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Let ${\partial }_{\pm }={\partial }_{\tau }\pm {\partial }_{\sigma }$ and $\mathcal{L}={\partial }_{+}X\cdot {\partial }_{-}X+i{\psi }_1\cdot {\partial }_{+}{\psi }_1+i{\psi }_2\cdot {\partial }_{-}{\psi }_2$. For constant Grassmann-odd parameters, set $\delta X=i({ϵ}_2{\psi }_1-{ϵ}_1{\psi }_2)$, $\delta {\psi }_1=-{ϵ}_2{\partial }_{-}X$, $\delta {\psi }_2={ϵ}_1{\partial }_{+}X$. In the first chiral sector the variation of the fermionic term cancels $i{ϵ}_2{\partial }_{+}{\psi }_1\cdot {\partial }_{-}X$ from the bosonic term, leaving $i{ϵ}_2{\partial }_{-}({\psi }_1\cdot {\partial }_{+}X)$. The second sector gives the other total derivative. This proves off-shell invariance when the boundary flux vanishes. Closure on fermions uses their free equations of motion. A constant parameter must also preserve the chosen boundary conditions and spin structure.

A covector is an element of the dual space, hence a linear functional on a vector space. In finite-dimensional coordinates, it acts as a row multiplied by a coordinate vector column. The row $\left(\begin{array}{cc}2& -1\end{array}\right)$ acts on $\left(\begin{array}{c}3\\ 1\end{array}\right)$ to give five. The terminology emphasizes its role as a scalar-valued measurement and its behavior under a change of basis.

For example, the total derivative of $f(x,y)=x^2+y^2$ at $(1,2)$ sends a small displacement $h$ to the linear change $df(h)=2h_x+4h_y$. This covector describes the first-order response. The gradient $\nabla f=(2,4)$ represents the same rule through the chosen Euclidean inner product. In differential geometry, the cotangent space collects these scalar-valued measurements of tangent vectors.

For ungauged chiral superfields, the two-derivative supersymmetric action is a real full-superspace integral plus a holomorphic half-superspace integral:The real Kähler potential $K$ determines the Kähler metric and kinetic terms; the superpotential $W$ determines Yukawa interactions and the F-term scalar potential. For canonical normalization, $K={\sum }_i{\Phi }_i^{†}{\Phi }_i$. Berezin integration is normalized to extract the highest component, $\int d^2\theta {\theta }^2=1$, and the conjugate measure is chosen so the canonical term is $F_i^{\ast }{F}_i$. The D-term and F-term highest components vary by spacetime total derivatives, making the action invariant under supersymmetry. Gauging would replace the canonical bilinear by its gauge-covariant version and add a gauge kinetic F-term; no gauge multiplet is needed for the chiral theory here.

For canonical $K$, extract the ${\theta }^2{\bar{\theta }}^2$ component of ${\Phi }_i^{†}{\Phi }_i$. The part depending on the auxiliary fields is $F_i^{\ast }{F}_i$. To extract the F-term of the superpotential, expand around the complex scalar fields:All higher terms contain at least three identical-type Grassmann variables and vanish. In the convention $(\theta {\psi }_i)(\theta {\psi }_j)=-\frac{1}{2}{\theta }^2{\psi }_i{\psi }_j$, the highest component is $W_i{F}_i-\frac{1}{2}{W}_{ij}{\psi }_i{\psi }_j$. ThereforeHere $W_i=\partial W/\partial {\varphi }_i$ and $W_{ij}={\partial }_i{\partial }_{j}W$. The Yukawa interaction is displayed to distinguish it from the terms actually containing $F$. With signature $(+---)$, the other canonical terms are ${\partial }_{\mu }{\varphi }_i^{\ast }{\partial }^{\mu }{\varphi }_i+i{\bar{\psi }}_i{\bar{\sigma }}^{\mu }{\partial }_{\mu }{\psi }_i$, up to a total derivative.

For variations vanishing at both endpoints, differentiating the action and using integration by parts givesThe fundamental lemma of the calculus of variations therefore yields the Euler-Lagrange equationsFor the specified Lagrangian, the gauge transformation changes it byFor arbitrary $F$, it is a total derivative precisely when $\boxed{\alpha =-1}$. The corresponding change in action is then an endpoint term whose variation is zero. In general time-dependent notation, the equations areThe electric field $-\nabla \varphi -{\partial }_t\mathbf{A}$ and magnetic field $\nabla \times \mathbf{A}$ are unchanged by the gauge transformation with $\alpha =-1$. Although the original potentials have no time dependence, their transformed versions generally do; allowing this is essential to the stated invariance.

The matter part of the massless closed-string vertex operator isIts symmetric trace-free polarization is the graviton, while its antisymmetric polarization is the Kalb–Ramond field, or B-field. The mass-shell condition $k^2=0$ and transversality make $V_{ϵ}$ a primary operator of conformal weights $(1,1)$, as required for an integrated string vertex operator.

The graviton polarization has the linearized gauge redundancywhile the antisymmetric polarization obeysIn either case the change in the integrated vertex is a worldsheet total derivative, hence vanishes on a closed worldsheet; in covariant language it is BRST-exact. This string-state gauge redundancy is the vertex-operator form of linearized target-space diffeomorphism or two-form gauge invariance.

The conserved number operator isSubstitution of $\Psi =\sqrt{\rho }{e}^{-i\theta }$ into the Landau-Ginzburg theory givesThe imaginary term is a total derivative, soThus $\rho$ is the number density and is the canonical momentum conjugate to $\theta$. The canonical commutation relation is $[\theta (\mathbf{x}),\rho (\mathbf{y})]=i{\delta }^{(d)}(\mathbf{x}-\mathbf{y})$. Consequently, for ${\theta }_0=\int d^{d}x\theta$ in volume $V$,Equivalently, the averaged phase $\bar{\theta }={\theta }_0/V$ obeys $\Delta \bar{\theta }\Delta N\ge 1/2$. This number-phase conjugacy means that a state of sharp $N$ has no sharp phase, whereas a phase-selected state exhibiting spontaneous symmetry breaking must superpose different number sectors. Such sectors become effectively degenerate in the thermodynamic limit.

Stability requires $\lambda >0$. The classical potential $r\rho +\lambda {\rho }^2$ has a symmetry-breaking minimum when $r<0$, atWriting $\rho =\bar{\rho }+\delta \rho$ and discarding constants and total derivatives gives the quadratic LagrangianThe density fluctuation is a gapped amplitude mode, while the phase is the prospective Goldstone boson.

In the CP1 spinor representation,where the ${\sigma }_i$ are Pauli matrices. The transformation $z\mapsto e^{i\alpha (x,t)}z$ leaves $\mathbf{n}$ unchanged. For $U\in SU(2)$,with $R$ in the special orthogonal group $SO(3)$, so $z\mapsto Uz$ induces $n_i\mapsto R_{ij}{n}_j$.

A rotationally invariant first-order term is the Ferromagnetic Wess–Zumino termUnder the phase redundancy it changes as ${\mathcal{L}}_{\mathrm{WZ}}\mapsto {\mathcal{L}}_{\mathrm{WZ}}-\kappa \dot{\alpha }$. The change is a total derivative, so the action has the required invariance.

For $z=(e^{-i\varphi }\cos (\theta /2),\sin (\theta /2){)}^T$,Thus $i\kappa z^{†}\dot{z}=(\kappa /2)(1+\cos \theta )\dot{\varphi }$. Dropping the total derivative $(\kappa /2)\dot{\varphi }$ and writing $s=\kappa /2$ givesThe leading rotationally invariant gradient energy is $({\rho }_s/2)(\nabla \mathbf{n}{)}^2$, so an effective Lagrangian is

The total derivative contributes only the boundary termThis is fixed by the endpoint data, so its first variation vanishes. It cannot change the Euler-Lagrange equations or which path minimizes $F$.

For a direct check, let $h=g_x\dot{x}+g_y\dot{y}$. ThenThe equality uses commuting mixed partial derivatives. Thus $\boxed{dg/dt\text{ contributes zero to both Euler-Lagrange equations}}$, as in total time derivatives preserve fixed-endpoint Euler-Lagrange equations.

The needed partial derivatives areSubstitution into the Euler-Lagrange equations givesThe ${\gamma }_1$ terms cancel in the first equation, leavingThese are accompanied by $x(0)=y(0)=0$ and $x(T)=y(T)=S$. The cancellation is consistent with part (b): the term proportional to ${\gamma }_1$ is the total derivative of $\alpha {\gamma }_1(x+\beta x^2/2)$.

Impose $y=x$ before varying. Writing $\delta ={\delta }_1+{\delta }_2$, the restricted integrand isIts middle term is the total derivative of $\alpha ({\gamma }_1+{\gamma }_2)(x+\beta x^2/2)$, so it can be dropped for fixed endpoints by part (b). The remaining Euler-Lagrange equation is $\delta \ddot{x}=\alpha \beta$. Integrating twice and imposing $x(0)=0$, $x(T)=S$ givesThis is the unique global minimizer among paths on the specified line. To prove it, add any admissible variation $\eta$ with $\eta (0)=\eta (T)=0$. The change in fuel, after dropping the fixed boundary term, isThe first integral vanishes by integration by parts and the equation for $x$. Equality forces $\eta =0$. No monotonicity or restriction to the segment between endpoints was imposed in the question; the minimizing coordinate can initially decrease for sufficiently large $T$.

If $L^{\prime }=L+dF(q,t)/dt$, the action changes by an endpoint term:In the principle of stationary action with fixed endpoints, $\delta q(t_i)=\delta q(t_f)=0$ and the endpoint times are fixed. Therefore $\delta (S^{\prime }-S)=0$, and the two actions give the same bulk Euler-Lagrange equations.

For a Lagrangian density, adding ${\partial }_{\mu }{K}^{\mu }$ changes the action by ${\int }_{\partial \Omega }{K}^{\mu }{n}_{\mu }d\Sigma$, by the divergence theorem. Its variation vanishes for compactly supported field variations, or when the appropriate boundary data are held fixed. These total divergences in a variational principle can therefore be discarded when deriving bulk equations under those conditions.

The qualification about boundaries matters: if $K^{\mu }$ depends on field derivatives, their variations on the boundary must also be controlled. A total derivative can change the canonical momenta or the chosen representative of a Noether current, even while leaving the bulk dynamics unchanged.

Choose the real representative $v={\psi }_0=\sqrt{\mu /g}$. For $\psi =v+\chi$,and the fourth power through quadratic order isIn $\mu |\psi {|}^2-g|\psi {|}^4/2$, the linear terms and the $|\chi {|}^2$ terms cancel because $g{v}^2=\mu$. The remaining potential contribution is ${\mu }^2/(2g)-\mu (\chi +{\chi }^{\ast }{)}^2/2$.

The time-derivative contribution isIts first term is a total derivative and contributes only a temporal boundary term for fixed endpoint fields. Since $v$ is uniform, the gradient contribution is simply $-{\hslash }^2|\nabla \chi {|}^2/(2M)$. Dropping that boundary term and the constant, the quadratic condensate fluctuation Lagrangian isThe omitted cubic and quartic interactions describe scattering between these small fluctuations; they do not enter the linear normal mode calculation.

The two continuous, time-independent symmetries are time translation and a common rotation of both angles. Their Noether charges are respectively the total energy and vertical angular momentum:For the second symmetry, $\theta \mapsto \theta +ϵ$ and $\phi \mapsto \phi +ϵ$, so the Noether theorem gives the sum of their conjugate momenta. Independent rotations are not symmetries of the spring coupling.

There are also discrete symmetries: simultaneous reflection $(\theta ,\phi )\mapsto (-\theta ,-\phi )$, and time reversal in classical mechanics, which reverses both angular velocities. These leave the dynamics invariant but provide no additional infinitesimal Noether charges. Combined with common rotations, the reflections give the full planar orthogonal symmetry of the angular geometry. If $I_1=I_2$, interchange of the two angular coordinates is an additional discrete symmetry of the reduced dynamics.

There is a further useful time-dependent symmetry, up to a total derivative, on a continuous lift of the angles: adding the same constant angular velocity, $(\theta ,\phi )\mapsto (\theta +vt,\phi +vt)$. It changes $L$ byIts conserved quantity is $\boxed{I_1\theta +I_2\phi -tJ}$. It is the integration constant for free common rotation, rather than a new globally single-valued angle charge on the configuration torus. The globally defined continuous charges are $E$ and $J$ above.

A total derivative accounts for every dependence of a quantity on the variable of differentiation. For a differentiable function $F(t,x_1,\dots ,x_n)$ evaluated along differentiable coordinate functions $x_i(t)$, the multivariable chain rule givesAll the partial derivatives on the right are evaluated at $(t,x_1(t),\dots ,x_n(t))$. The first term captures explicit dependence on $t$; the remaining terms capture changes transmitted through the coordinates.

For example, take $F(t,x)=t{x}^2$ and $x(t)=t$. The partial derivative with respect to $t$, holding $x$ fixed, is $x^2$, whose value on the path is $t^2$. The total derivative is insteadAt $t=2$, the partial derivative is $4$ and the total derivative is $12$. The difference is the contribution from changing $x$.

If the composite function is continuously differentiable on $[a,b]$, the fundamental theorem of calculus givesThis is the sense in which an integrated total derivative contributes endpoint terms. The endpoint statement concerns the whole composite function along the path. ^[ref].