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GATE 2023 AE – Question 11

Engineering Mathematics · Calculus: Functions of several variables, partial derivatives, gradient, divergence, curl and directional derivatives · 1 mark · Multiple choice

The direction in which a scalar field $\phi(x,y,z)$ has the largest rate of change at a point with position vector $\vec r=x\hat i+y\hat j+z\hat k$ is the direction of

  1. $\nabla\phi$
  2. $\nabla\times(\phi\vec r)$
  3. $\phi\vec r$
  4. $(\nabla\phi\cdot d\vec r)\vec r$

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Correct answer: (A) $\nabla\phi$

Explanation

The **directional derivative** of $\phi$ in the direction of a unit vector $\hat n$ is
$$\frac{d\phi}{ds}=\nabla\phi\cdot\hat n=|\nabla\phi|\cos\theta,$$
where $\theta$ is the angle between $\hat n$ and $\nabla\phi$.

This is largest when $\cos\theta=1$, that is, when $\hat n$ points **along $\nabla\phi$**. So the direction of greatest rate of increase is the direction of the gradient, with magnitude $|\nabla\phi|$.

Answer **$\nabla\phi$** (A).