Let F(x, y,z) = (-xe² +3x, -y e² + arctan(x² + z²), 2 e²) and S the portion of the sphereacoording 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.

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…

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.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter7: Locus And Concurrence

Section7.2: Concurrence Of Lines

Problem 7E: Which lines or line segments or rays must be drawn or constructed in a triangle to locate its a...

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Question

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 $F (x, y, z) = (- x e^{z} + 3 x, - y e^{z} + \arctan (x^{2} + z^{2}), 2 e^{z})$ .

Also, $S = \{(x, y, z) | x^{2} + y^{2} + z^{2} = 1, z \geq 0\}$ .

To determine the integral $\iint_{S} F \cdot η d S$ .

By Gauss' theorem, $\iint_{S} F \cdot η d S = ∭_{D} \nabla \cdot F d V$ .

Here, D is a hemisphere of radius 1.

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