Question

The direction in which a scalar field \(\phi(x,y,z)\) has the largest rate of change is along:

• A. \(\nabla \phi\)
• B. \(\nabla \times (\phi \vec{r})\)
• C. \(\phi \vec{r}\)
• D. \((\nabla \phi \cdot d\vec{r}) \vec{r}\)

Hint:

For any scalar field, the gradient always points in the direction of steepest increase. Its magnitude is the maximum rate of change at that point.

Answer 1

The Correct Option is A

Step 1: Gradient definition.
For a scalar field \(\phi(x,y,z)\), the gradient vector is defined as: \[ \nabla \phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k}. \]

Step 2: Direction of steepest ascent.
- The directional derivative of \(\phi\) in unit direction \(\hat{u}\) is \[ D_{\hat{u}}\phi = \nabla \phi \cdot \hat{u}. \] - The maximum value of this dot product occurs when \(\hat{u}\) is aligned with \(\nabla \phi\).

Step 3: Eliminate other options.
- (B) Curl of \(\phi \vec{r}\) is unrelated.
- (C) \(\phi \vec{r}\) is scaling of position vector, not gradient.
- (D) Expression is not standard, no guarantee of steepest change.

Final Answer:
\[ \boxed{\nabla \phi} \]