10B (Vector Calculus)

Parametrize the upward-oriented paraboloid by

X (r, θ) = (r cos θ, r sin θ, 1 - r^{2}), 0 \leq r \leq 1, 0 \leq θ \leq 2 π .

Then

X_{r} \times X_{θ} = (2 r^{2} cos θ, 2 r^{2} sin θ, r) .

On the surface,

F = (r^{2} cos^{2} θ, - r^{2} sin^{2} θ, (1 - r^{2})^{4}) .

The cubic trigonometric terms integrate to zero over

0 \leq θ \leq 2 π

\int_{S} F \cdot d S = 2 π \int_{0}^{1} r (1 - r^{2})^{4} d r = \frac{π}{5} .

ii

Close the surface with the unit disk

D

in the plane

z = 0

. Its outward normal is

- e_{z}

, but

F \cdot (- e_{z}) = - z^{4} = 0

there. Also

\nabla \cdot F = 2 x - 2 y + 4 z^{3} .

The

x

and

y

terms integrate to zero by symmetry. The divergence theorem therefore gives

\int_{S} F \cdot d S = \int_{0}^{2 π} \int_{0}^{1} \int_{0}^{1 - r^{2}} 4 z^{3} r d z d r d θ = 2 π \int_{0}^{1} r (1 - r^{2})^{4} d r = \frac{π}{5} .

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