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CSS Applied Mathematics QUESTION #1293
Question 1
If \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) and \(r = |\vec{r}|\), then \(\nabla r^n\) equals:
  • \(nr^{n-2}\)
  • \(nr^{n-1}\vec{r}\)
  • \(nr^{n-2}\vec{r}\)✔️
  • \(nr^n\vec{r}\)
Correct Answer Explanation
Using the chain rule: \(\nabla r^n = n r^{n-1}\nabla r\). Since \(\nabla r = \dfrac{\vec{r}}{r}\), we get \(\nabla r^n = n r^{n-1}\cdot\dfrac{\vec{r}}{r} = nr^{n-2}\vec{r}\).