Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Chapter 6.6, Problem 6P
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Answer questions 8.3.3 and 8.3.4 respectively
8.3.4 .WP An article in Medicine and Science in Sports and
Exercise [“Electrostimulation Training Effects on the Physical Performance of Ice Hockey Players” (2005, Vol. 37, pp.
455–460)] considered the use of electromyostimulation (EMS) as
a method to train healthy skeletal muscle. EMS sessions consisted of 30 contractions (4-second duration, 85 Hz) and were carried
out three times per week for 3 weeks on 17 ice hockey players.
The 10-meter skating performance test showed a standard deviation of 0.09 seconds. Construct a 95% confidence interval of the
standard deviation of the skating performance test.
8.6.7 Consider the tire-testing data in Exercise 8.2.3. Compute a 95% tolerance interval on the life of the tires that has confidence level 95%. Compare the length of the tolerance interval with the length of the 95% CI on the population mean. Which interval is shorter? Discuss the difference in interpretation of these two intervals.
8.6.2 Consider the natural frequency of beams described in
Exercise 8.2.8. Compute a 90% prediction interval on the
diameter of the natural frequency of the next beam of this type
that will be tested. Compare the length of the prediction interval
with the length of the 90% CI on the population mean.
8.6.3 Consider the television tube brightness test described in
Exercise 8.2.7. Compute a 99% prediction interval on the brightness of the next tube tested. Compare the length of the prediction
interval with the length of the 99% CI on the population mean.
Chapter 6 Solutions
Mathematical Methods in the Physical Sciences
Ch. 6.3 - If A=2ijk,B=2i3j+k,C=j+k, find (AB)C,A(BC),(AB)C,...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - A force F=2i3j+k acts at the point (1,5,2). Find...Ch. 6.3 - Prob. 8PCh. 6.3 - Prob. 9PCh. 6.3 - In Figure 3.5, let r be another vector from O to...
Ch. 6.3 - Write out the twelve triple scalar products...Ch. 6.3 - (a) Simplify ( AB)2[(AB)B]A by using ( 3.9). (b)...Ch. 6.3 - Prove that the triple scalar product of (AB),(BC),...Ch. 6.3 - Prove the Jacobi identity: A(BC)+B(CA)+C(AB)=0....Ch. 6.3 - Prob. 15PCh. 6.3 - In the discussion of Figure 3.8, we found for the...Ch. 6.3 - Expand the triple product for a=(r) given in the...Ch. 6.3 - Two moving charged particles exert forces on each...Ch. 6.3 - The force F=i+3j+2k acts at the point (1,1,1). (a)...Ch. 6.3 - Prob. 20PCh. 6.4 - Verify equations (4.5) by writing out the...Ch. 6.4 - Let the position vector (with its tail at the...Ch. 6.4 - As in Problem 2, if the position vector of a...Ch. 6.4 - Prob. 4PCh. 6.4 - The position of a particle at time t is given by...Ch. 6.4 - The force acting on a moving charged particle in a...Ch. 6.4 - Sketch a figure and verify equation ( 4.12).Ch. 6.4 - In polar coordinates, the position vector of a...Ch. 6.4 - The angular momentum of a particle m is defined by...Ch. 6.4 - If V(t) is a vector function oft, find the...Ch. 6.6 - Find the gradient of w=x2y3z at (1,2,1).Ch. 6.6 - Starting from the point (1,1), in what direction...Ch. 6.6 - Find the derivative of xy2+yz at (1,1,2) in the...Ch. 6.6 - Find the derivative of zexcosy at (1,0,/3) in the...Ch. 6.6 - Find the gradient of =zsinyxz at the point...Ch. 6.6 - Find a vector normal to the surface x2+y2z=0 at...Ch. 6.6 - Find the direction of the line normal to the...Ch. 6.6 - (a) Find the directional derivative of =x2+sinyxz...Ch. 6.6 - (a) Given =x2y2z, find at (1,1,1). (b) Find the...Ch. 6.6 - For Problems 10 to 14, use a computer as needed to...Ch. 6.6 - For Problems 10 to 14, use a computer as needed to...Ch. 6.6 - For Problems 10 to 14, use a computer as needed to...Ch. 6.6 - For Problems 10 to 14, use a computer as needed to...Ch. 6.6 - For Problems 10 to 14, use a computer as needed to...Ch. 6.6 - Repeat Problem 14b for the following points and...Ch. 6.6 - Show by the Lagrange multiplier method that the...Ch. 6.6 - Find r, where r=x2+y2, using ( 6.7) and also using...Ch. 6.6 - As in Problem 17, find the following gradients in...Ch. 6.6 - As in Problem 17, find the following gradients in...Ch. 6.6 - As in Problem 17, find the following gradients in...Ch. 6.6 - Verify equation ( 6.8 ); that is, find f in...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Compute the divergence and the curl of each of the...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Calculate the Laplacian 2 of each of the following...Ch. 6.7 - Verify formulas (b), (c), (d), (g), (h), (i), (i),...Ch. 6.7 - For r=xi+yj+zk, evaluate (kr)Ch. 6.7 - For r=xi+yj+zk, evaluate rrCh. 6.7 - For r=xi+yj+zk, evaluate rrCh. 6.8 - Evaluate the line integral x2y2dx2xydy along each...Ch. 6.8 - Evaluate the line integral (x+2y)dx2xdy along each...Ch. 6.8 - Evaluate the line integral xydx+xdy from (0,0) to...Ch. 6.8 - Prob. 4PCh. 6.8 - Find the work done by the force F=x2yixy2j along...Ch. 6.8 - Prob. 6PCh. 6.8 - For the force field F=(y+z)i(x+z)j+(x+y)k, find...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Verify that each of the following force fields is...Ch. 6.8 - Given F1=2xi2yzjy2k and F2=yixj (a) Are these...Ch. 6.8 - Which, if either, of the two force fields...Ch. 6.8 - For the force field F=yi+xj+zk, calculate the work...Ch. 6.8 - Show that the electric field...Ch. 6.8 - For motion near the surface of the earth, we...Ch. 6.8 - Consider a uniform distribution of total mass m...Ch. 6.9 - Write out the equations corresponding to ( 9.3 )...Ch. 6.9 - In Problems 2 to 5 use Greens theorem [formula (...Ch. 6.9 - In Problems 2to5useGree n stheorem[formula(9.7)]...Ch. 6.9 - In Problems 2 to 5 use Greens theorem [formula (...Ch. 6.9 - In Problems 2 to 5 use Greens theorem [formula (...Ch. 6.9 - For a simple closed curve C in the plane show by...Ch. 6.9 - Use Problem 6 to show that the area inside the...Ch. 6.9 - Use Problem 6 to find the area inside the curve...Ch. 6.9 - Apply Greens theorem with P=0,Q=12x2 to the...Ch. 6.9 - Evaluate each of the following integrals in the...Ch. 6.9 - Evaluate each of the following integrals in the...Ch. 6.9 - Evaluate each of the following integrals in the...Ch. 6.10 - Evaluate both sides of ( 10.17) if V=r=ix+jy+kz,...Ch. 6.10 - Given V=x2i+y2j+z2k, integrate Vnd over the whole...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - Evaluate each of the integrals in Problems 3 to 8...Ch. 6.10 - If F=xi+yj, calculate Fnd over the part of the...Ch. 6.10 - Evaluate Vnd over the curved surface of the...Ch. 6.10 - Given that B= curl A, use the divergence theorem...Ch. 6.10 - A cylindrical capacitor consists of two long...Ch. 6.10 - Draw a figure similar to Figure 10.6 but with q...Ch. 6.10 - Obtain Coulombs law from Gausss law by considering...Ch. 6.10 - Suppose the density of a fluid varies from point...Ch. 6.10 - The following equations are variously known as...Ch. 6.11 - Do case (b) of Example 1 above.Ch. 6.11 - Given the vector A=x2y2i+2xyj. (a) Find A (b)...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Use either Stokes' theorem or the divergence...Ch. 6.11 - Vnd over the entire surface of the volume in the...Ch. 6.11 - (curlV)nd over the part of the surface z=9x29y2...Ch. 6.11 - Vnd over the entire surface of a cube in the first...Ch. 6.11 - Vdr around the circle (x2)2+(y3)2=9,z=0, where...Ch. 6.11 - (2xi2yj+5k)nd over the surface of a sphere of...Ch. 6.11 - (yixj+zk)dr around the circumference of the circle...Ch. 6.11 - cydx+zdy+xdz, where C is the curve of intersection...Ch. 6.11 - What is wrong with the following proof that there...Ch. 6.11 - Prob. 17PCh. 6.11 - Find vector fields A such that V=curlA for each...Ch. 6.11 - Find vector fields A such that V= curl A for each...Ch. 6.11 - Find vector fields A such that V=curlA for each...Ch. 6.11 - Find vector fields A such that V=curlA for each...Ch. 6.11 - Find vector fields A such that V=curlA for each...Ch. 6.12 - Prob. 1MPCh. 6.12 - If A and B are the diagonals of a parallelogram,...Ch. 6.12 - The force on a charge q moving with velocity...Ch. 6.12 - Prob. 4MPCh. 6.12 - Use Greens theorem (Section 9) to do Problem 8.2.Ch. 6.12 - Prob. 6MPCh. 6.12 - Let F=2i3j+k act at the point (5,1,3). (a) Find...Ch. 6.12 - Prob. 8MPCh. 6.12 - Let F=i5j+2k act at the point (2,1,0). Find the...Ch. 6.12 - Given u=xy+sinz, find (a) the gradient of u at...Ch. 6.12 - Given =z23xy, find (a) grad ; (b) the directional...Ch. 6.12 - Given u=xy+yz+zsinx, find (a) u at (0,1,2); (b)...Ch. 6.12 - Given =x2yz and the point P(3,4,1), find (a) at...Ch. 6.12 - If the temperature is T=x2xy+z2, find (a) the...Ch. 6.12 - Show that...Ch. 6.12 - Given F1=2xzi+yj+x2k and F2=yixj: (a) Which F, if...Ch. 6.12 - Find the value of Fdr along the circle x2+y2=2...Ch. 6.12 - Is F=yi+xzj+zk conservative? Evaluate Fdr from...Ch. 6.12 - Given F1=2yi+(z2x)j+(y+z)k,F2=yi+2xj: (a) Is F1...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...Ch. 6.12 - In Problems 20 to 31, evaluate each integral in...
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- Answer question S8 stepwisearrow_forwardAnswer questions 8.2.11 and 8.2.12 respectivelyarrow_forward8.4.2 An article in Knee Surgery, Sports Traumatology, Arthroscopy [“Arthroscopic Meniscal Repair with an Absorbable Screw: Results and Surgical Technique” (2005, Vol. 13, pp. 273–279)] showed that only 25 out of 37 tears (67.6%) located between 3 and 6 mm from the meniscus rim were healed. a. Calculate a two-sided 95% confidence interval on the proportion of such tears that will heal. b. Calculate a 95% lower confidence bound on the proportion of such tears that will heal. 8.4.3 An article in the Journal of the American Statistical Association [“Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling” (1990, Vol. 85, pp. 972–985)] measured the weight of 30 rats under experiment controls. Suppose that 12 were underweight rats. a. Calculate a 95% two-sided confidence interval on the true proportion of rats that would show underweight from the experiment. b. Using the point estimate of p obtained from the preliminary sample, what sample size is needed to be 95%…arrow_forward
- 8.4.8 Use the data from Exercise 8.4.2 to compute the two-sided Agresti-Coull CI on the proportion of tears that heal. Compare and discuss the relationship of this interval to the one computed in Exercise 8.4.2.arrow_forwardAnswer questions 8.3.7 and 8.4.1 respectivelyarrow_forwardDon't do 14. Please solve 19arrow_forward
- 8.4.7 Use the data from Exercise 8.4.5 to compute the two-sided Agresti-Coull CI on the proportion of digits read correctly. Compare and discuss the relationship of this interval to the one computed in Exercise 8.4.5.arrow_forward8.6.5 Consider the fuel rod enrichment data described in Exercise 8.2.11. Compute a 90% prediction interval on the enrichment of the next rod tested. Compare the length of the prediction interval with the length of the 99% CI on the population mean.arrow_forward8.4.4 The Arizona Department of Transportation wishes to survey state residents to determine what proportion of the population would like to increase statewide highway speed limits from 65 mph to 75 mph. How many residents does the department need to survey if it wants to be at least 99% confident that the sample proportion is within 0.05 of the true proportion? 8.4.5 The U.S. Postal Service (USPS) has used optical character recognition (OCR) since the mid-1960s. In 1983, USPS began deploying the technology to major post offices throughout the country (www.britannica.com). Suppose that in a random sample of 500 handwritten zip code digits, 466 were read correctly. a. Construct a 95% confidence interval for the true proportion of correct digits that can be automatically read. b. What sample size is needed to reduce the margin of error to 1%? c. How would the answer to part (b) change if you had to assume that the machine read only one-half of the digits correctly?arrow_forward
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