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 4.8, Problem 2P
Using the two-variable Taylor series [say ( 2.7)] prove the following "second derivative tests" for maximum or minimum points of functions of two variables. If fx=fy=0 at (a,b), then (a,b) is a minimum point if at (a,b),fxx>0,fyy>0, and fxxfyy>f2xy;
(a,b) is a maximum point if at (a,b),fxx<0,fyy<0, and fxxfyy>f2xy; (a,b) is neither a maximum nor a minimum point if fxxfyy<f2xy. (Note that this includes fxxfyy<0, that is, fxx and fyy of opposite sign.)
Hint: Let fxx=A,fxy=B,fyy=C; then the second derivative terms in the Taylor series are Ah2+2Bhk+Ck2; this can be written A(h+Bk/A)2+(C−B2/A)k2 Find out when this expression is positive for all small h,k [that is, all (x,y) near (a,b)]; also find out when it is negative for all small h, k, and when it has both positive and negative values for small h, k .
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i need help with this question i tried by myself and so i am uploadding the question to be quided with step by step solution and please do not use chat gpt i am trying to learn thank you.
i need help with this question i tried by myself and so i am uploadding the question to be quided with step by step solution and please do not use chat gpt i am trying to learn thank you.
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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