles arising posteriorly in the orbit.

The levator palpebrae superioris elevates the upper eyelid. It is controlled chiefly through the oculomotor nerve, with additional smooth-muscle contributions to lid position. Raising the lid and rotating the eyeball are different movements even though both affect visible gaze.

To compute two nonnegative integer powers in one while loop, maintain a separate accumulator, current base and remaining exponent for each exponentiation by squaring. The paired loop invariant is $p{x}^k=A^n$ and $q{y}^l=B^m$. For each positive remaining exponent, transfer its odd factor to its accumulator and halve the exponent while squaring its base. A zero remaining exponent is left unchanged. The mathematical integer $k+l$ is a loop variant: at least one positive term strictly decreases in every iteration. Both powers are therefore complete when the loop stops. The iteration count is $O(\log (\max (n,m)+1))$, with constant space and a constant number of multiplications per iteration. This is an exact-arithmetic identity; floating-point rounding introduces a separate error in a numerical implementation.

Michael Faraday's 1845 experiment combined solid carbon dioxide, diethyl ether and reduced pressure. Removal of vapour cooled the bath further. He reported a low alcohol-thermometer indication near $-166{}^{\circ }\mathrm{F}$, while explicitly discussing lag and uncertainty in the graduation at low temperature.^[ref]

The totient divisor-sum identity and multiplication of absolutely convergent Dirichlet series give the quotient for $\mathrm{Re}s>2$. The bound $\varphi (n)\le n$ proves convergence there. The simple pole of the Riemann zeta function at one makes the quotient unbounded as real $s\downarrow 2$. Since the coefficients are nonnegative, convergence at two would bound all these larger-real-part values, a contradiction. Thus the abscissa of absolute convergence of a Dirichlet series is two.

The prime-power formula and multiplicativity of Euler's totient function imply $\varphi (mn)\varphi (\gcd (m,n))=\varphi (m)\varphi (n)\gcd (m,n)$. For a prime with positive exponents $a,b$ in the inputs, both local products equal $p^{a+b+\min (a,b)-2}(p-1{)}^2$. If one exponent is zero, the equality is immediate from $\varphi (1)=1$.

For positive twice differentiable functions $y$, the Euler-Lagrange equation for this functional reduces to $y{y}^{\prime \prime }+1+y^{\prime 2}=0$. Thus $(y{y}^{\prime }{)}^{\prime }=-1$ and integration gives $(x-h{)}^2+y^2=R^2$. The upper circular arc is a graph wherever its radicand is positive. A fixed horizontal endpoint with free height imposes $F_{y^{\prime }}=y^{\prime }/(y\sqrt{1+y^{\prime 2}})=0$, so that endpoint must lie above the centre. Distinct horizontal endpoints cannot both satisfy this natural boundary conditions for a free endpoint. The Beltrami identity alone also admits constant functions, which are excluded by the full Euler-Lagrange equation.

If a smooth functional has integrand $f(x,v)$ independent of the function value, a weak local minimizer satisfies the displayed first integral almost everywhere. Its first variation gives $\int f_v(x,u^{\prime }){\phi }^{\prime }=0$ for every compactly supported test function. Thus the distributional derivative of $f_v(x,u^{\prime })$ vanishes, making it constant on the interval. Equivalently, any smooth compactly supported function with integral zero is the derivative of such a test function, so the first-variation identity says that $f_v(x,u^{\prime })$ agrees with a single constant against all tests. This reduces the Euler-Lagrange equation from a second-order equation to an algebraic relation in $u^{\prime }$.

For an autonomous scalar integral $\int f(u^{\prime })$, the first variation gives the integrated Euler-Lagrange equation $f^{\prime }(u^{\prime })=c$. If $u$ is $C^1$ and each fiber of $f^{\prime }$ is finite, the connected image of the interval under the continuous derivative $u^{\prime }$ lies in a finite set and is therefore a singleton. The fixed endpoints then determine exactly one affine solution. If $f$ is strictly convex, the strict Jensen's inequality applied to $u^{\prime }$ additionally makes this solution the unique global minimizer among fixed-endpoint Lipschitz functions. Without convexity the same unique smooth stationary solution can be a maximizer, for example at zero for $f(v)=1/(1+v^{2008})$ and zero endpoint values.

Intersection of $o+\lambda r$ with the plane $n\cdot (p-a)=0$, when $n\cdot r\ne 0$. For a ray, require $\lambda \ge 0$. A zero denominator requires separate parallel or coplanar handling.

For constant density and a conservative force per mass $-\nabla \Phi$, gravity and centrifugal acceleration can be incorporated into the displayed reduced pressure. The resulting Euler equations in rotating axes are ${\mathbf{u}}_t+(\mathbf{u}\cdot \nabla )\mathbf{u}+2\mathbf{\Omega }\times \mathbf{u}=-\nabla P/\rho$. The sign of the centrifugal contribution follows from $\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{x})=-\nabla |\mathbf{\Omega }\times \mathbf{x}{|}^2/2$. Constant density is what permits combination into one pressure gradient without extra density-gradient terms.

Hippocampal sclerosis is a pathological pattern of neuronal loss and gliosis in the hippocampus. It is associated with some forms of temporal-lobe epilepsy, although the anatomical finding and the full clinical syndrome are distinct.

Identifying tissue change can constrain mechanisms of network dysfunction, but does not by itself reconstruct the timing or every cause of the person's seizures.