Explanation Concept used: The divergence theorem states that, Let S be a smooth, orientable surface that encloses a solid region R in ℝ 3 . If F → is a continuous vector field whose components have continuous partial derivatives in an open set containing R , then ∬ S F → ⋅ N → d s = ∭ R div F → d V ; where N → is the outward unit normal field for the surface S . Consider the vector function ∬ S F → ⋅ N → d s for F → = ( x + 10 y 2 z 2 ) i ^ + ( y + 10 x 2 z 2 ) j ^ + ( z + 10 x 2 y 2 ) k ^ and the hemispherical surface S : z = 1 − x 2 − y 2 together with the disk x 2 + y 2 ≤ 1 in the x y − plane

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

See similar textbooks

Concept explainers

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic…

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along…

Triple Integral

Examples:

Videos

Question

Chapter 13.7, Problem 36PS

To determine

To calculate: The value of the surface integral ∬SF→⋅N→ds for F→=(x+10y2z2)i^+(y+10x2z2)j^+(z+10x2y2)k^ and the hemispherical surface S:z=1−x2−y2 together with the disk x2+y2≤1 in the xy− plane.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.7, Problem 35PS

Chapter 13.7, Problem 37PS

Students have asked these similar questions

For the given function, find (a) the equation of the secant line through the points where x has the given values and (b) the equation of the tangent line when x has the first value. y= f(x) = x²+x; x=-1,x=2 a. Which of the following formulas can be used to find the slope of the secant line? ○ A. 2-(-1) f(2) f(-1) 2+(-1) C. 1(2)+(-1) The equation of the secant line is 1(2)+(-1) О в. 2+(-1) f(2)-(-1) D. 2-(-1)

please do not use chat gpt

Use technology to find f'(4), f'(16), f'(-5) for the given function when the derivative exists. f(x) = -2x² + +10x

Chapter 13 Solutions

Calculus

Ch. 13.1 - Prob. 1PS Ch. 13.1 - Prob. 2PS Ch. 13.1 - Prob. 3PS Ch. 13.1 - Prob. 4PS Ch. 13.1 - Prob. 5PS Ch. 13.1 - Prob. 6PS Ch. 13.1 - Prob. 7PS Ch. 13.1 - Prob. 8PS Ch. 13.1 - Prob. 9PS Ch. 13.1 - Prob. 10PS

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 the surface integral ∬ S F → ⋅ N → d s for F → = ( x + 10 y 2 z 2 ) i ^ + ( y + 10 x 2 z 2 ) j ^ + ( z + 10 x 2 y 2 ) k ^ and the hemispherical surface S : z = 1 − x 2 − y 2 together with the disk x 2 + y 2 ≤ 1 in the x y − plane.

Solution Summary: The author explains the divergence theorem, which states that S is a smooth, orientable surface that encloses the solid region R in R3.

Concept explainers

Videos

To calculate: The value of the surface integral ∬SF→⋅N→ds for F→=(x+10y2z2)i^+(y+10x2z2)j^+(z+10x2y2)k^ and the hemispherical surface S:z=1−x2−y2 together with the disk x2+y2≤1 in the xy− plane.

Want to see the full answer?

Chapter 13 Solutions