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For $\overline{\mathbf{r}}=a^2\mathbf{r}/r^2$, the chain rule gives ${\nabla }_{\mathbf{r}}=(a^2/r^2)(I-2\hat{\mathbf{r}}{\hat{\mathbf{r}}}^T){\nabla }_{\overline{\mathbf{r}}}$. The radial term is killed by crossing with $\mathbf{r}$, giving the identity. Purely angular vector spherical harmonics are therefore unchanged as angular basis functions. This is distinct from pushing forward a physical vector field by the inversion Jacobian.

For squared loss $E={\sum }_k(t_k-z_k{)}^2/2$, let $e_k=(t_k-z_k)g_k^{\prime }(x_k)$ and $e_j=g_j^{\prime }(x_j){\sum }_k{w}_{jk}{e}_k$. The chain rule gives $\partial E/\partial w_{jk}=-e_k{z}_j$ and $\partial E/\partial w_{ij}=-e_j{z}_i$. A gradient descent step of size $\eta$ uses $\Delta w_{jk}=\eta e_k{z}_j$ and $\Delta w_{ij}=\eta e_j{z}_i$, with every derivative evaluated at the same pre-update weights.

On an open set where $u>0$, put $v=\sqrt{u}$. The chain rule gives $\nabla u=2v\nabla v$, so the quadratic eikonal equation is equivalent to $|\nabla v|=1$. For constant positive data $v=v_0$ on a smooth initial curve, the tangential derivative vanishes and the two allowed normal derivatives are ${\partial }_{n}v=\pm 1$. Locally the two solutions are $v=v_0\pm d$, where $d$ is oriented distance along the normal rays; hence $u=(v_0\pm d{)}^2$ wherever the chosen $v$ stays positive and the normal-coordinate map remains invertible.

There are two separate obstructions to using this initial-data chart. A characteristic caustic occurs when normal rays focus and the projection Jacobian determinant vanishes. Alternatively $v$ can reach zero while the geometric rays remain separated: the square-root transformation then ceases to apply and the original Charpit's equations have a degenerate zero-gradient state. A continuation of $u$ beyond either obstruction needs separate justification; squaring an algebraic expression alone does not preserve a prescribed positive square-root branch.

Where $q(x)\ne 0$, the product rule and chain rule give the displayed real derivative. Its stationary points therefore satisfy $3q+(x-a)q^{\prime }=0$. The denominator uses the square of the real cube root, which stays positive for negative $q$. Points with $q=0$ must be checked in the original function: a simple zero of $q$ with $x\ne a$ creates a vertical tangent, not a stationary point. If the prefactor also vanishes, a cancellation can change differentiability and must be analyzed separately.

A coordinate chart $x$ trivializes the tangent bundle by expressing $v\in T_{p}M$ in the coordinate basis. For $y=f(x)$, the chain rule gives the displayed transition. Smoothness and fibrewise linear invertibility of the Jacobian matrix construct the natural smooth vector bundle structure. The chain rule also supplies the transition cocycle.

At a fixed point $p$ of an $n$-dimensional smooth manifold $M$ without boundary, the tangent space $T_{p}M$ is the real vector space of all tangent vectors at $p$. In manifold charts $x=(x^1,\dots ,x^n)$ and $y=(y^1,\dots ,y^n)$, a tangent vector has componentsThe coordinate basis transforms inversely, so the tangent vector is unchanged. This invertible Jacobian matrix also preserves vector addition and scalar multiplication, giving a vector space of dimension $n$ independently of coordinates. ^[ref]

For example, at $(x,y)=(1,0)$ in ${\mathbb{R}}^2$, use the alternative manifold chart $u=x$, $w=y+x^2$. The Jacobian matrix sends $(v_x,v_y)=(2,3)$ to $(v_u,v_w)=(2,7)$. Indeed, the smooth curve $(x(t),y(t))=(1+2t,3t)$ gives $w(t)=1+7t+4t^2$. Conversely, ${\partial }_u{|}_p={\partial }_x{|}_p-2{\partial }_y{|}_p$ and ${\partial }_w{|}_p={\partial }_y{|}_p$, so $2{\partial }_u{|}_p+7{\partial }_w{|}_p=2{\partial }_x{|}_p+3{\partial }_y{|}_p$. The numbers change; their action on every smooth function does not.

For an embedded submanifold $M\subseteq {\mathbb{R}}^N$, the tangent space can be identified with the vector subspaceA proof outline explains why every coordinate vector is possible: in a manifold chart $x$, lift the short straight path $x(p)+ta$ using $x^{-1}$. Its tangent vector has components $a$. For a regular local parametrization $X:U\subseteq {\mathbb{R}}^n\to M$, the same chain rule gives $T_{p}M=\mathrm{im}D{X}_q$, where $X(q)=p$. This is the tangent space from a local parametrization.

For the sphere $S_R^2=\{p\in {\mathbb{R}}^3:p\cdot p=R^2\}$, differentiating that equation gives $p\cdot v=0$. Conversely, for any nonzero $v$ with $p\cdot v=0$, the smooth curvelies on the sphere and has ${\gamma }_v^{\prime }(0)=v$; use a constant smooth curve for $v=0$. Thus $T_p{S}_R^2=\{v:p\cdot v=0\}$, not merely a subset of that plane. At $p=(0,0,R)$ this vector subspace is $v_z=0$, while the affine tangent plane drawn through $p$ is $z=R$.

This also explains a mechanical constraint without requiring a force calculation. In the ideal mathematical model of a particle confined to a stationary sphere, its allowed velocities at $p$ are exactly $T_p{S}_R^2$: the radial component must vanish, but either independent tangential direction is available. For a general stationary holonomic constraint $F(q)=0$ with independent equations, the regular level set theorem gives $T_{q}M=\ker D{F}_q$. These are instantaneous kinematic possibilities; they do not determine the actual subsequent motion. On a general smooth manifold, tangent spaces at different base points cannot be identified canonically; an affine connection supplies parallel transport when such comparisons are needed.

A tangent vector at a point $p$ of a smooth manifold without boundary records an instantaneous velocity through $p$. Its base point is part of the data; measuring its length additionally requires a Riemannian metric. Intrinsically, a smooth curve $\gamma$ with $\gamma (0)=p$ defines its tangent vector by how every local smooth function changes along it:This is a derivation at a point. Two smooth curves through $p$ represent the same tangent vector if these derivatives agree for every such smooth function; equivalently, their coordinate velocity vectors agree in one manifold chart. The chain rule makes that agreement independent of the chosen manifold chart. ^[ref]

For an embedded submanifold of Euclidean space, the intrinsic tangent vector is represented by the ordinary derivative ${\gamma }^{\prime }(0)$, with its base point $p$ retained. A drawn arrow starting at $p$ represents that tangent vector; its endpoint $p+{\gamma }^{\prime }(0)$ need not lie on the embedded submanifold. Nor is a tangent vector required to be a unit tangent vector: a stationary smooth curve gives the zero vector, which belongs to the tangent space but has no direction to normalize.

Here are explicit calculations. In ${\mathbb{R}}^2$, $\gamma (t)=(t,t^2)$ and $\eta (t)=(t,2t^2)$ have different routes but the same tangent vector $(1,0)$ at $(0,0)$. For $f(x,y)=3x+y$, both give $v(f)=3$. For a circle of radius $R$, $c(t)=(R\cos (\omega t),R\sin (\omega t))$ has tangent vector $(0,R\omega )$ at $(R,0)$: changing $\omega$ changes the speed or reverses travel without changing the circle.

A reparametrization $\beta (s)=\gamma (h(s))$ with $h(0)=0$ gives, by the chain rule, ${\beta }^{\prime }(0)=h^{\prime }(0){\gamma }^{\prime }(0)$. A regular change of curve parameter has $h^{\prime }(0)\ne 0$. The singular choice $h(s)=s^3$ instead givesBoth describe the same parabola, whose tangent line at the origin is still the horizontal axis. The zero velocity diagnoses this singular change of curve parameter, whose inverse is not differentiable at zero; it does not show that the geometric tangent line is absent. The regular curve $\gamma$ supplies its nonzero direction.

The diagram is an analytical illustration of these exact parametric curves, with equal parameter increments marked on each route. It separates geometry from traversal: the tangent line depends on the route, while a velocity vector also depends on how the curve parameter advances.

For a Planck function expressed as spectral radiance per frequency interval,$h$ is the Planck constant, $c$ the speed of light and $k_B$ the Boltzmann constant. Differentiating at fixed frequency givesThe proof uses ${\partial }_{T}x=-x/T$ and the chain rule. At low frequency, the multiplier tends to one, recovering the Rayleigh-Jeans law. At high frequency it grows approximately as $x$: the same fractional temperature change has a larger fractional brightness effect on the declining spectral tail.

For an original example choose $T=2.725\mathrm{K}$ and $\delta T=1\mathrm{mK}$. At $100\mathrm{GHz}$, $x=1.76119$, so the first-order fractional residual is $7.80415\times 10^{-4}$. At $300\mathrm{GHz}$, $x=5.28357$, it is $1.94881\times 10^{-3}$. Both spectra are exact blackbodies: this residual is a temperature mismatch, not evidence of a nonthermal spectrum. Fitting temperature before assessing remaining spectral residuals therefore removes a physically distinct degree of freedom. The figure divides by a fixed peak brightness rather than by the rapidly vanishing high-frequency tail.

Suppose $a(t)>0$ for positive times. The strictly increasing new time $T(t)={\int }_0^{t}a(s)ds$ converts the scalar conservation law $u_t+a(t){\partial }_{x}F(u)=0$ into $U_T+{\partial }_{x}F(U)=0$, with $u(x,t)=U(x,T(t))$. The chain rule proves the claim for classical solutions; a change of variables in the weak formulation proves it for weak solutions and preserves the time orientation of the Kruzhkov entropy inequality. An isolated zero of $a$ at the initial endpoint does not obstruct the positive-time transformation.

The Rankine-Hugoniot condition becomes $d{x}_s/dt=a(t)[F(u)]/[u]$, while characteristic curves obey $dx/dt=a(t)F^{\prime }(u)$. Thus a straight shock in the new time may accelerate in physical time. For $a(t)=t$, a constant-speed trajectory in $T=t^2/2$ is a parabola in $(x,t)$. This separates genuine changes in wave structure from a common time-dependent speed factor.

A topological $n$-manifold without boundary is a topological space $M$ satisfying three conditions: it is Hausdorff, it is second countable, and every point $p\in M$ lies in an open set $U\subseteq M$ with a homeomorphism $\phi :U\to V$ onto an open set $V\subseteq {\mathbb{R}}^n$. Here $n\ge 0$ is a fixed dimension of a manifold, and ${\mathbb{R}}^n$ has the usual topology of Euclidean space. The pair $(U,\phi )$ is a coordinate chart. Unless explicitly qualified, “topological manifold” here means without boundary.^[ref]

The Hausdorff condition requires disjoint open neighbourhoods for distinct points. Second countability requires a countable basis of a topology. Neither follows merely from the local homeomorphisms: non-Hausdorff manifolds fail the first condition, while an uncountable disjoint union of copies of the real line fails the second. Indeed, a basis of a topology must supply a different nonempty member inside each copy, so it cannot be countable.

The circle and the real line are both one-dimensional topological manifolds, with the same local model. Their global topologies differ: the circle is compact, while the real line is not. By contrast, the junction of a topological tripod already fails the local test. Thus local coordinate charts specify what kind of small pieces are allowed; global topological invariants distinguish how those pieces fit together.^[ref]

The usefulness of a topological manifold is that local questions about continuity can be transported into Euclidean space, while compactness, connectedness, and the fundamental group still record global information. For calculus, further structure is needed: a smooth atlas requires smooth transition maps, letting the chain rule reconcile derivatives computed in different coordinate charts. A topological atlas by itself only guarantees homeomorphisms between coordinates.^[ref]

For a uniform rod of length $L$ and twist per unit length $\tau$, the end-rotation angle is $\phi =L\tau$. The chain rule gives $K_{\tau }=L{K}_{\phi }$, where $K_{\phi }$ is the torsional stiffness. If a source defines a normalized torque $M=N/(2\pi )$, its derivative is $dM/d\tau =K_{\tau }/(2\pi )$.

Let $p={\varphi }_Y$, $r={\varphi }_{YY}$ and $w_2=C_1(\xi )r+C_2(\xi )p^2$. Two applications of the chain rule give ${\mathcal{D}}_Y^2{w}_2=C_1{\varphi }_{YYYY}+2(C_1^{\prime }+C_2)p{\varphi }_{YYY}+(C_1^{\prime }+2C_2)r^2+(C_1^{\prime \prime }+5C_2^{\prime })p^{2}r+C_2^{\prime \prime }{p}^4$. In a Fredholm solvability condition, the fourth term generally contributes to the coefficient of ${\varphi }_Y^2{\varphi }_{YY}$. It cannot be omitted merely because the fast-coordinate correction functions have been denoted by $C_j(x)$.

A transport equation carries a quantity along the flow of a velocity field $b$. For a smooth solution of $u_t+b\cdot \nabla u=0$, the value of $u$ is constant along each characteristic curve. With constant speed $c$, the chain rule gives $\frac{d}{dt}u(x_0+ct,t)=u_t+c{u}_x=0$, hence $u(x,t)=g(x-ct)$. This is exact translation: it preserves the shape rather than smoothing it, as the periodic heat and transport comparison shows.

For $0<a<1$, the Joukowski map $z=\zeta +a^2/\zeta$ maps the exterior unit circle to an ellipse of horizontal and vertical semiaxes $1+a^2$ and $1-a^2$. The Milne-Thomson circle theorem gives $W(\zeta )=U(e^{-i\alpha }\zeta +e^{i\alpha }/\zeta )$ for a uniform flow inclined by $\alpha$. The chain rule produces the displayed complex velocity in the physical plane. Since $z\sim \zeta$, the far-field flow retains its speed and direction. Its two stagnation points areobtained from $\zeta =\pm e^{i\alpha }$. The map derivative is nonzero on and outside the unit circle, so these zeros are preserved without introducing a boundary singularity.

An $\mathbb{R}$-linear degree-one graded derivation of the exterior algebra of smooth differential forms that equals the differential on functions and squares to zero is unique. A smooth bump function and the graded Leibniz rule first establish locality. Then $d(d{x}^i)=0$, and applying the graded Leibniz rule to $\alpha ={\sum }_I{a}_{I}d{x}^I$ gives the displayed coordinate formula. The chain rule makes that formula coordinate-independent.

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.

For a point mass of constant mass $m$ observed in an inertial frame, the net work along its actual path equals its change in kinetic energy:The middle expression is a line integral; the dot product selects the component of force along the displacement. Positive net work increases speed, negative net work decreases it, and zero net work leaves the endpoint speeds equal. Equal endpoint speeds do not imply equal velocities: a perpendicular force can continually change direction while doing zero work. ^[ref]

The bridge to Newton's second law is short: take the dot product of $m\dot{\mathbf{v}}={\mathbf{F}}_{\mathrm{net}}$ with velocity. The chain rule gives $d(m|\mathbf{v}{|}^2/2)/dt={\mathbf{F}}_{\mathrm{net}}\cdot \mathbf{v}$, equating the rate of change of kinetic energy to instantaneous power. Integrating gives the theorem. No conservative force assumption entered this argument; friction also contributes to the net work.

For the harmonic oscillator with spring stiffness $k=2N/m$, release a mass $m=0.5\mathrm{kg}$ from rest at $x=0.3\mathrm{m}$. The spring exerts $F=-kx$. Its work before the first passage through $x=0$ isThus $m{v}^2/2=0.09\mathrm{J}$ gives speed $0.6m/s$ there. The velocity is negative on this passage because the position is decreasing. The energy calculation alone determines its magnitude, so the direction still requires information about the motion.

For a time-independent potential energy $V$, the conservative force contributes work $-\Delta V$. Separating all other work as $W_{\mathrm{other}}$ gives $\Delta (T+V)=W_{\mathrm{other}}$. Consequently conservation of mechanical energy requires zero net other work; the work-energy theorem itself does not require that conservation. With damping, mechanical energy decreases even though the theorem remains valid. ^[ref]