Discussion
The direction in which a scalar field \(\phi(x, y, z)\) has the largest rate of change at any point with position vector \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) is the same as that of the vector
Answer: A
The direction is that of the gradient, \(\nabla\phi\).
The rate of change of \(\phi\) in the direction of a unit vector \(\hat{n}\) is the directional derivative \(\nabla\phi \cdot \hat{n} = |\nabla\phi|\cos\alpha\), where \(\alpha\) is the angle between \(\hat{n}\) and the gradient. It is largest when \(\alpha = 0\), that is, along \(\nabla\phi\), and its largest value is \(|\nabla\phi|\).
\(\nabla \times (\phi\vec{r})\) equals \(\nabla\phi \times \vec{r}\), which is perpendicular to the gradient, so along it the rate of change is zero. \(\phi\vec{r}\) points along the position vector, which has nothing to do with how \(\phi\) varies. \((\nabla\phi \cdot d\vec{r})\vec{r}\) is also a multiple of \(\vec{r}\), scaled by the change in \(\phi\), and again points radially rather than along the gradient.
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