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Just one of these four quantities is a scalar. Which?

  • A.\( \nabla \times \nabla\phi \)
  • B.\( \nabla \cdot \mathbf{A} \)
  • C.\( \nabla \times \mathbf{A} \)
  • D.\( \nabla\phi \)

Answer: B

The divergence contracts the operator with the field and returns a scalar rate of outflow per unit volume. The curl and the gradient both return vectors, and the curl of a gradient is a vector too, albeit the zero vector. Keeping track of which operations change the rank is what stops ill-formed identities being written

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