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The directional derivative of a scalar V along a unit vector \( \mathbf{a} \) is

  • A.\( \nabla V \times \mathbf{a} \)
  • B.\( |\nabla V| \)
  • C.\( \nabla \cdot (V\mathbf{a}) \)
  • D.\( \nabla V \cdot \mathbf{a} \)

Answer: D

It is the projection of the gradient onto the chosen direction, and it is greatest when the direction coincides with the gradient and zero when it is perpendicular. That perpendicular case is movement along a surface of constant V.

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