Answered: Evaluate the following surface…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Evaluate the following surface integral.

∫∫_S (x - y + z) dS; S is the entire surface, including the base, of
the hemisphere x² + y² + z² = 4, for z ≥ 0.

Expert Solution

Step 1

Given: $x^{2} + y^{2} + z^{2} = 4$ , $z \leq 0$

To find: $I = \iint_{S} (x + y + z) d S$

Step 2

Parametrize the surface S of the hemisphere which implies

$r (θ, ϕ) = 2 \sin ϕ \cos θ i + 2 \sin ϕ \sin θ j + 2 \cos ϕ k$

$r_{θ} = \frac{\partial}{\partial θ} (2 \sin ϕ \cos θ i + 2 \sin ϕ \sin θ j + 2 \cos ϕ k) r_{θ} = - 2 \sin ϕ \sin θ i + 2 \sin ϕ \cos θ j r_{ϕ} = \frac{\partial}{\partial ϕ} (2 \sin ϕ \cos θ i + 2 \sin ϕ \sin θ j + 2 \cos ϕ k) r_{ϕ} = 2 \cos ϕ \cos θ i + 2 \cos ϕ \sin θ j - 2 \sin ϕ k$

The cross product of the above vectors is

$r_{θ} \times r_{ϕ} = |\begin{matrix} i & j & k \\ - 2 \sin ϕ \sin θ & 2 \sin ϕ \cos θ & 0 \\ 2 \cos ϕ \cos θ & 2 \cos ϕ \sin θ & - 2 \sin ϕ \end{matrix}| r_{θ} \times r_{ϕ} = i (- 4 \sin^{2} \cos θ) - j (- 4 \sin^{2} ϕ \sin θ) + 4 (- \sin ϕ \cos ϕ \sin^{2} θ - \sin ϕ \cos ϕ \cos^{2} θ) k r_{θ} \times r_{ϕ} = 4 (- \sin^{2} ϕ \cos θ) i + 4 (\sin^{2} ϕ \sin θ) j - 4 (\sin ϕ \cos ϕ) k$

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