Mathematical Methods in the Physical Sciences
Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
bartleby

Concept explainers

Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the se…
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integratio…
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects th…
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
bartleby

Videos

Textbook Question
Book Icon
Chapter 4.8, Problem 10P

Repeat Problem 9 if the bottom is to be three times as thick as the sides.

An aquarium with rectangular sides and bottom (and no top) is to hold 5 gal. Find its proportions so that it will use the least amount of material.

Expert Solution & Answer
Check Mark

Want to see the full answer?

Check out a sample textbook solution
Blurred answer
Previous
Chapter 4.8, Problem 9P
Next
Chapter 4.8, Problem 11P

Chapter 4 Solutions

Mathematical Methods in the Physical Sciences

Ch. 4.1 - If u=x2/x2+y2, find u/x,u/y.Ch. 4.1 - Prob. 2PCh. 4.1 - If z=lnu2+v2+w2, find z/u,z/v,z/w.Ch. 4.1 - For w=x3y32xy+6, find 2w/ax2 and 2w/ay2 at the...Ch. 4.1 - For w=8x4+y42xy2, find 2w/x2 and 2w/y2 at the...Ch. 4.1 - For u=excosy, (a) verify that 2u/xy=2u/yx; (b)...Ch. 4.1 - Prob. 7PCh. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...
Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - Prob. 12PCh. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - Prob. 18PCh. 4.1 - Prob. 19PCh. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - Prob. 24PCh. 4.2 - Find the two-variable Maclaurin series for the...Ch. 4.2 - Find the two-variable Maclaurin series for the...Ch. 4.2 - Find the two-variable Maclaurin series for the...Ch. 4.2 - Find the two-variable Maclaurin series for the...Ch. 4.2 - Find the two-variable Maclaurin series for the...Ch. 4.2 - Find the two-variable Maclaurin series for the...Ch. 4.2 - Prob. 7PCh. 4.2 - Find the two-variable Maclaurin series for excosy...Ch. 4.3 - Consider a function f(x,y) which can be expanded...Ch. 4.4 - Use differentials to show that, for very large...Ch. 4.4 - Use differentials to show that, for large n and...Ch. 4.4 - The thin lens formula is 1i+1o=1f, where f is the...Ch. 4.4 - Do Problem 3 if i=12 when o=18, to find i if...Ch. 4.4 - Let R be the resistance of R1=25 ohms and R2=15...Ch. 4.4 - The acceleration of gravity can be found from the...Ch. 4.4 - Coulombs law for the force between two charges q1...Ch. 4.4 - About how much (in percent) does an error of 1 in...Ch. 4.4 - Show that the approximate relative error (df)/f of...Ch. 4.4 - A force of 500 nt is measured with a possible...Ch. 4.4 - Show how to make a quick estimate (to two decimal...Ch. 4.4 - As in Problem 11, estimate (2.05)2+(1.98)23.Ch. 4.4 - Without using a computer or a calculator, estimate...Ch. 4.4 - Estimate the change in f(x)=0xett2+0.51dt if x...Ch. 4.4 - For an ideal gas of N molecules, the number of...Ch. 4.4 - The operating equation for a synchrotron in the...Ch. 4.4 - Here are some other ways of obtaining the formula...Ch. 4.5 - Given z=xey,x=cosht,y=cost, find dz/dt.Ch. 4.5 - Prob. 2PCh. 4.5 - Given r=ep2q2,p=es,q=es, Find dr/ds.Ch. 4.5 - Given x=lnu2v2,u=t2,v=cost, find dx/dt.Ch. 4.5 - If we are given z=z(x,y) and y=y(x), show that the...Ch. 4.5 - Given z=(x+y)5,y=sin10x, find dz/dx.Ch. 4.5 - Given c=sin(ab),b=ae2a, find dc/da.Ch. 4.5 - Prove the statement just after (5.2), that dz...Ch. 4.6 - If pva=C (where a and C are constants), find dv/dp...Ch. 4.6 - If yexy=sinx find dy/dx and d2y/dx2 at (0,0).Ch. 4.6 - If xy=yx, find dy/dx at (2,4).Ch. 4.6 - If xey=yex, find dy/dx and d2y/dx2 for y1.Ch. 4.6 - If xy3yx3=6 is the equation of a curve, find the...Ch. 4.6 - In Problem 5 find d2y/dx2 at (1,2).Ch. 4.6 - If y3x2y=8 is the equation of a curve, find the...Ch. 4.6 - In Problem 7 find d2y/dx2 at (3,1).Ch. 4.6 - For the curve x2/3+y2/3=4, find the equations of...Ch. 4.6 - For the curve xey+yex=0, find the equation of the...Ch. 4.6 - In Problem 10, find y at the origin.Ch. 4.7 - If x=yz and y=2sin(y+z), find dx/dy and d2x/dy2.Ch. 4.7 - IfP=rcostandrsint2ter=0,finddP/dt.Ch. 4.7 - Ifz=xeyandx=cosht,y=coss,findz/sandz/t.Ch. 4.7 - Prob. 4PCh. 4.7 - Prob. 5PCh. 4.7 - Prob. 6PCh. 4.7 - If x=rcos and y=rsin, find (y/)r and (y/)x. Also...Ch. 4.7 - If xs2+yt2=1 and x2s+y2t=xy4, find...Ch. 4.7 - Verify (7.16) in three ways: (a) Differentiate...Ch. 4.7 - Prob. 10PCh. 4.7 - Finish Example 4 above, both for the general case...Ch. 4.7 - Ifw=x+ywithx3+xy+y3=sandx2y+xy2=t,findw/s,w/t.Ch. 4.7 - Prob. 13PCh. 4.7 - Ifu=x2+y2+xyzandx4+y4+z4=2x2y2z2+10, find (u/x)z...Ch. 4.7 - Prob. 15PCh. 4.7 - Let w=x2+xy+z2 (a) If x3+x=3t,y4+y=4t,z5+z=5t,...Ch. 4.7 - ...Ch. 4.7 - Ifm=a+bandn=a2+b2find(b/m)nand(m/b)a.Ch. 4.7 - Prob. 19PCh. 4.7 - Prob. 20PCh. 4.7 - Prob. 21PCh. 4.7 - If w=f(ax+by), show that bwxawy=0. Hint: Let...Ch. 4.7 - Prob. 23PCh. 4.7 - Prob. 24PCh. 4.7 - The formulas of this problem are useful in...Ch. 4.7 - Given f(x,y,z)=0 and g(x,y,z)=0, find a formula...Ch. 4.7 - Given u(x,y) and y(x,z), show that uxz=uxy+uyxyxz.Ch. 4.7 - Prob. 28PCh. 4.8 - Use the Taylor series about x=a to verify the...Ch. 4.8 - Using the two-variable Taylor series [say ( 2.7)]...Ch. 4.8 - Use the facts stated in Problem 2 to find the...Ch. 4.8 - Use the facts stated in Problem 2 to find the...Ch. 4.8 - Use the facts stated in Problem 2 to find the...Ch. 4.8 - Use the facts stated in Problem 2 to find the...Ch. 4.8 - Given z=yx2y2x2, show that z has neither a maximum...Ch. 4.8 - A roof gutter is to be made from a long strip of...Ch. 4.8 - An aquarium with rectangular sides and bottom (and...Ch. 4.8 - Repeat Problem 9 if the bottom is to be three...Ch. 4.8 - Find the most economical proportions for a tent as...Ch. 4.8 - Find the shortest distance from the origin to the...Ch. 4.8 - Given particles of masses m, 2m, and 3m at the...Ch. 4.8 - Repeat Problem 13 for masses m1,m2,m3 at...Ch. 4.8 - Find the point on the line through (1,0,0) and...Ch. 4.8 - To find the best straight line fit to a set of...Ch. 4.8 - Repeat Problem 16 for each of the following sets...Ch. 4.9 - What proportions will maximize the area shown in...Ch. 4.9 - What proportions will maximize the volume of a...Ch. 4.9 - Find the largest rectangular parallelepiped (box)...Ch. 4.9 - Find the largest box (with faces parallel to the...Ch. 4.9 - Find the point on 2x+3y+z11=0 for which 4x2+y2+z2...Ch. 4.9 - Prob. 6PCh. 4.9 - Repeat Problem 6 if the plane is ax+by+cz=d. A box...Ch. 4.9 - A point moves in the (x,y) plane on the line...Ch. 4.9 - Find the largest triangle that can be inscribed in...Ch. 4.9 - Complete Example 4 above.Ch. 4.9 - Find the shortest distance from the origin to the...Ch. 4.9 - Find the right triangular prism of given volume...Ch. 4.10 - Find the shortest distance from the origin to...Ch. 4.10 - Prob. 2PCh. 4.10 - Prob. 3PCh. 4.10 - Find the shortest distance from the origin to each...Ch. 4.10 - Find the shortest distance from the origin to each...Ch. 4.10 - Find the shortest distance from the origin to each...Ch. 4.10 - Prob. 7PCh. 4.10 - If the temperature at the point (x,y,z) is T=xyz,...Ch. 4.10 - The temperature T of the disk x2+y21 is given by...Ch. 4.10 - The temperature at a point (x,y,z) in the ball...Ch. 4.10 - The temperature of a rectangular plate bounded by...Ch. 4.10 - Find the largest and smallest values of the sum of...Ch. 4.10 - Find the largest and smallest values of the sum of...Ch. 4.10 - Do Example 3b using Lagrange multipliers.Ch. 4.11 - In the partial differential equation...Ch. 4.11 - As in Problem 1, solve 22zx2+2zxy102zy2=0 by...Ch. 4.11 - Prob. 3PCh. 4.11 - Verify the chain rule formulas Fx=Frrx+Fx, and...Ch. 4.11 - Solve equations (11.11) to get equations (11.12)....Ch. 4.11 - Reduce the equation x2d2ydx2+2xdydx5y=0 to a...Ch. 4.11 - Change the independent variable from x to by...Ch. 4.11 - Change the independent variable from x to u=2x in...Ch. 4.11 - If x=escost,y=essint, show that...Ch. 4.11 - Given du=Tdspdv, find a Legendre transformation...Ch. 4.11 - Given L(q,q) such that dL=pdq+pdq, find H(p,q) so...Ch. 4.11 - Using du in Problem 10, and the text method of...Ch. 4.11 - As in Problem 12, find three more Maxwell...Ch. 4.12 - Ify=0xsint2dt,finddy/dx.Ch. 4.12 - If s=uv1ettdt, find s/v and s/u and also their...Ch. 4.12 - If z=ainxcosxsinttdt, find dzdx.Ch. 4.12 - Use L'HĂ´pitals rule to evaluate...Ch. 4.12 - If u=xyxsinttdt, find ux,uy, and yx at x=/2,y=....Ch. 4.12 - Ifw=xy2x3ydulnu,findwx,wy,andyxatx=3,y=1.Ch. 4.12 - Ifuvet2dt=xanduv=y,finduxy,uy,andyxuatu=2,v=0.Ch. 4.12 - If0xes2ds=u,finddxdu.Ch. 4.12 - If y=0sinxtdt, find dy/dx (a) by evaluating the...Ch. 4.12 - Find dy/dx explicitly if y=01exu1udu.Ch. 4.12 - Find ddx3xx2(xt)dt by evaluating the integral...Ch. 4.12 - Findddxxx2duln(x+u).Ch. 4.12 - Findddx1/x2/xsinxttdt.Ch. 4.12 - Given that 0dxy2+x2=2y, differentiate with respect...Ch. 4.12 - Given that 0eaxsinkxdx=ka2+k2 differentiate with...Ch. 4.12 - In kinetic theory we have to evaluate integrals of...Ch. 4.12 - Complete Example 4 to obtain (12.10)....Ch. 4.12 - Showthatu(x,y)=yf(t)dt(xt)2+y2satisfiesuxx+uyy=0.Ch. 4.12 - Showthaty=0xf(u)sin(xu)dusatisfiesy+y=f(x).Ch. 4.12 - (a) Show that y=0xf(xt)dt satisfies (dy/dx)=f(x)....Ch. 4.13 - Prob. 1MPCh. 4.13 - (a) Given the point (2,1) in the (x,y) plane and...Ch. 4.13 - In Problems 3 to 6, assume that x, y and r, are...Ch. 4.13 - In Problems 3 to 6, assume that x,y and r, are...Ch. 4.13 - In Problems 3 to 6, assume that x, y and r, are...Ch. 4.13 - In Problems 3 to 6, assume that x, y and r, are...Ch. 4.13 - About how much (in percent) does an error of 1 in...Ch. 4.13 - Assume that the earth is a perfect sphere. Suppose...Ch. 4.13 - Ifz=xyand2x3+2y3=3t2,finddz/dt3x2+3y2=6t.Ch. 4.13 - Ifw=(rcos)rsin,findw/.Ch. 4.13 - If x2a2+y2b2=1, find dydx and d2ydx2 by implicit...Ch. 4.13 - Prob. 12MPCh. 4.13 - ...Ch. 4.13 - If w=f(x,s,t),s=2x+y,t=2xy, find (w/x)y in terms...Ch. 4.13 - Prob. 15MPCh. 4.13 - Ifz=1xfyx,provethatxzx+yzy+z=0.Ch. 4.13 - Find the shortest distance from the origin to the...Ch. 4.13 - Find the shortest distance from the origin to the...Ch. 4.13 - Find by the Lagrange multiplier method the largest...Ch. 4.13 - Find the largest and smallest values of...Ch. 4.13 - Find the hottest and coldest points on a bar of...Ch. 4.13 - Find the hottest and coldest points of the region...Ch. 4.13 - Findddt0sintsin1xxdx.Ch. 4.13 - Findddxt=1/xt=2/xcoshxttdt.Ch. 4.13 - Findddx11/xexttdt.Ch. 4.13 - Findddx0x2sinxttdt.Ch. 4.13 - Showthatddxcosxsinx1t2dt=1.Ch. 4.13 - In discussing the velocity distribution of...Ch. 4.13 - The time dependent temperature at a point of a...Ch. 4.13 - Evaluated2dx20x0xf(s,t)dsdt.

Additional Math Textbook Solutions

Find more solutions based on key concepts
To simplify the expression 4+3(c−12)

Pre-Algebra Student Edition

Solve each formula for the given letter . [2.3] What percent of 60 is 42? [2.4]

Elementary and Intermediate Algebra: Concepts and Applications (7th Edition)

1. combination of numbers, variables, and operation symbols is called an algebraic______.

Algebra and Trigonometry (6th Edition)

Voting A random sample of likely voters showed that 49 planned to support Measure X. The margin of error is 3 p...

Introductory Statistics

In Hamilton County, Ohio, the mean number of days needed to sell a house is 86 days (Cincinnati Multiple Listin...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Explain the meaning of the term “statistically significant difference” in statistics terminology.

Intro Stats, Books a la Carte Edition (5th Edition)

Knowledge Booster
Background pattern image
Math
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
Text book image
Elementary Geometry For College Students, 7e
Geometry
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Cengage,
Text book image
Elementary Geometry for College Students
Geometry
ISBN:9781285195698
Author:Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:Cengage Learning
Text book image
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,
Text book image
Glencoe Algebra 1, Student Edition, 9780079039897...
Algebra
ISBN:9780079039897
Author:Carter
Publisher:McGraw Hill
Text book image
PREALGEBRA
Algebra
ISBN:9781938168994
Author:OpenStax
Publisher:OpenStax
Text book image
Algebra: Structure And Method, Book 1
Algebra
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:McDougal Littell
SEE MORE TEXTBOOKS
Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY