Explanation Given : The surface is the part of sphere x 2 + y 2 + z 2 = 1 where 1 2 ≤ z ≤ 3 2 , inward pointing normal Calculation : Step 1: We parametrize S by the following parametrization. Φ ( θ , ϕ ) = ( cos θ sin ϕ , sin θ sin ϕ , cos ϕ ) D : 0 ≤ θ ≤ 2 π , ϕ 0 ≤ ϕ ≤ ϕ 1 Step 2: The angles ϕ 0 and ϕ 1 are determined by cos ϕ 0 = 3 2 ⇒ ϕ 0 = π 6 cos ϕ 1 = 1 2 ⇒ ϕ 1 = π 3 Step 3: The normal vector pointing to the inside of the sphere is n = T θ × T ϕ = − sin ϕ ( c o s θ sin ϕ , sin θ sin ϕ , c o s ϕ ) . Also, that for π 6 ≤ ϕ ≤ π 3 , − sin ϕ cos ϕ < 0 therefore the z-component is negative and the normal points to the inside of the sphere. Step 3: Evaluate the dot product F • n ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Videos

Question

Chapter 17.5, Problem 8E

To determine

To calculate: The value of $\int \int_{S} F . d s$ for $F = 〈 x, y, z 〉,$ over the part of sphere $x^{2} + y^{2} + z^{2} = 1$ where $\frac{1}{2} \leq z \leq \frac{\sqrt{3}}{2}$ , inward pointing normal.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 17.5, Problem 7E

Chapter 17.5, Problem 9E

Students have asked these similar questions

Graphically, explain the various forms of linear functions

2. Find the Bezier surface equation using the 9 control points shown below. Use the u and v directions shown. It is required to show all the calculation processes for finding Bernstein polynomials. Find the surface tangent, twist and normal vectors at point u=0.5 and v=0.5. (40 points) y 10 9 8 7 6 5 4 3 2 Poo и 1 1 2 3 4 5 6 7 8 9 10 10 X

A cable runs along the wall from C to P at a cost of $24 per meter, and straight from P to M at a cost of $26 per meter. If M is 10 meters from the nearest point A on the wall where P lies, and A is 72 meters from C, find the distance from C to P such that the cost of installing the cable is minimized and find this cost. C 72 P A 10 M

Chapter 17 Solutions

Calculus - Standalone book

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

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 value of ∫ ∫ S F . d s for F = 〈 x , y , z 〉 , over the part of sphere x 2 + y 2 + z 2 = 1 where 1 2 ≤ z ≤ 3 2 , inward pointing normal.

Solution Summary: The author explains how to calculate the value of S F. d s for x, y, and z over the part of sphere.

Videos

To calculate: The value of $\int \int_{S} F . d s$ for $F = 〈 x, y, z 〉,$ over the part of sphere $x^{2} + y^{2} + z^{2} = 1$ where $\frac{1}{2} \leq z \leq \frac{\sqrt{3}}{2}$ , inward pointing normal.

Want to see the full answer?

Chapter 17 Solutions

The value of ∫ ∫ S F . d s for F = 〈 x , y , z 〉 , over the part of sphere x 2 + y 2 + z 2 = 1 where 1 2 ≤ z ≤ 3 2 , inward pointing normal.

Solution Summary: The author explains how to calculate the value of S F. d s for x, y, and z over the part of sphere.

Videos

To calculate: The value of ∫∫SF.ds for F=〈x,y,z〉, over the part of sphere x2+y2+z2=1 where 12≤z≤32, inward pointing normal.

Want to see the full answer?

Chapter 17 Solutions

To calculate: The value of $\int \int_{S} F . d s$ for $F = 〈 x, y, z 〉,$ over the part of sphere $x^{2} + y^{2} + z^{2} = 1$ where $\frac{1}{2} \leq z \leq \frac{\sqrt{3}}{2}$ , inward pointing normal.