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Let F(x, y,z) = (-xe² +3x, -ye² + arctan(x? + z), 2e²) and S the portion of the sphere acoording to S= {x, y, 2) : r² + y² + z² = 1, z > 0). Calcule F.nds, where n is the unit normal outside the sphere. Use the Gauss' Theorem.

Let F(x, y,z) = (-xe² +3x, -ye² + arctan(x? + z), 2e²) and S the portion of the sphere acoording to S= {x, y, 2) : r² + y² + z² = 1, z > 0). Calcule F.nds, where n is the unit normal outside the sphere. Use the Gauss' Theorem.

Advanced Engineering Mathematics
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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Let F(x, y,z) = (-xe² +3x, -y e² + arctan(x² + z²), 2 e²) and S the portion of the sphere acoording to S= x, y, 2) : x² + y² + z² = 1, z > 0}. Calcule f F.nds, where n is the unit normal outside the sphere. Use the Gauss' Theorem.
Transcribed Image Text:Let F(x, y,z) = (-xe² +3x, -y e² + arctan(x² + z²), 2 e²) and S the portion of the sphere acoording to S= x, y, 2) : x² + y² + z² = 1, z > 0}. Calcule f F.nds, where n is the unit normal outside the sphere. Use the Gauss' Theorem.
Expert Solution
Step 1

Given that Fx,y,z=-xez+3x, -yez+arctanx2+z2, 2ez.

Also, S=x,y,z| x2+y2+z2=1, z≥0.

To determine the integral ∬SF·ηdS.

By Gauss' theorem, ∬SF·ηdS=∭D∇·F dV.

Here, D is a hemisphere of radius 1.

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