Question

The gradient of a scalar field is ______.

• A. a vector
• B. a scalar
• C. a second order tensor
• D. always zero

Hint:

Write out the definition of gradient using partial derivatives and check what type of quantity it forms.

Answer 1

The Correct Option is A

This question checks the basic definition of the gradient operator in vector calculus, applied to a scalar field.

1. a vector: For a scalar field \( \phi(x,y,z) \), the gradient is defined as \( \nabla \phi = \dfrac{\partial \phi}{\partial x}\,\hat{i} + \dfrac{\partial \phi}{\partial y}\,\hat{j} + \dfrac{\partial \phi}{\partial z}\,\hat{k} \). This combination of three partial derivatives, each attached to a direction, is by definition a vector. Its magnitude gives the steepest rate of increase of \( \phi \) at a point, and its direction points along that steepest increase, perpendicular to the level surface of \( \phi \) through that point. This is the correct choice.
2. a scalar: Wrong, since the gradient carries direction information as well as magnitude, which a plain scalar cannot hold.
3. a second order tensor: Wrong, a second order tensor arises from the gradient of a vector field, not a scalar field. Taking the gradient of a scalar only raises the order by one, from a scalar (order 0) to a vector (order 1).
4. always zero: Wrong, the gradient is zero only at stationary points of the field, such as local maxima, minima, or saddle points, not everywhere in general.

So the gradient of a scalar field is a vector, which is option A.