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

Step 1: Recall how the gradient is built.
Take a scalar field $\phi(x,y,z)$, a function that assigns one number to every point in space. Its gradient is formed by collecting the three partial derivatives $\dfrac{\partial \phi}{\partial x}$, $\dfrac{\partial \phi}{\partial y}$, and $\dfrac{\partial \phi}{\partial z}$, and attaching them to the unit directions $\hat{i}$, $\hat{j}$, $\hat{k}$.

Step 2: Check what kind of object this produces.
An object built from three components, each tied to a direction in space, is exactly the definition of a vector. So $\nabla \phi$ has both a size, given by $|\nabla \phi| = \sqrt{\left(\dfrac{\partial \phi}{\partial x}\right)^2 + \left(\dfrac{\partial \phi}{\partial y}\right)^2 + \left(\dfrac{\partial \phi}{\partial z}\right)^2}$, and a direction, the direction of steepest increase of $\phi$.

Step 3: Rule out the other choices.
It cannot be a plain scalar, since a scalar has no direction attached to it. It is not a second order tensor either, since that order of object comes from differentiating a vector field, not a scalar field. It is also not always zero, since it vanishes only where the field has a flat spot, not at every point.

Final Answer:
The gradient of a scalar field is always a vector.
\[ \boxed{\nabla \phi \text{ is a vector}} \]