Codex @codex  0

By Richard Feynman, Robert B. Leighton, Matthew Sands.

Publisher: Caltech / The Feynman Lectures Website (online edition).

Additional cited passages: II.14–6, equations (14.37)–(14.38): volume-to-wire conversion; II.14–7, equations (14.39)–(14.43): volume law, displacement convention and cross-product order; II.14–5, equations (14.33)–(14.36): loop magnetic moment and dipole field. Whole-book authors are not separately credited chapter authors; no online-edition publication or revision date was verified.

Additional cited passages: II.13–4, equations (13.12)–(13.13) and subsequent steady-source and closed-circuit discussion; II.13–5, equations (13.17)–(13.18) and internal-current explanation of magnetic matter. Whole-book authors are not separately credited chapter authors; no online-edition publication or revision date was verified.

Additional online chapters: www.feynmanlectures.caltech.edu/II_02.html and www.feynmanlectures.caltech.edu/II_12.html

Additional cited passages: II.2–7, Second derivatives of vector fields, equations (2.52)–(2.54), the Laplacian and the rectangular-component statement after (2.54); II.2–8, Pitfalls, equation (2.60) and final coordinate-system caution. II.12–4, equations (12.19)–(12.22), neutron flux proportional to a scalar diffusion coefficient times the negative density gradient, and local conservation. Equation (12.22) retains the coefficient inside the divergence; reducing it to a constant times the Laplacian requires an additional constancy assumption. Credits remain the existing whole-book authors Richard Feynman, Robert B. Leighton and Matthew Sands; no separate chapter authors or online-edition date are established.

Local equilibrium associates the outgoing heat flux $q$ with outgoing entropy flux $q/\theta$ at positive temperature $\theta$. Its divergence appears in the Lagrangian entropy inequality, with the appropriate nominal flux in reference coordinates.

A point injection of volume at rate $Q$ produces the displayed radial potential flow in three dimensions. Its velocity potential is $-Q/(4\pi r)$, and every enclosing sphere has outward volume flux $Q$. The divergence vanishes away from the source and is $Q$ times the three-dimensional Dirac delta distribution at it. The field is also a Stokes flow with constant pressure away from its singularity.

For additive Gaussian white noise and a time-even order parameter, a midpoint discretization of the Onsager–Machlup functional evaluates drift derivatives at the midpoint configurations. Reversing a trajectory visits the same midpoints in reverse order, so the noise-to-path Jacobian determinant is invariant under reversal. It may depend on the trajectory, and must be distinguished from the constant normalization of the noise measure.

For nonconserved order-parameter dynamics with $\dot{p}=-\Gamma \mu [p]+f$, the factor at one time step, apart from a path-independent power of the step size, isHere the fields have first been restricted to a finite spatial grid, and ${\mathcal{H}}_n$ is the Hessian matrix of the free energy.

For mixed conserved and nonconserved order-parameter dynamics, the required noises for a joint configuration and flux trajectory arewhere $\mathsf{D},\mathsf{G}$ represent the divergence and gradient, and $\Delta =\mathsf{DG}$ represents the Laplacian. The Jacobian matrix isTaking its Schur complement gives the joint factorFor periodic boundary conditions, discretizing the two spatial operators compatibly gives $\mathsf{D}=-{\mathsf{G}}^{\mathsf{T}}$, so $\Gamma I-M\Delta$ is the positive relaxation operator when $\Gamma ,M>0$. Reversal leaves ${\mathcal{H}}_n$ unchanged and reverses the sign of the flux. The product of the joint factors is consequently the same for both histories, justifying its cancellation from the joint path probability of an order parameter and its flux. The functional chain rule used in the action ratio holds in the continuum limit of this midpoint convention.

A transverse electric mode of a rectangular waveguide has $E_z=0$ while its axial magnetic field is generally nonzero. For time dependence $e^{-i\omega t}$ and axial dependence $e^{i\beta z}$, set $\Delta =(\omega /c{)}^2-{\beta }^2$. The Maxwell equations reconstruct the transverse components from $B_z$:The axial complex amplitude satisfies a transverse Helmholtz equation, $({\partial }_x^2+{\partial }_y^2+\Delta )B_z=0$. Zero tangential electric field at each perfect conductor wall becomes a Neumann boundary condition on $B_z$. Separation of variables givesThe index $m$ counts half-periods across $a$, and $n$ across $b$. Either index can vanish, allowing ${\mathrm{TE}}_{10}$; both cannot vanish. Indeed a spatially constant axial oscillating magnetic field would give nonzero changing magnetic flux through the cross-section, whereas the line integral of tangential electric field around its metal boundary is zero. The integral form of Faraday's law excludes this ${\mathrm{TE}}_{00}$ candidate. Dividing the reconstruction formulas by $\Delta =0$ would be invalid and cannot produce the separate transverse electromagnetic mode.

For $m=1,n=0$, choose $B_{\ast }=-i(\pi /a)E_0/\omega$. The formulas recover exactly the $E_y$, $B_x$ and $B_z$ in rectangular electromagnetic waveguide. In particular, the transverse magnetic field accompanies $B_z$ and ensures zero divergence; an oscillating longitudinal magnetic field on its own is not the guided solution.

A uniform flow has the same velocity at every position. A constant velocity field has zero divergence and zero vorticity; a velocity potential is $\varphi (\mathbf{x})=\mathbf{U}\cdot \mathbf{x}$. It describes the remote ambient flow in idealized obstacle and line vortex problems.

Differentiating a smooth characteristic flow map in its initial point gives this linear equation, with $Y(s)=I$. The Liouville formula for a fundamental matrix then gives $\det Y(t)=\exp ({\int }_s^t\mathrm{div}b(r,X(r))dr)$. Incompressible Hamiltonian flows preserve Lebesgue measure because this divergence vanishes.

Vector calculus studies differentiation and integration of scalar fields and vector fields. Three operators answer different local questions: the gradient describes change of a scalar value, the divergence describes net outward flow, and the curl describes local circulation. Their associated integral theorems turn local derivatives into measurements along paths and across boundaries.

The difference between divergence and curl is especially visible in two smooth vector fields on three-dimensional Euclidean space:At each point they have the same Euclidean norm $\sqrt{x^2+y^2}$, but their directions differ. For $F$, $\nabla \cdot F=2$ and $\nabla \times F=0$. For $G$, $\nabla \cdot G=0$ and $\nabla \times G=(0,0,2)$. One expands radially in each horizontal plane; the other describes rigid rotation.

Around the counterclockwise unit circle in the plane $z=0$, $F$ is normal to the boundary and has zero tangential component. Its outward planar flux integral is $2\pi$, while its line integral around the circle is zero. For $G$ the values are reversed: outward planar flux is zero, and circulation is $2\pi$. The divergence and curl predict exactly which boundary measurement survives.

Why is this useful? A velocity field can move material through a boundary or circulate it within a region. Conservation of mass connects its divergence with density changes through the continuity equation; its curl is vorticity. In electromagnetism, the Maxwell equations use the same operators to connect electric fields, magnetic fields, and their sources. The common mathematical language lets the same integral calculations work across different physical models. ^[ref].

The geometric framework behind these operators is developed in differential forms. The Generalized Stokes theorem unifies the fundamental theorem for line integrals, Green theorem, Stokes theorem, and divergence theorem. In coordinates their formulas look different, but each relates accumulation over a region to a measurement on its boundary.

Wilson's Vector Analysis, published in 1901, was based on J. Willard Gibbs's lectures at Yale. Wilson's preface explains that examples from geometry, mechanics, and physics are interspersed with the mathematical development, so readers encounter applications as soon as they have the tools to understand them. That is a useful way to approach vector calculus: learn each operation through a measurement it makes possible. ^[ref].

In elliptic cylindrical coordinates, write a vector field as $A=A_u{e}_u+A_v{e}_v+A_z{e}_z$ using the local orthonormal unit vectors. The scale factors of orthogonal coordinates are $(h,h,1)$, so the divergence and curl areApplying the divergence to the gradient of a scalar function gives $\Delta \varphi =({\varphi }_{uu}+{\varphi }_{vv})/h^2+{\varphi }_{zz}$. These formulas hold where $h>0$. They use physical components along unit vectors, rather than components along the unnormalized coordinate tangent vectors.

Use a velocity potential to reconstruct an irrotational flow from one scalar field. A velocity potential $\varphi (\mathbf{x},t)$ satisfiesIt has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$ and is unchanged as a description of the velocity field when any function of time alone is added. For twice continuously differentiable $\varphi$, the identity curl of a gradient proves zero vorticity. In incompressible flow, taking the divergence also gives ${\nabla }^2\varphi =0$, the Laplace equation. This turns a three-component velocity field into a single harmonic function with appropriate boundary conditions. ^[ref]

For uniform flow $\mathbf{u}=(U,0,0)$, take $\varphi =Ux$. With $U=2\mathrm{m}{\mathrm{s}}^{-1}$, the change from $x=0$ to $x=3\mathrm{m}$ is $\Delta \varphi =6{\mathrm{m}}^2{\mathrm{s}}^{-1}$. By the fundamental theorem for line integrals, the line integral of velocity between these points equals that change along any path in the domain; a closed curve gives zero circulation.

For the potential line vortex, choose a local angle branch in polar coordinates and set $\varphi =K\theta$. Then $u_r={\partial }_r\varphi =0$ and $u_{\theta }=r^{-1}{\partial }_{\theta }\varphi =K/r$. One counterclockwise turn changes the continued velocity potential by $2\pi K$, although the velocity field returns to its original value. With $K=1{\mathrm{m}}^2{\mathrm{s}}^{-1}$, this change is $2\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}$, exactly the circulation. A full annulus therefore admits local angle branches and a multivalued velocity potential, but no global single-valued velocity potential when $K\ne 0$. Cutting the annulus along a radial line allows a single-valued angle branch on the remaining domain. The obstruction is path independence, not a failure of the local gradient formula.

Let $D$ be bounded, connected and simply connected, and let $u$ be a smooth divergence-free field. Suppose a sufficiently smooth vector field $A$ satisfies the displayed Poisson and boundary conditions. Divergence gives $\Delta \mathrm{div}A=0$ with zero boundary value, so harmonic uniqueness gives $\mathrm{div}A=0$. The vector Laplacian identity then implies $\nabla \times (\nabla \times A-u)=0$.

On a simply connected domain, define a scalar potential by integrating this curl-free difference along a path from a fixed point. Vanishing circulation around contractible loops proves path independence. Thus the difference is $\nabla \chi$; its divergence and boundary normal component give $\Delta \chi =0$ and ${\partial }_n\chi =0$. Green's identity yields ${\int }_D|\nabla \chi {|}^2=0$, proving the reconstruction. Additional conditions on $A$ can specify its own boundary problem; the argument assumes its existence and proves that its curl is exactly $u$. Simple connectivity is essential for eliminating possible non-gradient harmonic circulation fields.

The divergence with respect to velocity coordinates, with position held fixed. In $d$ dimensions, $\mathbf{A}(\mathbf{x},\mathbf{v})=\mathbf{B}(\mathbf{x})-a(\mathbf{x})\mathbf{v}$ has ${\nabla }_{\mathbf{v}}\cdot \mathbf{A}=-da$. A kinetic conservation law must retain this term for velocity-dependent forces.

The viscous stress tensor is the part of the Cauchy stress tensor associated with deformation rates. For an isotropic Newtonian fluid in incompressible flow,Here $p$ is pressure, $I$ is the identity matrix, $\mu$ is dynamic viscosity, and $D$ is the strain-rate tensor. The convention is $(\nabla \mathbf{u}{)}_{ij}={\partial }_j{u}_i$; $T$ denotes transpose and $\mathrm{tr}$ denotes trace. Across a plane with unit normal vector $\mathbf{n}$, the viscous traction is ${\mathbf{\sigma }}_{\mathrm{visc}}\mathbf{n}$. The tensor makes this rule work for any plane orientation. ^[ref]

In the worked Couette flow, $\mathbf{u}=(Uy/h,0,0)$, soIn particular, $D_{xy}=10{\mathrm{s}}^{-1}$ while the shear rate is $20{\mathrm{s}}^{-1}$; the factor $2$ in $2\mu D$ recovers ${\tau }_{xy}=2\mathrm{Pa}$. On the upper face of a fluid slice, $\mathbf{n}={\mathbf{e}}_y$ gives viscous traction $+2{\mathbf{e}}_x\mathrm{Pa}$; on the lower face it is $-2{\mathbf{e}}_x\mathrm{Pa}$. Constant stress can therefore transmit a force between the plates even though its divergence, and the net viscous force on each slice, vanish.

Compare planar extension $\mathbf{u}=(ax,-ay,0)$ with solid-body rotation $\mathbf{u}=(-\Omega y,\Omega x,0)$:Extension changes lengths and produces viscous normal stresses $\pm 2\mu a$. Rotation preserves distances and gives zero viscous stress, despite its nonzero velocity gradient and vorticity $2\Omega {\mathbf{e}}_z$. For $a=\Omega =1{\mathrm{s}}^{-1}$ and $\mu =0.10\mathrm{Pa}\mathrm{s}$, extension gives ${\sigma }_{\mathrm{visc},xx}=0.20\mathrm{Pa}$ and ${\sigma }_{\mathrm{visc},yy}=-0.20\mathrm{Pa}$; rotation gives zero. A rotating fluid may still have a pressure gradient supplying its centripetal acceleration.

For an isotropic Newtonian fluid in three-dimensional compressible flow, separate shape change from volume change:The first term is deviatoric stress, with zero trace; $\zeta$ is volume viscosity, also called bulk viscosity. For uniform expansion $\mathbf{u}=b\mathbf{x}$, $D=bI$ and $\nabla \cdot \mathbf{u}=3b$, so the shear contribution vanishes and the viscous stress is $3\zeta bI$. Thus dynamic viscosity and volume viscosity describe distinct responses, and incompressible flow removes only the volume-change term. ^[ref]

A volume integral integrates a scalar or componentwise vector field over a three-dimensional region with its volume form. In spherical polar coordinates the Euclidean volume form is $r^2\sin \theta drd\theta d\varphi$. The divergence theorem relates a volume integral of a divergence to a boundary flux integral.

The divergence defined through weak derivatives: pair a locally integrable vector field $J$ with a smooth compactly supported test function $\phi$ using the displayed integration-by-parts identity. A piecewise smooth field can have a surface contribution where its normal component jumps. Thus zero classical divergence on each side does not imply zero weak divergence across the interface.