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8. This is a counter-intuitive example! Consider the vector field F(2,1,2) = ( 1 I y Z (x² + y² + z²) ³/2' (x² + y²+22)³/2 + (x² + y² + 2²)³/2 (a) Calculate div F. Are there any points in R3 where div F is not defined? I (b) Note that if r is the position vector for the point (x, y, z), then this vector field is F(r) = ||||3 Sketch a little bit of this vector field near the point (1,2,2). Starting at (1, 2, 2), as x, y, and z increase, will ||F|| increase or decrease? Starting at (1,2,2), as x, y, and z decrease, will ||F|| increase or decrease? Imagine a small sphere centered at the point (1,2,2), and consider the flow of F through this sphere. Explain why div F = 0 at this point.

8. This is a counter-intuitive example! Consider the vector field F(2,1,2) = ( 1 I y Z (x² + y² + z²) ³/2' (x² + y²+22)³/2 + (x² + y² + 2²)³/2 (a) Calculate div F. Are there any points in R3 where div F is not defined? I (b) Note that if r is the position vector for the point (x, y, z), then this vector field is F(r) = ||||3 Sketch a little bit of this vector field near the point (1,2,2). Starting at (1, 2, 2), as x, y, and z increase, will ||F|| increase or decrease? Starting at (1,2,2), as x, y, and z decrease, will ||F|| increase or decrease? Imagine a small sphere centered at the point (1,2,2), and consider the flow of F through this sphere. Explain why div F = 0 at this point.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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Please don't forget to sketch a portion of the vector field for part b).

8. This is a counter-intuitive example! Consider the vector field F(2,1,2) = ( 1 I y Z (x² + y² + z²) ³/2' (x² + y²+22)³/2 + (x² + y² + 2²)³/2 (a) Calculate div F. Are there any points in R3 where div F is not defined? I (b) Note that if r is the position vector for the point (x, y, z), then this vector field is F(r) = ||||3 Sketch a little bit of this vector field near the point (1,2,2). Starting at (1, 2, 2), as x, y, and z increase, will ||F|| increase or decrease? Starting at (1,2,2), as x, y, and z decrease, will ||F|| increase or decrease? Imagine a small sphere centered at the point (1,2,2), and consider the flow of F through this sphere. Explain why div F = 0 at this point.
Transcribed Image Text:8. This is a counter-intuitive example! Consider the vector field F(2,1,2) = ( 1 I y Z (x² + y² + z²) ³/2' (x² + y²+22)³/2 + (x² + y² + 2²)³/2 (a) Calculate div F. Are there any points in R3 where div F is not defined? I (b) Note that if r is the position vector for the point (x, y, z), then this vector field is F(r) = ||||3 Sketch a little bit of this vector field near the point (1,2,2). Starting at (1, 2, 2), as x, y, and z increase, will ||F|| increase or decrease? Starting at (1,2,2), as x, y, and z decrease, will ||F|| increase or decrease? Imagine a small sphere centered at the point (1,2,2), and consider the flow of F through this sphere. Explain why div F = 0 at this point.
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