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Calculus on Wave Fronts

Oliver Knill

Abstract

We define a deformation of the exterior derivative that is a bounded operator and preserves the symmetries of the geometry. It satisfies a modified wave equation that honors the strong Huygens principle in all dimensions.

Calculus on Wave Fronts

Abstract

We define a deformation of the exterior derivative that is a bounded operator and preserves the symmetries of the geometry. It satisfies a modified wave equation that honors the strong Huygens principle in all dimensions.
Paper Structure (7 sections, 9 theorems, 28 equations, 5 figures)

This paper contains 7 sections, 9 theorems, 28 equations, 5 figures.

Table of Contents

  1. Deformed exterior derivative
  2. Deformed Wave equation
  3. Results
  4. Harmonic forms
  5. Physics
  6. Questions
  7. Conclusion

Key Result

Theorem 1

a) The form $u(t,p)=d_t f(p)$ solves $(D_{tt} + L) =0$ with initial position $u(0,p)=0$ and initial velocity $\lim_{t \to 0} \frac{d}{dt} u_t(0,p) = df(p)$. b) The form $u(t,p)=\phi_q(tD) df(p)$ solves $(d_{tt} + L)=0$ with initial position $u(0,p)=df(p)$ and initial velocity $u_t(0,p)=0$.

Figures (5)

  • Figure 1: A non-local exterior derivative of a $1$-form $f$ on a 2-manifold evaluates a line integral of $f$ along a wave front. By Green's theorem applied to a coordinate patch, it is an integral of curl.
  • Figure 2: On a general Riemannian 2-manifold we can evaluate a line integral along a wave front $W_t(p)$ and get so a notion of a discrete derivative for $1$-forms. By Green's theorem, it is a double integral over the ball $B_t(p) = \{ x \in M, d(x,p) = t \}$ at least if $t$ is smaller than the radius of injectivity.
  • Figure 3: In a manifold with boundary, the wave front are reflected at the boundary. We still can show that the total integral of $df$ is zero.
  • Figure 4: R2-D2 formula for curvature for 2 manifolds in interior or boundary.
  • Figure 5: R2-D2 formula at boundary of polygon or Koch region.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1: Wave equation
  • proof
  • Theorem 2: Strong Huygens
  • proof
  • Theorem 3: Harmonic forms
  • Corollary 1
  • ...and 5 more