 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013/4/a/solution

\nabla T = 2 (x, y) / (x^{2} + y^{2})

and at

(1, 2)

it is

(2/5, 4/5)

. The directional derivative along a unit vector is maximal in the gradient direction. Therefore

\hat{n}_{increase} = \frac{( 1 , 2 )}{5}, ∣\nabla T ∣ = \frac{2}{5} .

The direction is radially outward from the origin, perpendicular to the circular level curve through the point.

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