Discussion
If \(r^{2} = x^{2} + y^{2} + z^{2}\), grad \(r^{n}\) is :
Answer: C
The gradient is \(n r^{n-2} \vec{r}\).
For a function of \(r\) alone, \(\nabla f(r) = f'(r)\,\hat{r}\), and \(\hat{r} = \vec{r}/r\). With \(f = r^{n}\):
\[ \begin{aligned} \nabla r^{n} &= n r^{n-1}\,\hat{r} \\ &= n r^{n-1}\,\frac{\vec{r}}{r} = n r^{n-2}\,\vec{r} \end{aligned} \]
The option \(r^{n-1}\vec{r}\) drops the factor n and uses \(\vec{r}\) where \(\hat{r}\) belongs, which changes the power of r. The option \(n(n-2)r^{n}\vec{r}\) has the wrong dimensions, and 0 would hold only for a constant function, i.e. n = 0.
No comment is present. Be the first to comment.
Loading…
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development