 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-2/13/b/i/solution

Z = z (x, y)

, an upward normal is

(- z_{x}, - z_{y}, 1)

. At

x = y = 1

, let

q = e^{- 1/2}

; then

z_{x} = q

and

z_{y} = 0

. Normalizing gives the unit normal vector

n = \frac{( - q , 0 , 1 )}{1 + q ^{2}} = \frac{( - e ^{- 1/2} , 0 , 1 )}{1 + e ^{- 1}} .

Its negative is the equally valid downward normal.

 New to topics? Read the docs here!