Foundations

Scientific relationships · 16 / 16

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

1. Definite integral and signed accumulation → Limit of a real function
   Uses definition
   Relationship sources · 1

   • Calculus Volume 1, §5.2: Definite integral and net signed area ↗

   Linked statements

   • For a continuous real function on [a,b], the definite integral is the limit of sums f(xᵢ*)Δx over successively finer equal subdivisions. It measures signed accumulation; negative contributions subtract rather than become positive area.
2. Fundamental theorem of calculus → Definite integral and signed accumulation
   Uses definition
   Relationship sources · 1

   • Calculus Volume 1, §5.3: Theorems 5.4 and 5.5 ↗

   Linked statements

   • If f is continuous on [a,b], the accumulation F(x)=∫ₐˣf(t)dt satisfies F′(x)=f(x) for a<x<b. For any antiderivative G of f, ∫ₐᵇf(x)dx=G(b)−G(a).
3. Double integrals on simple regions → Definite integral and signed accumulation
   Uses definition
   Relationship sources · 1

   • Calculus Volume 3, 5.2 ↗

   Linked statements

   • On D={(x,y):a≤x≤b, g₁(x)≤y≤g₂(x)}, with continuous g₁≤g₂ and continuous f on D, ∬D f dA=∫ₐᵇ∫g₁(x)ᵍ₂(x) f(x,y)dy dx.
4. Darboux sums and Riemann integrability → Definite integral and signed accumulation
   Uses definition
   Relationship sources · 1

   • Introduction to Analysis, chapter 11: The Riemann integral ↗

   Linked statements

   • For bounded real f on [a,b], lower and upper Darboux sums use each subinterval's infimum and supremum. The Riemann integral exists when the supremum of lower sums equals the infimum of upper sums; equivalently, for every ε>0 some partition has U(f,P)−L(f,P)<ε.
5. Integral test and p-series → Definite integral and signed accumulation
   Uses definition
   Relationship sources · 1

   • Calculus Volume 2: Divergence and integral tests ↗

   Linked statements

   • For positive continuous decreasing f on [1,∞), Σₙ₌₁∞f(n) and ∫₁∞f(x)dx converge or diverge together. Consequently Σn⁻ᵖ converges exactly when p>1.
6. Composite trapezoidal quadrature → Definite integral and signed accumulation
   Uses definition
   Relationship sources · 1

   • Fundamentals of Numerical Computation: Numerical integration ↗

   Linked statements

   • On n equal subintervals with h=(b−a)/n, the trapezoidal estimate is T=h[f(a)/2+Σᵢ₌₁ⁿ⁻¹f(a+ih)+f(b)/2]. For C² f on the finite interval its error is O(h²) as h→0.
7. Curve length and curvature → Definite integral and signed accumulation
   Uses definition
   Relationship sources · 1

   • Calculus Volume 3: Arc length and curvature ↗

   Linked statements

   • A regular C¹ plane or space curve r(t), a≤t≤b, has length ∫ₐᵇ‖r′(t)‖dt. For a C² regular space curve its curvature is κ=‖r′×r″‖/‖r′‖³, equivalently ‖dT/ds‖ with unit tangent T and arc length s.
8. Uniform limits and Riemann integration → Definite integral and signed accumulation
   Uses definition
   Relationship sources · 1

   • Introduction to Analysis, chapter 12: The Riemann integral ↗

   Linked statements

   • If Riemann-integrable fₙ on a finite interval [a,b] converge uniformly to f, then f is Riemann integrable and ∫ₐᵇfₙ dx→∫ₐᵇf dx.
9. Poisson integral for a disk → Definite integral and signed accumulation
   Uses definition
   Relationship sources · 3

   • Notes on Diffy Qs: Dirichlet problem in the circle and the Poisson integral ↗
   • Math 120B (Spring 2020), Lecture 19: Harmonic functions and the Poisson kernel ↗
   • Complex Analysis (Spring 2026), Lecture 7: The Poisson integral formula ↗

   Linked statements

   • For continuous real 2π-periodic boundary data g, u(r,θ)=(2π)⁻¹∫₋π^π[(1−r²)/(1−2r cos(θ−α)+r²)]g(α)dα is harmonic on 0≤r<1 and extends continuously to u(1,θ)=g(θ). It is the unique such harmonic solution on the closed unit disk.
10. Composite Simpson rule and bounded error → Definite integral and signed accumulation
    Uses definition
    Relationship sources · 1

    • Calculus Volume 2: Numerical integration: Simpson's rule and error ↗

    Linked statements

    • With even n≥2 and h=(b−a)/n, Sₙ=(h/3)[f(a)+f(b)+4Σodd i f(a+ih)+2Σeven i f(a+ih)], summing interior indices. For f∈C⁴[a,b], |∫ₐᵇf−Sₙ|≤(b−a)h⁴ max|f⁽⁴⁾|/180. It is exact for polynomials of degree at most three.
11. Fourier mean-square convergence and Parseval → Definite integral and signed accumulation
    Uses definition
    Relationship sources · 1

    • Basic Analysis, section 11.8: Fourier series ↗

    Linked statements

    • For a complex 2π-periodic Riemann-integrable f, cₙ=(2π)⁻¹∫₋π^πf(x)e⁻ⁱⁿˣdx and s_N=Σ|n|≤N cₙeⁱⁿˣ satisfy ∫|f−s_N|²→0. Parseval gives (2π)⁻¹∫₋π^π|f|²=Σₙ∈Z|cₙ|². This is mean-square convergence, which does not assert pointwise convergence everywhere.
12. Daily mean top-of-atmosphere insolation → Definite integral and signed accumulation
    Derived from
    Relationship sources · 1

    • General Solar Position Calculations, first-page solar-zenith relation ↗

    Linked statements

    • For solar flux S perpendicular to the rays at the current orbital distance,latitudeφ and declinationδ give cosZ=sinφsinδ+cosφcosδcosh. Integrating positive illumination over a rotation gives meanQ=(S/π)[h₀sinφsinδ+cosφcosδsinh₀]. For ordinary sunrise/sunset h₀=arccos(−tanφtanδ); polar-day/night limits use h₀=π or0 with the appropriate geometry.

    Rotation-average energy is the stated finite integral.

13. Scalar seismic moment and moment magnitude → Definite integral and signed accumulation
    Related to
    Relationship sources · 1

    • Finite Faults, Introduction and Scientific Details, scalar moment, Mw conversion and model ambiguity ↗

    Linked statements

    • For a planar shear-fault source with uniform shear modulus μ and consistently directed slip magnitude D, scalar moment is M₀=μAD̄, equivalently μ∫_A D dA. Moment magnitude uses Mw=(2/3)[log₁₀(M₀/(1N m))−9.1]. Source moment describes fault-slip size and is distinct from radiated energy or intensity at a site.

    The area-weighted source sum is a bounded integral.

14. Seismic rays, interface refraction and travel time → Definite integral and signed accumulation
    Derived from
    Relationship sources · 1

    • 12.002 Lecture11, printed17–19, interface rays, critical angle and uniform-sphere travel times ↗

    Linked statements

    • In a high-frequency ray model, travel time along a pathγ is t=∫_γ ds/v. Across a stationary planar interface, matching phase along the interface gives sinθ₁/v₁=sinθ₂/v₂ for propagating branches, with angles from the normal. For v₂>v₁ the transmitted grazing angle occurs at sinθ_c=v₁/v₂. Reflection and converted elastic branches require their own speeds and boundary conditions.

    Travel time integrates path slowness.

15. Gravity forward models for a buried line and strip → Definite integral and signed accumulation
    Derived from
    Relationship sources · 1

    • 12.002 Lectures27–28, printed55–59, reference gravity, line/strip/sheet anomalies and resolution ↗

    Linked statements

    • For an infinite line parallel to y at horizontal coordinate x₀ and depth b>0 with signed excess mass per lengthm_l, its downward surface anomaly is Δg=2Gm_lb/[(x−x₀)²+b²]. Integrating lines from x₁ to x₂ gives a thin strip model Δg=2GΔρH[arctan((x₂−x)/b)−arctan((x₁−x)/b)]. Greater burial broadens a fixed line-mass anomaly and lowers its peak magnitude.

    The line and strip kernels are stated integrals.

16. Steady geotherm with internal heat production → Definite integral and signed accumulation
    Related to
    Relationship sources · 1

    • 12.002 Lecture13, printed29–31, conductive conservation, diffusivity, length scale and steady geotherm ↗

    Linked statements

    • With depthz positive downward, constant conductivityk and uniform volumetric heat productionH, a stationary conductive slab obeys kT_zz+H=0. If T(0)=T_s and outward/upward surface heat flux is q_s=kT_z(0), then T(z)=T_s+(q_s/k)z−Hz²/(2k). At H=0 the gradient is constant; internal production makes upward flux decrease with depth in this model.

    Integrating a constant-coefficient second-order equation fixes the geotherm.