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CSS Applied Mathematics QUESTION #1319
Question 1
The gradient operator \(\nabla\) in Cartesian coordinates acts on a scalar field \(\phi\) to give:
  • \(\dfrac{\partial\phi}{\partial x}+\dfrac{\partial\phi}{\partial y}+\dfrac{\partial\phi}{\partial z}\)
  • \(\dfrac{\partial\phi}{\partial x}\hat{i}+\dfrac{\partial\phi}{\partial y}\hat{j}+\dfrac{\partial\phi}{\partial z}\hat{k}\)✔️
  • \(\left(\dfrac{\partial^2\phi}{\partial x^2}+\dfrac{\partial^2\phi}{\partial y^2}+\dfrac{\partial^2\phi}{\partial z^2}\right)\)
  • \(\phi\left(\hat{i}+\hat{j}+\hat{k}\right)\)
Correct Answer Explanation
\(\nabla\phi=\text{grad}\,\phi=\dfrac{\partial\phi}{\partial x}\hat{i}+\dfrac{\partial\phi}{\partial y}\hat{j}+\dfrac{\partial\phi}{\partial z}\hat{k}\). It is a vector field giving the direction and rate of maximum increase of the scalar field \(\phi\).