For r=x i+y j+z k with r =x2+y2+z2, find for = r .

GATE 2018 ME (ME2) – Question 38

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities, line, surface and volume integrals · 2 marks · Multiple choice

For $\vec r=x\hat i+y\hat j+z\hat k$ with $|\vec r|=\sqrt{x^2+y^2+z^2}$, find $\nabla\phi$ for $\phi=\ln|\vec r|$.

$\vec r$
$\vec r/|\vec r|$
$\vec r/(\vec r\cdot\vec r)$
$\vec r/|\vec r|^3$

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Correct answer: (C) $\vec r/(\vec r\cdot\vec r)$

Explanation

For $r=\sqrt{x^2+y^2+z^2}$, $\partial\ln r/\partial x=x/r^2$ and similarly for y and z. Hence $$\nabla\ln|\vec r|=\frac{x\hat i+y\hat j+z\hat k}{x^2+y^2+z^2}=\frac{\vec r}{\vec r\cdot\vec r}.$$ This holds away from the origin. Answer **C**.