Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 4.10, Problem 13P
Find the largest and smallest values of the sum of the acute angles that a line through the origin makes with the three coordinate planes.
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Students have asked these similar questions
1. (20 pts) Determine whether the following statements are true (T) or false (F)? (A
reasoning is required.)
(1) Let V be the set of all ordered pairs of real numbers. Consider the following
addition and scalar multiplication operations on u =
u= (u1, u2) and v =
(v1, v2): u + v = (U₁ + V₁, U₂ + v₂), ku = (ku₁, u₂). Is V a vector space
under the above operations?
U2
(2) The set Mmxn of all m×n matrices with the usual operations of addition
and scalar multiplication is a vector space.
α
(3) The dimension of the vector space of all matrices A = [a b] in R2×2 with
a+d=0 is 4.
(4) The coordinate vector of p(x) = 2-x+x² in P3 relative to the basis S =
{1, 1+x, x + x2} is [4 -2 1].
(5) If a 6×4 matrix A has a rank 3, then the dimension of N(A) is 3.
Scenario
Sales of products by color follow a peculiar, but predictable, pattern that determines how many units will sell in
any given year. This pattern is shown below
Product Color
1995
1996
1997
Red
28
42
21
1998
23
1999
29
2000
2001
2002
Unit Sales
2003 2004
15
8
4
2
1
2005 2006
discontinued
Green
26
39
20
22
28
14
7
4
2
White
43
65
33
36
45
23
12
Brown
58
87
44
48
60
Yellow
37
56
28
31
Black
28
42
21
Orange
19
29
Purple
Total
28
42
21
49
68
78
95
123
176
181
164
127
24
179
Questions
A) Which color will sell the most units in 2007?
B) Which color will sell the most units combined in the 2007 to 2009 period?
Please show all your analysis, leave formulas in cells, and specify any assumptions you make.
5. (20%) The linear transformation L: P3 → P2 defined by L(f(x)) = f'(x)+
f(0).
(a) Find the representing matrix A of L with respect to the ordered basis
{x2, x, 1} for P3, and the ordered basis {2,1 - x} for P2.
(b) Find the coordinates of the f(x) = 2x² +2 in P3 with respect to the
ordered basis {x2,-x, 1}, and find the coordinates of L(f(x)) with respect
to the ordered basis {2,1-x}
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.
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