The direction in which a scalar field
( , , )x y z has the largest rate of change at
any point with position vector
ˆˆˆr xi yj zk= + +
is the same as that of the vector
Options:
(A)
(B) () r
(C) r
(D) (𝛻$\phi$ • dr⃗)r⃗
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2023 Q.11)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to determine the direction in which a smooth 3D scalar field $\phi(x, y, z)$ possesses the maximum spatial rate of change at any given point.
**2. Step-by-Step Mathematical Derivation**:
1. **Directional Derivative Formulation**:
The directional derivative of scalar field $\phi$ in the direction of an arbitrary unit vector $\hat{u}$ is:
D_{\hat{u}}\phi = \nabla \phi \cdot \hat{u} = |\nabla \phi| \cos\theta
where $\theta$ is the angle between the gradient vector $\nabla \phi$ and the direction $\hat{u}$.
2. **Maximization**:
Since $-1 \le \cos\theta \le 1$, the maximum value of $D_{\hat{u}}\phi$ occurs when $\cos\theta = 1$, which requires $\theta = 0$.
Therefore, the maximum rate of change occurs when $\hat{u}$ is aligned precisely in the direction of $\nabla \phi$.
3. **Magnitude**:
The maximum rate of change is equal to the magnitude $|\nabla \phi|$ itself.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) along the gradient vector ∇$\phi$**: ✅ **CORRECT**. The gradient vector points along the path of steepest ascent, maximizing the directional derivative.
- **(B) along the tangent to the level surface**: ❌ **INCORRECT**. Tangent to a level surface has zero rate of change ($D\phi = 0$).
- **(C) opposite to the gradient vector**: ❌ **INCORRECT**. This is the direction of steepest descent (minimum rate of change, $-|\nabla \phi|$).
- **(D) along the curl of $\phi$**: ❌ **INCORRECT**. The curl of a scalar field is undefined.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Gradient $\nabla \phi$ points perpendicular to level surfaces $\phi = c$, in the direction of maximum rate of increase. The directional derivative is maximum along $\nabla \phi$ and zero tangent to the level surface.