Calculus: Early Transcendental Functions
7th Edition
ISBN: 9781337552516
Author: Ron Larson, Bruce H. Edwards
Publisher: Cengage Learning
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Chapter 15.6, Problem 4E
To determine
The physical interpretation of the flux of F across S and how to calculate it.
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Calculus 11th Edition - Ron Larson
Chapter 3.9 - Differentials
Find the tangent line approximationT to the graph of f at the given point. Thencomplete the table.
Heat transfer Fourier’s Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = -k∇T, which means that heat energy flows from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/(m-s-K). A temperature function for a region D is given. Find the net outward heat flux ∫∫S F ⋅ n dS = -k∫∫S ∇T ⋅ n dS across the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume k = 1.
T(x, y, z) = 100e-x2 - y2 - z2; D is the sphere of radius a centered at the origin.
Heat transfer Fourier’s Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = -k∇T, which means that heat energy flows from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/(m-s-K). A temperature function for a region D is given. Find the net outward heat flux ∫∫S F ⋅ n dS = -k∫∫S ∇T ⋅ n dS across the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume k = 1.
T(x, y, z) = 100 + x2 + y2 + z2;;D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1}
Chapter 15 Solutions
Calculus: Early Transcendental Functions
Ch. 15.1 - Vector Field Define a vector field in the plane...Ch. 15.1 - Prob. 2ECh. 15.1 - Potential Function Describe how to find a...Ch. 15.1 - Prob. 4ECh. 15.1 - Prob. 5ECh. 15.1 - In Exercise 5-8, match the vector field with its...Ch. 15.1 - In Exercise 5-8, match the vector field with its...Ch. 15.1 - In Exercise 5-8, match the vector field with its...Ch. 15.1 - Prob. 9ECh. 15.1 - Prob. 10E
Ch. 15.1 - Prob. 11ECh. 15.1 - Prob. 12ECh. 15.1 - Sketching a Vector Field In Exercises 9-14, find F...Ch. 15.1 - Prob. 14ECh. 15.1 - Prob. 15ECh. 15.1 - Prob. 16ECh. 15.1 - Prob. 17ECh. 15.1 - Prob. 18ECh. 15.1 - Finding a Conservative Vector Field In Exercises...Ch. 15.1 - Prob. 20ECh. 15.1 - Prob. 21ECh. 15.1 - Prob. 22ECh. 15.1 - In Exercises 19-28, find the conservative vector...Ch. 15.1 - Prob. 24ECh. 15.1 - Prob. 25ECh. 15.1 - In Exercises 19-28, find the conservative vector...Ch. 15.1 - In Exercises 19-28, find the conservative vector...Ch. 15.1 - Prob. 28ECh. 15.1 - Prob. 29ECh. 15.1 - Prob. 30ECh. 15.1 - Prob. 31ECh. 15.1 - Prob. 32ECh. 15.1 - Prob. 33ECh. 15.1 - Prob. 34ECh. 15.1 - Prob. 35ECh. 15.1 - In Exercises 29-36, determine whether the vector...Ch. 15.1 - In Exercises 37-44, determine whether the vector...Ch. 15.1 - In Exercises 37-44, determine whether the vector...Ch. 15.1 - Prob. 39ECh. 15.1 - Prob. 40ECh. 15.1 - Prob. 41ECh. 15.1 - Prob. 42ECh. 15.1 - Prob. 43ECh. 15.1 - Prob. 44ECh. 15.1 - Find curl F for the vector field at the given...Ch. 15.1 - Find Curl F for the vector field at the point...Ch. 15.1 - Find Curl of the vector field F at the given point...Ch. 15.1 - Find Curl of the vector field F at the given point...Ch. 15.1 - Prob. 49ECh. 15.1 - Prob. 50ECh. 15.1 - Determine whether the vector field F is...Ch. 15.1 - Determine whether the vector field F is...Ch. 15.1 - Determine whether the vector field F is...Ch. 15.1 - Determine whether the vector field F is...Ch. 15.1 - Determine whether the vector field F is...Ch. 15.1 - Determine whether the vector field F is...Ch. 15.1 - Prob. 57ECh. 15.1 - Prob. 58ECh. 15.1 - Prob. 59ECh. 15.1 - Prob. 60ECh. 15.1 - Find the divergence of the vector field at the...Ch. 15.1 - Find the divergence of the vector field at the...Ch. 15.1 - Prob. 63ECh. 15.1 - Prob. 64ECh. 15.1 - Prob. 65ECh. 15.1 - Prob. 66ECh. 15.1 - Prob. 67ECh. 15.1 - Prob. 68ECh. 15.1 - Prob. 69ECh. 15.1 - In Exercise 69 and 70, find curl (FxG)=x(FxG)...Ch. 15.1 - Prob. 71ECh. 15.1 - In Exercises 71 and 72, curl (curlF)=x(xF)...Ch. 15.1 - Prob. 73ECh. 15.1 - Divergence of a Cross Product In Exercises 73 and...Ch. 15.1 - Prob. 75ECh. 15.1 - Prob. 76ECh. 15.1 - In parts (a) - (h), prove the property for vector...Ch. 15.1 - Prob. 78ECh. 15.2 - CONCEPT CHECK Line integral What is the physical...Ch. 15.2 - Prob. 2ECh. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Prob. 4ECh. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Prob. 6ECh. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Evaluating a Line Integral In Exercises 9-12, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 9-12, (a)...Ch. 15.2 - Prob. 11ECh. 15.2 - Prob. 12ECh. 15.2 - Prob. 13ECh. 15.2 - Evaluating a Line Integral In Exercises 13-16, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 13-16, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 13-16, (a)...Ch. 15.2 - Prob. 17ECh. 15.2 - Prob. 18ECh. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Prob. 23ECh. 15.2 - Mass In Exercises 23 and 24, find the total mass...Ch. 15.2 - Prob. 25ECh. 15.2 - Prob. 26ECh. 15.2 - Prob. 27ECh. 15.2 - Mass In Exercises 25-28, find the total mass of...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Prob. 31ECh. 15.2 - Prob. 32ECh. 15.2 - Prob. 33ECh. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Prob. 35ECh. 15.2 - Prob. 36ECh. 15.2 - Prob. 37ECh. 15.2 - Work In Exercises 37-42, find the work done by the...Ch. 15.2 - Prob. 39ECh. 15.2 - Work In Exercises 37-42, find the work done by the...Ch. 15.2 - Prob. 41ECh. 15.2 - Work In Exercises 37-42, find the work done by the...Ch. 15.2 - Work In Exercises 43-46, determine whether the...Ch. 15.2 - Work In Exercises 43-46, determine whether the...Ch. 15.2 - Prob. 45ECh. 15.2 - Prob. 46ECh. 15.2 - Prob. 47ECh. 15.2 - Prob. 48ECh. 15.2 - Prob. 49ECh. 15.2 - Prob. 50ECh. 15.2 - Prob. 51ECh. 15.2 - Prob. 52ECh. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Prob. 54ECh. 15.2 - Prob. 55ECh. 15.2 - Prob. 56ECh. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Prob. 59ECh. 15.2 - Prob. 60ECh. 15.2 - Prob. 61ECh. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Prob. 63ECh. 15.2 - Prob. 64ECh. 15.2 - Prob. 65ECh. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Prob. 67ECh. 15.2 - Prob. 68ECh. 15.2 - Prob. 69ECh. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Prob. 71ECh. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Prob. 73ECh. 15.2 - Prob. 74ECh. 15.2 - Prob. 75ECh. 15.2 - Prob. 76ECh. 15.2 - Prob. 77ECh. 15.2 - Prob. 78ECh. 15.2 - Work Find the work done by a person weighing 175...Ch. 15.2 - Prob. 80ECh. 15.2 - Prob. 81ECh. 15.2 - Prob. 82ECh. 15.2 - Prob. 83ECh. 15.2 - Prob. 84ECh. 15.2 - Prob. 85ECh. 15.2 - Prob. 86ECh. 15.2 - Prob. 87ECh. 15.3 - Fundamental Theorem of Line integrals Explain how...Ch. 15.3 - Prob. 2ECh. 15.3 - Prob. 3ECh. 15.3 - Prob. 4ECh. 15.3 - Prob. 5ECh. 15.3 - Prob. 6ECh. 15.3 - Prob. 7ECh. 15.3 - Line Integral of a Conservative Vector Field In...Ch. 15.3 - In Exercises 9-18, evaluate CFdr using the...Ch. 15.3 - Prob. 10ECh. 15.3 - In Exercises 9-18, evaluate CFdr using the...Ch. 15.3 - Prob. 12ECh. 15.3 - Prob. 13ECh. 15.3 - Prob. 14ECh. 15.3 - Prob. 15ECh. 15.3 - Prob. 16ECh. 15.3 - In Exercises 9-18, evaluate CFdr using the...Ch. 15.3 - Prob. 18ECh. 15.3 - Prob. 19ECh. 15.3 - Prob. 20ECh. 15.3 - Prob. 21ECh. 15.3 - Finding Work in a Conservative Force Field In...Ch. 15.3 - Prob. 23ECh. 15.3 - Evaluating a Line Integral In Exercises 23-32,...Ch. 15.3 - Evaluating a Line Integral In Exercises 23-32,...Ch. 15.3 - Evaluating a Line Integral In Exercises 23-32,...Ch. 15.3 - Prob. 27ECh. 15.3 - Evaluating a Line Integral In exercises 23-32,...Ch. 15.3 - Prob. 29ECh. 15.3 - Prob. 30ECh. 15.3 - Prob. 31ECh. 15.3 - Prob. 32ECh. 15.3 - Prob. 33ECh. 15.3 - Prob. 34ECh. 15.3 - Prob. 35ECh. 15.3 - Prob. 36ECh. 15.3 - Prob. 37ECh. 15.3 - Prob. 38ECh. 15.3 - Prob. 39ECh. 15.3 - Prob. 40ECh. 15.3 - Prob. 41ECh. 15.3 - Prob. 42ECh. 15.3 - Prob. 43ECh. 15.3 - Prob. 44ECh. 15.3 - Prob. 45ECh. 15.3 - Prob. 46ECh. 15.3 - Prob. 47ECh. 15.3 - Prob. 48ECh. 15.3 - Prob. 49ECh. 15.4 - CONCEPT CHECK Writing What does it mean for a...Ch. 15.4 - Prob. 2ECh. 15.4 - Prob. 3ECh. 15.4 - Prob. 4ECh. 15.4 - Prob. 5ECh. 15.4 - Verifying Greens Theorem In Exercises 5-8, verify...Ch. 15.4 - Prob. 7ECh. 15.4 - Verifying Greens Theorem In Exercises 5-8, verify...Ch. 15.4 - Prob. 9ECh. 15.4 - Prob. 10ECh. 15.4 - Evaluating a Line Integral Using Greens Theorem In...Ch. 15.4 - Evaluating a Line Integral Using Greens Theorem In...Ch. 15.4 - Evaluating a Line Integral Using Greens Theorem In...Ch. 15.4 - Evaluating a Line Integral Using Greens Theorem In...Ch. 15.4 - Prob. 15ECh. 15.4 - Prob. 16ECh. 15.4 - Prob. 17ECh. 15.4 - Prob. 18ECh. 15.4 - Evaluating a Line Integral Using Greens Theorem In...Ch. 15.4 - Prob. 20ECh. 15.4 - Prob. 21ECh. 15.4 - Evaluating a Line Integral Using Greens Theorem In...Ch. 15.4 - Evaluating a Line Integral Using Greens Theorem In...Ch. 15.4 - Evaluating a Line Integral Using Greens Theorem In...Ch. 15.4 - Work In Exercises 25-28, use Greens Theorem to...Ch. 15.4 - Prob. 26ECh. 15.4 - Prob. 27ECh. 15.4 - Work In Exercises 25-28, use Greens Theorem to...Ch. 15.4 - Prob. 29ECh. 15.4 - Prob. 30ECh. 15.4 - Prob. 31ECh. 15.4 - Prob. 32ECh. 15.4 - Prob. 33ECh. 15.4 - Using Greens Theorem to Verify a Formula In...Ch. 15.4 - Centroid In Exercises 35-38, use the results of...Ch. 15.4 - Prob. 36ECh. 15.4 - Prob. 37ECh. 15.4 - Prob. 38ECh. 15.4 - Prob. 39ECh. 15.4 - Area In Exercises 39-42, use the result of...Ch. 15.4 - Area In Exercises 39-42, use the result of...Ch. 15.4 - Area In Exercises 39-42, use the result of...Ch. 15.4 - Prob. 43ECh. 15.4 - Prob. 44ECh. 15.4 - Greens Theorem: Region with a Hole Let R be the...Ch. 15.4 - Greens Theorem: Region with a Hole Let R be the...Ch. 15.4 - Prob. 47ECh. 15.4 - Prob. 48ECh. 15.4 - Prob. 49ECh. 15.4 - Prob. 50ECh. 15.4 - Prob. 51ECh. 15.4 - Proof In Exercises 51 and 52, prove the identity,...Ch. 15.4 - Prob. 53ECh. 15.4 - Prob. 54ECh. 15.5 - Prob. 1ECh. 15.5 - Prob. 2ECh. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Prob. 4ECh. 15.5 - Prob. 5ECh. 15.5 - Prob. 6ECh. 15.5 - Prob. 7ECh. 15.5 - Prob. 8ECh. 15.5 - Prob. 9ECh. 15.5 - Prob. 10ECh. 15.5 - Prob. 11ECh. 15.5 - Prob. 12ECh. 15.5 - Prob. 13ECh. 15.5 - Prob. 14ECh. 15.5 - Graphing a Parametric Surface In Exercises 13-16,...Ch. 15.5 - Prob. 16ECh. 15.5 - Prob. 17ECh. 15.5 - Prob. 18ECh. 15.5 - Prob. 19ECh. 15.5 - Prob. 20ECh. 15.5 - Prob. 21ECh. 15.5 - Prob. 22ECh. 15.5 - Prob. 23ECh. 15.5 - Prob. 24ECh. 15.5 - Prob. 25ECh. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Prob. 27ECh. 15.5 - Prob. 28ECh. 15.5 - Prob. 29ECh. 15.5 - Prob. 30ECh. 15.5 - Prob. 31ECh. 15.5 - Prob. 32ECh. 15.5 - Prob. 33ECh. 15.5 - Prob. 34ECh. 15.5 - Prob. 35ECh. 15.5 - Prob. 36ECh. 15.5 - Prob. 37ECh. 15.5 - Prob. 38ECh. 15.5 - Prob. 39ECh. 15.5 - Prob. 40ECh. 15.5 - Prob. 41ECh. 15.5 - Prob. 42ECh. 15.5 - Prob. 43ECh. 15.5 - Prob. 44ECh. 15.5 - Prob. 45ECh. 15.5 - Prob. 46ECh. 15.5 - Prob. 47ECh. 15.5 - Prob. 48ECh. 15.5 - Prob. 49ECh. 15.5 - Prob. 50ECh. 15.5 - Prob. 51ECh. 15.5 - Prob. 52ECh. 15.5 - Prob. 53ECh. 15.5 - Hyperboloid Find a vector-valued function for the...Ch. 15.5 - Area Use a computer algebra system to graph one...Ch. 15.5 - Prob. 56ECh. 15.5 - Prob. 57ECh. 15.5 - Prob. 58ECh. 15.6 - Prob. 1ECh. 15.6 - Prob. 2ECh. 15.6 - Prob. 3ECh. 15.6 - Prob. 4ECh. 15.6 - Prob. 5ECh. 15.6 - Prob. 6ECh. 15.6 - Prob. 7ECh. 15.6 - Prob. 8ECh. 15.6 - Prob. 9ECh. 15.6 - Prob. 10ECh. 15.6 - Prob. 11ECh. 15.6 - Prob. 12ECh. 15.6 - Prob. 13ECh. 15.6 - Mass In Exercise 13-14, find the mass of the...Ch. 15.6 - Prob. 15ECh. 15.6 - Prob. 16ECh. 15.6 - Prob. 17ECh. 15.6 - Prob. 18ECh. 15.6 - Prob. 19ECh. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Prob. 21ECh. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Prob. 24ECh. 15.6 - Prob. 25ECh. 15.6 - Prob. 26ECh. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Prob. 28ECh. 15.6 - Prob. 29ECh. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Prob. 31ECh. 15.6 - Prob. 32ECh. 15.6 - Prob. 33ECh. 15.6 - Prob. 34ECh. 15.6 - Prob. 35ECh. 15.6 - Prob. 36ECh. 15.6 - Prob. 37ECh. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Prob. 39ECh. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Prob. 41ECh. 15.6 - Prob. 42ECh. 15.6 - Prob. 43ECh. 15.7 - CONCEPT CHECK Using Different Methods Suppose that...Ch. 15.7 - Classifying a Point in a Vector Field How do you...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 3-8,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 3-8,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 3-8,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 3-8,...Ch. 15.7 - Prob. 7ECh. 15.7 - Verifying the Divergence Theorem In Exercises 3-8,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Prob. 13ECh. 15.7 - Prob. 14ECh. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Prob. 22ECh. 15.7 - Prob. 23ECh. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - EXPLORING CONCEPTS Closed Surface What is the...Ch. 15.7 - Prob. 26ECh. 15.7 - Prob. 27ECh. 15.7 - Prob. 28ECh. 15.7 - Prob. 29ECh. 15.7 - Prob. 30ECh. 15.7 - Prob. 31ECh. 15.7 - Prob. 32ECh. 15.8 - Prob. 1ECh. 15.8 - Prob. 2ECh. 15.8 - Prob. 3ECh. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokes Theorem In Exercises 3-6, verify...Ch. 15.8 - Using Stokess Theorem In Exercises 7-16, use...Ch. 15.8 - Prob. 8ECh. 15.8 - Using Stokess Theorem In Exercises 7-16, use...Ch. 15.8 - Prob. 10ECh. 15.8 - Prob. 11ECh. 15.8 - Using Stokess Theorem In Exercises 7-16, use...Ch. 15.8 - Using Stokess Theorem In Exercises 7-16, use...Ch. 15.8 - Prob. 14ECh. 15.8 - Prob. 15ECh. 15.8 - Prob. 16ECh. 15.8 - Prob. 17ECh. 15.8 - Prob. 18ECh. 15.8 - Prob. 19ECh. 15.8 - Prob. 20ECh. 15.8 - Prob. 21ECh. 15 - Sketching a Vector Field In Exercises 1 and 2,...Ch. 15 - Sketching a Vector Field In Exercises 1 and 2,...Ch. 15 - Prob. 3RECh. 15 - Prob. 4RECh. 15 - Prob. 5RECh. 15 - Prob. 6RECh. 15 - Prob. 7RECh. 15 - Prob. 8RECh. 15 - Prob. 9RECh. 15 - Prob. 10RECh. 15 - Prob. 11RECh. 15 - Prob. 12RECh. 15 - Prob. 13RECh. 15 - Prob. 14RECh. 15 - Prob. 15RECh. 15 - Prob. 16RECh. 15 - Prob. 17RECh. 15 - Prob. 18RECh. 15 - Prob. 19RECh. 15 - Prob. 20RECh. 15 - Prob. 21RECh. 15 - Prob. 22RECh. 15 - Prob. 23RECh. 15 - Prob. 24RECh. 15 - Prob. 25RECh. 15 - Prob. 26RECh. 15 - Prob. 27RECh. 15 - Prob. 28RECh. 15 - Prob. 29RECh. 15 - Prob. 30RECh. 15 - Prob. 31RECh. 15 - Prob. 32RECh. 15 - Prob. 33RECh. 15 - Prob. 34RECh. 15 - Prob. 35RECh. 15 - Prob. 36RECh. 15 - Prob. 37RECh. 15 - Prob. 38RECh. 15 - Prob. 39RECh. 15 - Prob. 40RECh. 15 - Prob. 41RECh. 15 - Prob. 42RECh. 15 - Prob. 43RECh. 15 - Lateral Surface Area In Exercises 43 and44, find...Ch. 15 - Prob. 45RECh. 15 - Prob. 46RECh. 15 - Prob. 47RECh. 15 - Prob. 48RECh. 15 - Using the Fundamental Theorem of line Integrals In...Ch. 15 - Prob. 50RECh. 15 - Prob. 51RECh. 15 - Prob. 52RECh. 15 - Prob. 53RECh. 15 - Prob. 54RECh. 15 - Prob. 55RECh. 15 - Prob. 56RECh. 15 - Prob. 57RECh. 15 - Prob. 58RECh. 15 - Work In Exercises 59 and 60, use Greens Theorem to...Ch. 15 - Work In Exercises 25-28, use Greens Theorem to...Ch. 15 - Prob. 61RECh. 15 - Prob. 62RECh. 15 - Prob. 63RECh. 15 - Prob. 64RECh. 15 - Prob. 65RECh. 15 - Prob. 66RECh. 15 - Prob. 67RECh. 15 - Prob. 68RECh. 15 - Prob. 69RECh. 15 - Prob. 70RECh. 15 - Prob. 71RECh. 15 - Prob. 72RECh. 15 - Prob. 73RECh. 15 - Prob. 74RECh. 15 - Prob. 75RECh. 15 - Prob. 76RECh. 15 - Prob. 77RECh. 15 - Prob. 78RECh. 15 - Prob. 79RECh. 15 - Prob. 80RECh. 15 - Prob. 81RECh. 15 - Prob. 82RECh. 15 - Using Stokess Theorem In Exercises 83 and 84, use...Ch. 15 - Prob. 84RECh. 15 - Prob. 85RECh. 15 - Prob. 86RECh. 15 - Heat Flux Consider a single heat source located at...Ch. 15 - Prob. 2PSCh. 15 - Prob. 3PSCh. 15 - Moments of Inertia Find the moments of inertia for...Ch. 15 - Prob. 5PSCh. 15 - Prob. 6PSCh. 15 - Prob. 7PSCh. 15 - Prob. 8PSCh. 15 - Prob. 9PSCh. 15 - Prob. 10PSCh. 15 - Area and Work How does the area of the ellipse...Ch. 15 - Prob. 12PS
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- Hooke’s Law In Exercises 65-68, use Hooke’s Law, which states that the distance a spring stretches (or compresses) from its natural, or equilibrium, length varies directly as the applied force on the spring. The coiled spring of a toy supports the weight of a child. The weight of a 25-pound child compresses the spring a distance of 1.9 inches. The toy does not work properly when a weight compresses the spring more than 3 inches. What is the maximum weight for which the toy works properly?arrow_forwardMeaning Of Rate Change: What is the common term for the rate of change of each of the following phenomenon? a. Directed distance as a function of time. b. Velocity as a function of time. c. Tax due as a function of income. d Profit as a function of dollars invested.arrow_forwardFrictional Force The frictional force F between the tires and the road required to keep a car on a curved section of a highway is directly proportional to the square of the speed s of the car. If the speed of the car is doubled, the force will change by what factor?arrow_forward
- Physics Experiment Consider a physics laboratory experiment designed to determine an unknown mass. A flexible metal meter stick is clamped to a table with 50 centimeters overhanging the edge (see figure). Known masses Mranging from 200grams to 2000grams are attached to the end of the meter stick. For each mass, the meter stick is displaced vertically and then allowed to oscillate. The average time t(in seconds) of one oscillation for each mass is recorded in the table. A model for the data that can be used to predict the time of one oscillation is t=38M+16,96510(M+5000). (a) Use this model to create a table showing the predicted time for each of the masses shown in the table above. (b) Compare the predicted times with the experimental times. What can you conclude? (c) Use the model to approximate the mass of an object for which t=1.056seconds.arrow_forward5-6 Yes or No? If No, give a reason aIs the average rate of change of a function between x=a and x=b the slope of the secant line through (a,f(a)) and (b,f(b))?arrow_forwardHeat transfer Fourier’s Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = -k∇T, which means that heat energy flows from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/(m-s-K). A temperature function for a region D is given. Find the net outward heat flux ∫∫S F ⋅ n dS = -k∫∫S ∇T ⋅ n dS across the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume k = 1. T(x, y, z) = 100 + x + 2y + z;D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1}arrow_forward
- Heat transfer Fourier’s Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = -k∇T, which means that heat energy flows from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/(m-s-K). A temperature function for a region D is given. Find the net outward heat flux ∫∫S F ⋅ n dS = -k∫∫S ∇T ⋅ n dS across the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume k = 1. T(x, y, z) = 100 + e-z;D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1}arrow_forwardFourier's Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature: that is, F=−k∇T, which means that heat energy flows from hot regions to cold regions. The constant k is called the conductivity, which has metric units ofJ/m-s-K or W/m-K. A temperature function T for a region D is given below. Find the net outward heat flux. A temperature function for a region D is given below Find the net outward heat flux dS∫∫S F•n dS=−k∫∫S ∇T•n dS across the boundary S of D. It may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k=1. T(x,y,z)=70 e-x^2-y^2-z^2 D is the sphere of radius a centered at the origin.arrow_forward
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