Calculus
Line Integrals
Цели урока
- Compute first-kind line integrals for arc length and mass along parametric curves
- Compute second-kind line integrals (work) and understand their orientation dependence
- Test fields for conservativeness using curl and compute potentials
- Relate circulation and normal flux to Stokes and Green theorems
Предварительные знания
- Parametric curves
- Gradient and partial derivatives
- Curl and divergence
Why does a spacecraft use the same fuel to go from A to B regardless of the path - and when does this path-independence break down?
- SpaceX trajectory optimization: work against gravity is path-independent (conservative field), but drag is not - the difference is the curl
- Tesla wiring layout: minimizing total wire length in 3D routing = minimizing the first-kind line integral of arc length
- Faraday's law in Maxwell's equations: electric field circulation around a loop = negative rate of change of magnetic flux
- PyTorch autograd: backpropagation computes gradient as a directed sum along the computational graph - structurally a line integral
Stokes, Green, and the unification of calculus
George Green (1793-1841) published his theorem in 1828. William Thomson (Lord Kelvin) restated the 3D version in 1850 in a letter to George Stokes (1819-1903). Stokes set it as an exam question in Cambridge, and 'Stokes theorem' was born from a student exercise. The line integral - work and flux along curves - was the core tool. By 1900, the language of line integrals and vector fields had unified mechanics, electromagnetics, and thermodynamics under one notation.
Line Integral of the First Kind
In 2019, Tesla reduced total cooling pipe length by 12 meters per Model 3 vehicle by optimizing wiring paths. The core tool: the first-kind line integral, which computes arc length and mass of non-uniform wires along arbitrary curves. One formula drove a 4% reduction in material cost across millions of units.
Setting f = 1 gives arc length L = integral_C ds. Setting f = rho (linear mass density) gives total mass. Setting f = T (temperature) gives the average temperature: T_avg = (1/L) * integral_C T ds.
Mass of a non-uniform helix
Variable density wire
Helix: r(t) = (cos t, sin t, t), t from 0 to 2*pi. Density rho(x,y,z) = 1 + z = 1 + t. Arc-length element: |r'(t)| = sqrt(sin^2 t + cos^2 t + 1) = sqrt(2). Mass = integral_0^{2pi} (1+t) * sqrt(2) dt = sqrt(2) * [t + t^2/2]_0^{2pi} = sqrt(2) * (2*pi + 2*pi^2). This is one application: winding coils with variable wire thickness.
The first-kind integral does not depend on the orientation of C - only on the set of points. The second-kind (work integral) reverses sign when orientation reverses. This distinction drives almost all sign errors in vector calculus problems.
Does the first-kind line integral change sign when the curve orientation is reversed?
ds = |r'(t)| dt >= 0 always. Reversing orientation swaps a and b, but simultaneously reverses dt, so ds stays positive. The first-kind integral is truly orientation-independent.
Line Integral of the Second Kind (Work)
SpaceX Falcon 9 performs roughly 2.8 GJ of mechanical work against gravity during an ascent to 70 km. That figure is a line integral of the second kind: the dot product of thrust with displacement, summed along the flight path. The integral changes sign if the rocket descends - direction matters here.
When F = grad(phi) (conservative field), work depends only on endpoints: A = phi(B) - phi(A). Path does not matter. Curl = 0 is the test for conservativeness on a simply connected domain.
Work by a non-conservative field
Rotational field F = (-y, x)
F = (-y, x). Work around the unit circle: r(t) = (cos t, sin t), t from 0 to 2*pi. A = integral_0^{2pi} (-sin t)(-sin t) + (cos t)(cos t) dt = integral_0^{2pi} 1 dt = 2*pi. Non-zero work on a closed path - this field is not conservative. The curl = 2 != 0 confirms it.
Field F = (2xy, x^2) is conservative with potential phi = x^2 y. Work along any path from (0,0) to (1,2)?
F is conservative with potential phi = x^2 y. Work = phi(1,2) - phi(0,0) = 2 - 0 = 2, regardless of path taken.
Conservative Fields and Path Independence
Google Maps computes shortest routes - but in physics and engineering, what matters is often the minimum-work path, not the shortest. Gravity is conservative: lifting an object 10 meters takes the same work regardless of the route taken. But fluid drag is not: path matters enormously. The mathematical difference is exactly the curl.
The angle form F = (-y, x) / (x^2+y^2) has zero curl on R^2 minus the origin, but is not conservative there: the work around the unit circle equals 2*pi, not zero. Simply connected domain is essential for the curl test.
To find the potential phi from a conservative field F = (P, Q): integrate P with respect to x to get phi = integral P dx + g(y). Then differentiate phi with respect to y and set equal to Q to find g'(y). Standard procedure in every electrostatics textbook.
Field F = (-y, x)/(x^2+y^2) on R^2 minus the origin has zero curl. Is it conservative?
On R^2 \ {0} (not simply connected): zero curl does not imply conservative. Work around S^1 = 2*pi != 0 proves the field is not conservative despite zero curl.
Applications: Flux, Circulation, and Physical Work
Maxwell's equations in 3D involve line integrals of E and B around closed loops. Faraday's law: the circulation of E around a loop equals the negative rate of change of magnetic flux through the loop. The circulation is a line integral. The flux is a surface integral. Green's theorem and Stokes' theorem connect these two.
Circulation in Faraday's law
Maxwell's equation as a line integral
Faraday's law: oint_C E.dr = -d/dt integral_S B.dS. The left side is the circulation of E around the closed loop C. The right side involves the flux of B through any surface bounded by C. This is exactly the setup for Stokes' theorem: oint_C E.dr = integral_S (curl E).dS. Equating: curl E = -dB/dt. This is how Maxwell's equations in differential form are derived from the integral ones.
In neural network backpropagation, the chain rule computes the gradient of the loss along the computational graph. This is structurally analogous to a line integral: summing contributions along a directed path. The 'field' is the partial derivative of each operation, and the 'work done' is the gradient.
In 2D, the flux of F = (P,Q) through a curve C (outward normal) is:
Outward normal n = (dy/ds, -dx/ds). Flux = integral_C F.n ds = integral_C P dy - Q dx. Green's theorem then equates this to the double integral of div(F) over the enclosed region.
Connections to other topics
Line integrals are the 1D foundation of all of vector calculus: Green, Stokes, Gauss theorems all live above them.
- Electrodynamics — Related topic
- Mechanics — Related topic
- Complex analysis — Related topic
- Differential forms — Related topic
Итоги
- First-kind: integral_C f ds = integral_a^b f(r(t)) |r'(t)| dt - orientation-independent, computes arc length and mass
- Second-kind: integral_C F.dr = integral_a^b (P x' + Q y' + R z') dt - changes sign on orientation reversal, computes work
- Conservative field test: curl F = 0 on simply connected domain iff F = grad(phi) iff work on all closed paths = 0
- Flux in 2D: integral_C P dy - Q dx (outward normal); circulation: integral_C P dx + Q dy (tangent)
- Zero curl on non-simply-connected domains is not enough: the angle form (x dy - y dx)/(x^2+y^2) has zero curl but non-zero circulation
Вопросы для размышления
- Gravity is conservative: why does this mean satellites can orbit indefinitely without fuel expenditure?
- The angle form has zero curl but non-zero circulation around the origin - what does this say about the topology of R^2 minus a point?
- Backpropagation computes gradients by summing along the computational graph. In what sense is this analogous to a line integral - and what is the 'field' being integrated?
Связанные уроки
- calc-19-gradient — Gradient fields and potential functions underlie conservative line integrals
- calc-25-green-theorem — Green's theorem converts boundary line integrals to area integrals
- calculus-27 — Classical Stokes theorem generalizes line integrals to surfaces
- stats-21
- la-06-transformations