For the scalar field \[{\text{u}} = \frac{{{{\text{x}}^2}}}{2} + \frac{{{{\text{y}}^2}}}{3},\] magnitude of the gradient at the point (1, 3) is A. \[\sqrt {\frac{{13}}{9}} \] B. \[\sqrt {\frac{9}{2}} \] C. \[\sqrt 5 \] D. \[\frac{9}{2}\]

” option2=”\[\sqrt {\frac{9}{2}} \]” option3=”\[\sqrt 5 \]” option4=”\[\frac{9}{2}\]” correct=”option1″]

The correct answer is $\boxed{\sqrt {\frac{{13}}{9}}}$.

The gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and its magnitude is the magnitude of the greatest rate of increase.

The gradient of the scalar field $\text{u} = \frac{{{{\text{x}}^2}}}{2} + \frac{{{{\text{y}}^2}}}{3}$ is given by

$$\nabla \text{u} = \left( \frac{\partial \text{u}}{\partial x}, \frac{\partial \text{u}}{\partial y} \right) = \left( \frac{x}{2}, \frac{y}{3} \right).$$

The magnitude of the gradient at the point $(1, 3)$ is then given by

$$\left| \nabla \text{u} \right| = \sqrt{\left( \frac{1}{2} \right)^2 + \left( \frac{3}{3} \right)^2} = \sqrt{\frac{{13}}{9}}.$$