Question

Let \( \varphi \) be a scalar function. Then, \( \nabla \varphi \) is

• A. always perpendicular to the surface of constant \( \varphi \)
• B. always parallel to the surface of constant \( \varphi \)
• C. the minimum rate of change of scalar \( \varphi \)
• D. always zero

Hint:

Recall that the gradient of a scalar field always points normal to its level surface.

Answer 1

The Correct Option is A

This question checks the basic geometric meaning of the gradient of a scalar field.

Pick any point on a level surface, the set of points where $\varphi$ stays constant. Moving along that surface does not change $\varphi$, so the directional derivative of $\varphi$ along any direction tangent to the surface is zero.

The gradient $\nabla \varphi$ points along the direction of fastest increase of $\varphi$, and its dot product with any tangent direction on the level surface equals that zero directional derivative. A vector whose dot product with every tangent vector of a surface is zero has to be normal to that surface.

1. always perpendicular to the surface of constant $\varphi$: correct, this follows directly from the zero directional derivative argument above.
2. always parallel to the surface of constant $\varphi$: wrong, a vector tangent to the surface would give a nonzero directional derivative along the surface, which contradicts $\varphi$ staying constant there.
3. the minimum rate of change of scalar $\varphi$: wrong, the gradient magnitude gives the maximum rate of change, not the minimum.
4. always zero: wrong, the gradient is zero only at special critical points, not in general.

So $\nabla \varphi$ is always perpendicular to the surface of constant $\varphi$, option (A).