Question
Answer the following about things that make 3D special in physics, for 10 points each.
[10m] Gravitational orbits and atoms can only exist in 3D since solving the Poisson equation in 3D gives force laws with this property. Only force laws with this form and linear force laws give rise to stable orbits by Bertrand’s theorem.
ANSWER: inverse-square [accept one over r-squared or one over d-squared or one over distance-squared or one over radius-squared; prompt on central forces]
[10h] 3D is the only dimension in which tensors like angular velocity and magnetic field can be represented as vectors, because bivectors and vectors have a form of duality named for this mathematician. More generally, that duality relates k-vectors to “n minus k”-vectors.
ANSWER: W. V. D. Hodge [or William Vallance Douglas Hodge; accept Hodge dual or Hodge star]
[10e] The duality of vectors and rank-2 tensors explains why this operation is defined in 3D. Taking this operation of two vectors outputs a vector that is perpendicular to both.
ANSWER: cross product [prompt on vector product; reject “dot product”]
<VD, Physics>
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