Explanation Divergence Theorem says about the relation between surface and volume integral. Let S be a closed smooth surface, which is boundary of V, with normal pointing outwards. Let F = 〈 x , y , z 〉 on a region containing V, then the flux is equal to triple integral of div F over the solid region V. ∬ S F ⋅ d S = ∭ V d i v ( F ) d V Given: F ( x , y , z ) = 〈 e z 2 , 2 y + sin ( x 2 z ) , 4 z + x 2 + 9 y 2 〉 , S is the region x 2 + y 2 ≤ z ≤ 8 − x 2 − y 2 . Formula Used: ∬ S F ⋅ d S = ∭ V d i v ( F ) d V Calculation: First we have to calculate the div F and we get: d i v ( F ) = ∂ ∂ x ( e z 2 ) + ∂ ∂ y ( 2 y + sin ( x 2 z ) ) + ∂ ∂ z ( 4 z + x 2 + 9 y 2 ) = 0 + 2 + 4 = 6 ∬ S F ⋅ d S = ∭ V ( 6 ) d x d y d z It is given that x 2 + y 2 ≤ z ≤ 8 − x 2 − y 2 Now, change the variables x , y and z in r , z and θ because and we get the polar coordinates: x = r cos θ , y = r sin θ , z = z and d V = r d z d r d θ x 2 + y 2 = r 2 So, the triple integral will become: ∬ S F ⋅ d S = ∭ V ( 6 ) r d z d r d θ So, the region can be written as: x 2 + y 2 ≤ z ≤ 8 − x 2 − y 2 r 2 ≤ z ≤ 8 − r 2 The intersection of the curve is: x 2 + y 2 = 8 − x 2 − y 2 2 x 2 + 2 y 2 = 8 x 2 + y 2 = 4 x 2 + y 2 = 2 2 So, the intersection of the curves is a circle of radius 2 centered at the origin...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Videos

Question

Chapter 18.3, Problem 16E

To determine

To calculate:

The flux $\iint_{S} F \cdot d S$ by using the Divergence Theorem.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 18.3, Problem 15E

Chapter 18.3, Problem 17E

Students have asked these similar questions

i need help please

(#1) Consider the solid bounded below by z = x² and above by z = 4-y². If we were to project this solid down onto the xy-plane, you should be able to use algebra to determine the 2D region R in the xy-plane for the purposes of integration. Which ONE of these limite of integration would correctly describe R? (a) y: x24x: -22 - (b) y: 22 x: 04-y² (c) y: -√√4-x2. →√√4x²x: −2 → 2 (d) z: 24-y² y: -2 → 2 (e) None of the above

X MindTap - Cenxxxx Answered: tat "X A 26308049 X 10 EKU-- SP 25: X E DNA Sequenc X b/ui/evo/index.html?elSBN=9780357038406&id=339416021&snapshotid=877369& GE MINDTAP , Limits, and the Derivative 40. Answer 5 4-5 t-10 5 f(x) = 2x - 4 if x ≤0 if x 0 10 ++ -4-3-2-1 f(x) = MacBook Pro Search or type URL 5 1234 x² +1 if x = 0 if x = 0 +

Chapter 18 Solutions

Calculus - Standalone book

Ch. 18.1 - Prob. 1PQ Ch. 18.1 - Prob. 2PQ Ch. 18.1 - Prob. 3PQ Ch. 18.1 - Prob. 4PQ Ch. 18.1 - Prob. 1E Ch. 18.1 - Prob. 2E Ch. 18.1 - Prob. 3E Ch. 18.1 - Prob. 4E Ch. 18.1 - Prob. 5E Ch. 18.1 - Prob. 6E

Ch. 18.1 - Prob. 7E Ch. 18.1 - Prob. 8E Ch. 18.1 - Prob. 9E Ch. 18.1 - Prob. 10E Ch. 18.1 - Prob. 11E Ch. 18.1 - Prob. 12E Ch. 18.1 - Prob. 13E Ch. 18.1 - Prob. 14E Ch. 18.1 - Prob. 15E Ch. 18.1 - Prob. 16E Ch. 18.1 - Prob. 17E Ch. 18.1 - Prob. 18E Ch. 18.1 - Prob. 19E Ch. 18.1 - Prob. 20E Ch. 18.1 - Prob. 21E Ch. 18.1 - Prob. 22E Ch. 18.1 - Prob. 23E Ch. 18.1 - Prob. 24E Ch. 18.1 - Prob. 25E Ch. 18.1 - Prob. 26E Ch. 18.1 - Prob. 27E Ch. 18.1 - Prob. 28E Ch. 18.1 - Prob. 29E Ch. 18.1 - Prob. 30E Ch. 18.1 - Prob. 31E Ch. 18.1 - Prob. 32E Ch. 18.1 - Prob. 33E Ch. 18.1 - Prob. 34E Ch. 18.1 - Prob. 35E Ch. 18.1 - Prob. 36E Ch. 18.1 - Prob. 37E Ch. 18.1 - Prob. 38E Ch. 18.1 - Prob. 39E Ch. 18.1 - Prob. 40E Ch. 18.1 - Prob. 41E Ch. 18.1 - Prob. 42E Ch. 18.1 - Prob. 43E Ch. 18.1 - Prob. 44E Ch. 18.1 - Prob. 45E Ch. 18.1 - Prob. 46E Ch. 18.1 - Prob. 47E Ch. 18.2 - Prob. 1PQ Ch. 18.2 - Prob. 2PQ Ch. 18.2 - Prob. 3PQ Ch. 18.2 - Prob. 4PQ Ch. 18.2 - Prob. 5PQ Ch. 18.2 - Prob. 1E Ch. 18.2 - Prob. 2E Ch. 18.2 - Prob. 3E Ch. 18.2 - Prob. 4E Ch. 18.2 - Prob. 5E Ch. 18.2 - Prob. 6E Ch. 18.2 - Prob. 7E Ch. 18.2 - Prob. 8E Ch. 18.2 - Prob. 9E Ch. 18.2 - Prob. 10E Ch. 18.2 - Prob. 11E Ch. 18.2 - Prob. 12E Ch. 18.2 - Prob. 13E Ch. 18.2 - Prob. 14E Ch. 18.2 - Prob. 15E Ch. 18.2 - Prob. 16E Ch. 18.2 - Prob. 17E Ch. 18.2 - Prob. 18E Ch. 18.2 - Prob. 19E Ch. 18.2 - Prob. 20E Ch. 18.2 - Prob. 21E Ch. 18.2 - Prob. 22E Ch. 18.2 - Prob. 23E Ch. 18.2 - Prob. 24E Ch. 18.2 - Prob. 25E Ch. 18.2 - Prob. 26E Ch. 18.2 - Prob. 27E Ch. 18.2 - Prob. 28E Ch. 18.2 - Prob. 29E Ch. 18.2 - Prob. 30E Ch. 18.2 - Prob. 31E Ch. 18.2 - Prob. 32E Ch. 18.2 - Prob. 33E Ch. 18.2 - Prob. 34E Ch. 18.2 - Prob. 35E Ch. 18.2 - Prob. 36E Ch. 18.3 - Prob. 1PQ Ch. 18.3 - Prob. 2PQ Ch. 18.3 - Prob. 3PQ Ch. 18.3 - Prob. 4PQ Ch. 18.3 - Prob. 5PQ Ch. 18.3 - Prob. 1E Ch. 18.3 - Prob. 2E Ch. 18.3 - Prob. 3E Ch. 18.3 - Prob. 4E Ch. 18.3 - Prob. 5E Ch. 18.3 - Prob. 6E Ch. 18.3 - Prob. 7E Ch. 18.3 - Prob. 8E Ch. 18.3 - Prob. 9E Ch. 18.3 - Prob. 10E Ch. 18.3 - Prob. 11E Ch. 18.3 - Prob. 12E Ch. 18.3 - Prob. 13E Ch. 18.3 - Prob. 14E Ch. 18.3 - Prob. 15E Ch. 18.3 - Prob. 16E Ch. 18.3 - Prob. 17E Ch. 18.3 - Prob. 18E Ch. 18.3 - Prob. 19E Ch. 18.3 - Prob. 20E Ch. 18.3 - Prob. 21E Ch. 18.3 - Prob. 22E Ch. 18.3 - Prob. 23E Ch. 18.3 - Prob. 24E Ch. 18.3 - Prob. 25E Ch. 18.3 - Prob. 26E Ch. 18.3 - Prob. 27E Ch. 18.3 - Prob. 28E Ch. 18.3 - Prob. 29E Ch. 18.3 - Prob. 30E Ch. 18.3 - Prob. 31E Ch. 18.3 - Prob. 32E Ch. 18.3 - Prob. 33E Ch. 18.3 - Prob. 34E Ch. 18.3 - Prob. 35E Ch. 18.3 - Prob. 36E Ch. 18.3 - Prob. 37E Ch. 18.3 - Prob. 38E Ch. 18.3 - Prob. 39E Ch. 18.3 - Prob. 40E Ch. 18.3 - Prob. 41E Ch. 18.3 - Prob. 42E Ch. 18 - Prob. 1CRE Ch. 18 - Prob. 2CRE Ch. 18 - Prob. 3CRE Ch. 18 - Prob. 4CRE Ch. 18 - Prob. 5CRE Ch. 18 - Prob. 6CRE Ch. 18 - Prob. 7CRE Ch. 18 - Prob. 8CRE Ch. 18 - Prob. 9CRE Ch. 18 - Prob. 10CRE Ch. 18 - Prob. 11CRE Ch. 18 - Prob. 12CRE Ch. 18 - Prob. 13CRE Ch. 18 - Prob. 14CRE Ch. 18 - Prob. 15CRE Ch. 18 - Prob. 16CRE Ch. 18 - Prob. 17CRE Ch. 18 - Prob. 18CRE Ch. 18 - Prob. 19CRE Ch. 18 - Prob. 20CRE Ch. 18 - Prob. 21CRE Ch. 18 - Prob. 22CRE Ch. 18 - Prob. 23CRE Ch. 18 - Prob. 24CRE Ch. 18 - Prob. 25CRE Ch. 18 - Prob. 26CRE Ch. 18 - Prob. 27CRE Ch. 18 - Prob. 28CRE Ch. 18 - Prob. 29CRE Ch. 18 - Prob. 30CRE Ch. 18 - Prob. 31CRE Ch. 18 - Prob. 32CRE Ch. 18 - Prob. 33CRE Ch. 18 - Prob. 34CRE Ch. 18 - Prob. 35CRE Ch. 18 - Prob. 36CRE Ch. 18 - Prob. 37CRE Ch. 18 - Prob. 38CRE Ch. 18 - Prob. 39CRE Ch. 18 - Prob. 40CRE Ch. 18 - Prob. 41CRE

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The flux ∬ S F ⋅ d S by using the Divergence Theorem.

Solution Summary: The author calculates the flux by using the Divergence Theorem. Let S be a closed smooth surface, with normal pointing outwards.

Videos

To calculate:

The flux $\iint_{S} F \cdot d S$ by using the Divergence Theorem.

Want to see the full answer?

Chapter 18 Solutions

The flux ∬ S F ⋅ d S by using the Divergence Theorem.

Solution Summary: The author calculates the flux by using the Divergence Theorem. Let S be a closed smooth surface, with normal pointing outwards.

Videos

To calculate: The flux ∬SF⋅dS by using the Divergence Theorem.

Want to see the full answer?

Chapter 18 Solutions

To calculate:

The flux $\iint_{S} F \cdot d S$ by using the Divergence Theorem.