 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016/10/a/solution

Parametrize the graph by

r (x, y) = (x, y, f (x, y))

over its projection

D

onto the

x y

(1, 0, f_{x})

and

(0, 1, f_{y})

, so their cross product is

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

d S = ∣ r_{x} \times r_{y} ∣ d x d y = 1 + f_{x}^{2} + f_{y}^{2} d x d y,

and

area (S) = \iint_{D} 1 + f_{x}^{2} + f_{y}^{2} d x d y .

The printed subscript

S

is shorthand for integrating over the projected parameter region:

d x d y

belongs to

D

, while the square-root factor converts projected area into actual surface area. The corresponding upward vector surface element of a graph is

(- f_{x}, - f_{y}, 1) d x d y

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