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Which quantity remains unchanged under this four-part decomposition?

  • A.The area of the element
  • B.The shape of the element, which is preserved because deformation and divergence cancel one another
  • C.The translation of the element as a whole
  • D.The rotation of the element, which is captured by the vorticity component of the decomposition used

Answer: C

The decomposition describes the gradient of the velocity, so a uniform translation, which has zero gradient, contributes nothing to any of the four components. That is why the quantities are all frame-independent under a change of the moving observer. Rotation, area and shape are each altered by particular components, which is exactly what the decomposition sets out to quantify.

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