Discussion
Let \(\varphi\) be a scalar function. Then, \(\nabla\varphi\) is
Answer: A
The gradient is always perpendicular to the surface of constant \(\varphi\).
Along any curve lying in a level surface \(\varphi = c\), the rate of change \(\nabla\varphi\cdot\vec{t}\) is zero, where \(\vec{t}\) is the tangent. So \(\nabla\varphi\) is orthogonal to every tangent direction of the surface, that is, normal to it.
It cannot be parallel to the surface for the same reason. Its magnitude is the maximum rate of change of \(\varphi\), not the minimum, and it vanishes only at stationary points, not always.
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