Choosing a more convenient surface The goal is to evaluateA = ∫∫S (∇ x F) ⋅ n dS, where F = ⟨yz, -xz, xy⟩ and S is thesurface of the upper half of the ellipsoid x2 + y2 + 8z2 = 1 (z ≥ 0).a. Evaluate a surface integral over a more convenient surface to find the value of A.b. Evaluate A using a line integral.

Answered: Choosing a more convenient surface The…

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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Choosing a more convenient surface The goal is to evaluate
A = ∫∫_S (∇ x F) ⋅ n dS, where F = ⟨yz, -xz, xy⟩ and S is the
surface of the upper half of the ellipsoid x² + y² + 8z² = 1 (z ≥ 0).
a. Evaluate a surface integral over a more convenient surface to find the value of A.
b. Evaluate A using a line integral.

Expert Solution

Step 1

Given: $A = \int \int_{S} (\nabla \times F) \cdot n d s, F = <y z, - x z, x y>$ where S is the surface of the upper half of the ellipsoid $x^{2} + y^{2} + 8 z^{2} = 1 (z \geq 0)$ .

To Find :

a) Surface integral over a more convenient surface to find the value of A.

b) A using a line integral.

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