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« PHYS290F Fall 2008

Help for Homework 6

Mon 11Oct2010 11:43AM

(This will be added to later.)

Problem 7 (5.3.5):

Note that for a geometric sequence of n terms:
S = a₁ + a₂ + ... + a_n
where a_n = a₁r^n-1, the sum S_n is given by
S_n = a₁ ( 1 - rⁿ)/(1 - r). This is true independent of whether any of the numbers involved are real or complex. Make sure that you have the correct r and n.

Problem 8 (5.3.6):

We actually already did the second part of this problem in class. For the first part, the idea is to write the de Moivre's theorem for n=4, and then expanding the 4th power product into the form of x+iy (where x, y are real). After that, just compare the real and imaginary parts of this (x, y) with the corresponding part of the RHS (i.e., cos 4θ and sin 4θ).

Then, calculate the real and imaginary parts of the sum from S_n and equate the corresponding part of the sequence (i.e., cos θ + cos 3θ + ... + cos (2n-1)θ for the real part and sin θ + sin 3θ + ... + sin (2n-1)θ for the imaginary part). You will have to use the trigonometric identities for addition and subtraction of sines and cosines to simplify the results.

Problem 10 (5.4.6):

This problem is for the parallel LRC circuit whereas we did the series LRC circuit in class. What is different for a parallel circuit as opposed to the series circuit is that the potential differences between the terminals of each element (L, R, and C) are the same (common) while, for the series circuit, the current through each element is the same (common). Otherwise, the relationships between the current and voltage for L, R, and C are common to both parallel and series circuits.

For example, V(t) = L (dI_L(t)/dt). Substituting V(t) = V₀ e^iωt and I_L(t) = I₁ e^iωt, we get I₁ = V₀/(iωL). Do the same for I₂ and I₃. This allows you to write the total current I₀ = I₁ + I₂ + I₃ in terms of V₀, ω, L, and R. This in turn makes it possible to calculate Z ≡ V₀/I₀. Once this is done, you can calculate its modulus r and polar angle (phase) φ in the same way as we did in class

Final Exam 7 - 9 pm on Tues., Dec. 16 in UNIV 303

Tue 16Dec2008 10:33AM

Our Final Exam is given on Tues. Dec. 16 at 7 - 9 pm in UNIV 303. It is a closed-book, 2-hour exam on the material corresponding to Chap. 3 and 7. You may bring the sheet I distributed on the differential operations in various coordinate systems (with no additional writing) and a non-programmable, non-graphing calculator in addition to pencils and erasers. Good luck!

Help for Homework 13

Thu 11Dec2008 10:49AM

Problem 1:

As I went over in class, if the electrostatic potential is V(r)=f(x), then E=-∇V will have only the x-component and its sign and the trend in its magnitude can be read off from the graph given for f(x). Similarly, we derived the local (differential) version of the Gauss' Law in class. According to this, the charge density ρ(r)=∇•E/ε₀. Just apply this to the E above and read off whatever you can on the charge density from the graph of f(x).

Problem 2:

(i) Use the parametrization: x=t, y=t².

(ii) Very similar to one of the problems from Homework 12. E.g., use a parametrization: x=cos(t), y=sin(t), where t goes from 0 to 2π to do the actual line integral of F. As directed, also calculate curl F and its surface integral over the unit disk.

Problem 3:

First, do the surface integral directly. Here, note that you can divide the 6 faces of the unit cube into 3 parallel pairs, and that for each pair the surface normal unit vector n points in the opposite directions. For example, the face parallel to the yz-plane at x=1 has dS=e_xdydz, and the one at x=0 has dS=-e_xdydz. That should tell you what the sum of the integrals for each pair is.

Second, calculate the divergence of V. Use the divergence theorem and equate the volume integral of this divergence with the required surface integra. The result should agree with that of the first part, of course.

Problem 4:

Make sure to draw a picture. Write the force vector F that is needed to keep the pendulum just in equilibrium, and do the line integral of F•dr from the bottom position to the position forming an angle θ with the vertical. If you note that the infinitesimal displacement dr in this process is always in the direction of the force and the magnitude is just L dθ, then the problem is very simple.

Problem 5:

To see if a vector field is conservative or not is equivalent to see if it is a gradient of some scalar function. So you can try to see if it is possible to construct a scalar function whose gradient is the given vector field. Another, simpler way is to calculate the curl of the given vector field because this is in turn equivalent to see if its curl is zero. [This is due to the Stokes Theorem.]

Problem 6:

This is very similar to Problem 3 (7.4.4) above. Just different values come up when you do the surface integrals for each of the paired faces of the unit cube. Note the x-component of W is odd in y and the y-component is odd in x, while the unit cube has a center at the origin. (This means that x, y, z range from -1/2 to +1/2 over its surface/volume.)

Help for Homework 12 Thu 04Dec2008 11:17AM Problem 1:

Please note that these equations refer to keeping the vector A fixed but rotating the coordinate axes by θ counterclockwise. This is equivalent to rotating the vector A clockwise by θ keeping the coordinate axes fixed.

Problem 2:

(a) Either use the rotation formulas of Problem 1, or note that e_r = (cosθ,sinθ), e_θ = (-sinθ,cosθ) in the Cartesian coordinates, and invert the relations.

(b) You can use the fact that e_r = (sinθcosφ,sinθsinφ,cosθ), e_θ = (cosθcosφ,cosθsinφ,-sinθ), e_φ = (-sinφ,cosφ,0), in the Cartesian system, and invert the relations.

Problem 3:

Here, I am asking you to do the problem without invoking the Stokes Theorem (which I have not yet covered in class). We will see soon that that Theorem can be used to solve this problem simply as well but that comes later.

First, write the integral in the canonical form of a line integral, that is, as the integral of a vector field f(x,y) in the form of ∫_Γf•dr. Now, write the vector field f as f = a (x,y) + b (-y,x). Note that the vector (x,y) is proportional to e_r and (-y,x) is proportional to e_θ. Since the path Γ is totally in e_θ direction, only b contributes to the integral. Then, the integral will be reduced something of the form ∫₀^2π b(-y,x)•e_θ r dθ. At this point, you can use x=r cosθ, y=r sinθ (r=1 for the unit circle), and do the integral over θ.

Problem 4:

Pick a path, parametrize it, and do the integral using the parameter as the integration variable, just as we did in class. You might ask why I am asking you to pick any path. It may be instructive to see if the integrand (in vector form) is of a special kind.

Problem 5:

Since the sphere is described by x²+y²+z²=R², you can write the height z as h(x,y). Then just take the gradient of h(x,y). Remember, the gradient of h is in the direction where h(x,y) changes most rapidly.

Proglem 6:

Just look at each component of ∇(φχ).

Differential Vector Operations Sheet Posted Thu 04Dec2008 9:40AM I have just posted a sheet (distributed in class as a hard copy) of some differential vector operations in various coordinate systems under Powerpoints (though it is just a PDF file). You are allowed to bring this sheet with NO additional writing of any kind on it to the Final Exam - though I am not sure if it will be directly helpful. This will be the only content material you are allowed to bring to the Final.

Wednesday office hours Tue 02Dec2008 3:12PM Alexei Svyatkovskiy will hold office hours from 10 - 12 on Wednesday, Dec. 3.

By the way, I just posted last year's Final Exam as a Practice Exam. However, you should be aware that the real Final will have 5 questions (like the two eariler exams) and also it only covers Chapters 3 and 7. (This doesn't mean that solving questions will not involve any material from other chapters already covered in the earlier two exams as Math is a cumulative subject. But there will be no question on the Final whose main topic is from other chapters.) Last year, a little bit of linear algebra was also covered, but this year, there is no coverage of that topic.

Help for Homework 11 Thu 20Nov2008 11:34AM Problem 1:

Take the coordinates of the upper right corner of the rectangle as (u,v). Then, since you want it to be on the ellipse, you have (u²/a²)+(v²/b²)=1. Thus, your constraint is g(u,v) = (u²/a²)+(v²/b²) - 1 = 0. The area of the rectangle is 4 uv. So you can maximize 4 uv (or just uv). So you can take f(u,v) = uv, and apply the method of Lagrange multipliers to F(u,v,λ) = f(u,v) - λ g(u,v). Don't forget that the length of the horizontal edge is 2u and that of the vertical edge is 2v.

Problem 2:

(a) The moment of inertia about an axis is, if you recall from mechnaics, the integral of ρ_mr² where r is the distance from the axis. So both the domain of integration (a disk) and the integrand are polar-coordinate friendly for this part. So there should not be any question that you should use that system. Just don't forget to count all powers of r correctly, including the Jacobian of the polar coordinates.

(b) On the surface, the integrand appears more xy-friendly than polar coordinates. However, noting x = r cosθ, y = r sinθ, it really is not too bad in polar coordinates either. In fact, since the domain of integration (a disk) is definitely circularly symmetric, it turns out to be much easier to use the polar coordinates here too.

Problems 3 and 4:

Since the ellipse and ellipsoid are so close to circle and sphere, the first impulse should be transforming the domains to those more symmetric ones. You can do this by introducing variable changes such as y' = (a/b) y, and z' = (a/c) z. Once that is done, you should then be able to change (x,y') or (x,y',z') system into the polar or spherical coordinate system with no difficulty. Just don't forget that there are two stages of Jacobians to take into account here. First, the introductinon of the primed variables yields a set of Jacobian(s). Then, changing that Cartesian system into polar or spherical system requires another set.

Problem 5:

(a) and (b) In both these parts, you are to obtain answers by giving x, y, and z values of v(t), a(t), and L. These values will be expressed in terms of the time derivatives of r and θ as well as trigonometric functions of θ. Use known trigonometric identities whereever possible to simplify.

(c) Since the unit vectors in polar coordinates (in xy-plane) can be expressed in terms of their (x,y) coordinates as e_r = ( cosθ, sinθ ) and e_θ = ( -sinθ, cosθ ), your answer to the acceleration in part (a) can in fact be converted to a very simple answer using the polar coordinates. Central force means that the acceleration is purely the radial direction, i.e., the coefficient of e_θ is zero. Consider how that condition translates to the derivative of L you calculated in (b). This result, of course, leads to the famous Kepler's second law of planetary orbits.

Problem 6:

Some of you may find the second part more challenging. Just recall that a line going through the origin is representd by an equation of the form y = mx. If this line is to be perpendicular to another line of slope n, then the slpe m = - 1/n. (Remember why?)

Wednesday Office Hours Mon 17Nov2008 5:18PM Mr. Lim will hold office hours from 8 - 10 am on Wednesday, Nov. 19.

Help for Homework 10 Mon 10Nov2008 5:05PM First, our graders will hold office hours on Wednesdays, 10 am - 12 pm, from this week onward, instead of on Fridays. I will continue to hold office hours on Mondays and Thursdays, 10:30 - 11:30 am. Please take advantage of these times. This week, the grader holding the office hours is Mr. Alexey Svyatkovskiy.

Problem 1:

If you let g(x) = x² except that g(0) = g(2π) = 2 π², then in the Fourier series expression for this g(x), the coefficients are given by: a_m = (1/π) ∫₀^2π x² cos(mx) dx and b_m = (1/π) ∫₀^2π x² sin(mx) dx. Evaluate these and just equate the Fourier series (the big summation) with 2 π² either at x=0 or at x=2π, then you will get Equation (2), one that tormented many mathematicians in the 18th century.

Problem 2:

The reason these relations are called orthonality is because you can interpret the integrals in this problem as dot products. If a dot product of two vectors are zero, then the two vectors are called orthogonal. In order to actually calculate these integrals, it is useful to use the trigonometric identities:

cos A cos B = (1/2) ( cos(A+B) + cos (A-B) ),
sin A sin B = (1/2) ( cos(A-B) - cos (A+B) ),
sin A cos B = (1/2) ( sin(A+B) + sin (A-B) ).

Problem 3:

Use integration by parts.

Problem 4:

Equation (3) comes from adding waves which are out of phase that is the phase equivalent in the path length difference by the amount k⋅Δr where r indexes the location of the slit where the wave passed through. That's the bit of physics, but mathematically, you just perform the integral in (3), and will get the important part of the intensity from the single-slit diffraction pattern. In elementary texts on physical optics, this calculation is usually performed by adding phasors using geometry, a much more painful process than the simple integration as in this problem.

Problem 5:

The simplest way to do this problem is probably to use the polar coordinates as discussed already in class. Set x = r cosθ, y = r sinθ, and use the method of Lagrange multipliers with variables r, θ, and, say, λ. If you visualize the problem with hyperbolas (xy = C) just touching the Lemniscate, it will be relatively easy for you to see why the extema you get are maxima or something else.

Homework 8 Scoring Correction Thu 06Nov2008 8:20PM

There was miscommunication between myself and the grader for grading Problem 1(a) and 2(a). Whereas I told you that you will not get penalized for not including the double pole in the contour integral calculation, points were in fact taken off. To correct for this, I have added 5 points to all homework submitted. I am sorry for the error. The increased scores are already reflected in my grade book - don't worry.

Correction on the contour integraton Wed 29Oct2008 9:02PM Class, I goofed. While 1/(z-z₀)² integrated around any loop that contain z₀ is indeed zero, the same for f(z)/(z-z₀)² is in general NOT zero. We did discuss in class that (2 π i) f'(z₀) = ∫_C f(z)/(z-z₀)² where f(z) is analytic in the whole domain and the contour C contains z₀. So, in Homework 8 Problems 1(a) and 2(a), in fact the double pole at z=-1/2 does contribute a non-zero constant. This constant is derived by first calculating the integral as if the double pole were a simple pole at z₀, and then differentiating the result with respect to z₀ and then evaluating the result at z₀ equal to the actual location of the double pole. This is more difficult than I think is reasonable at this point in this course. So, long and short of this is that I should not have included a double pole in the integrand of the homework problem. You will get full credits for those problems if you omitted contribution from the double pole - but I just want you to know that I gave you a wrong explanation and result in class in general for double and higher order poles. Those generally do contribute to the integral (in a more complicated way, involving derivatives of the residues) unless they are of the form of (constant)/(z-z₀)ⁿ (where n is 2 or greater). I apologize for this error.

Help for Homework 8 Sat 25Oct2008 11:50PM

Problem 1:

(a) The integrand has 4 points of singularity, of which 3 are simple poles and one is a pole of second order. The integral can be evaluated by considering those simple poles that fall inside the closed contour of integration plus the contribution C from the double pole at -1/2. Specifically, first identify which simple poles are inside the unit circle centered at the origin, and second, calculate the residues at each of those simple poles, and lastly, multiply the sum of those residues by 2π i. The contribution C from the double pole is more complicated and will be explained in class on Friday..

To remind you, in order to find the residue at, say, z₁, factor out 1/(z-z₁) from the integrand and substitute z₁ into z in the remaining parts of the integrand. (For example, the residue of f(z)=1/(z²+1) at z=i is 1/(2i).)

(b) As we saw from the previous Homework, tan(2z)=sin(2z)/cos(2z) has simple poles where cos(2z)=0. These poles are strictly on the real line (i.e., they are all real numbers). Again, first identify which simple pole(s) are within the unit circle centered at the origin, second, calculate the residues at each, and then multiply the sum of those residues by 2π i.

(In calculating the residues, it may be useful to expand cos(2z) in Taylor series about each pole up to the first non-vanishing term. Just don't forget that the argument of cos is 2z here, not just z as in the previous homework.)

Problem 2:

(a) The difference here from (1)(a) is that only 1 simple pole is within the contour. Thus this problem is simpler thatn (1)(a). The double pole is still inside the contour, so the same double pole contribution C must still be added.

(b) Again, look at how many (if any) simple poles fall within the contour (the circle of radius one centered at z=(3/4)i.

Problem 3:

First part: Expand ln(1+z) and 1/(1+z) in Taylor series about z=0. Show that the former series differentiated term by term is the same as the latter series. Then, calculate the radii of convergence for the two Taylor series and show that they are identical.

Second part: Do the same for sin(z) and cos(z).

Problem 4:

Integrate f(z)=z² from the origin to 1+i by the two different paths and show that the results are the same. The first path is composed of two parts, 0 to i on the imaginary axis and then from i to 1+i parallel to the real axis. The second path is a straight line from the origin to 1+i (at 45 degrees to the real axis).

Problem 5:

This is essentially the same as the first part of Problem 3. Only, it is worded in the reverse direction; show, e.g., that the Taylor series of 1/(1+z) about z=0, integrated term by term, becomes the Taylor series expansion of ln(1+z). Show that the radii of convergence of the two series are the same. In fact, show in general that the radii of convergence of a Taylor series and the series obtained by integrating it term-by-term are identical.

Correction in Problem 2 of Homework 8 Tue 21Oct2008 3:50PM In Problem 2 of Homework 8, the contour should have been a unit circle centered at 3i/4 instead of i/2. Please make a correction in your copy. I am sorry for the error.

Help for Homework 7 Sun 19Oct2008 10:14AM Lecture on last Friday was dedicated to discussing Homework 7 and relates material. So I will not write all the details out in here, but some synopsis of the discussion is given below:

Problem 1:

(a) This is a problem that shows a geometric implication of analyticity of a function of a complex variable. In (a), consider an equation u(x,y) = C (and v(x,y) = D) where C (and D) are some chosen constant(s). Say, u(x,y) = 1. This equation then related x and y, or defines y=y(x) implicitly. So, if you now rewrite the equation as u(x,y(x)) = 1, then you can take the (total) derivative of u with respect to x. (This corresponds to differentiating u with respect to x when y is constrained to vary along the curve u(x,y) = 1.) This results in u_x + u_yy' = 0. You can then solve this to obtain the slope y' = dy/dx of the curve y=y(x). You can do the same for v(x,y).

(b) Two lines are perpendicular to each other if the product of their slopes is -1. E.g, a line with a slope of 2 (such as y=2x) and one with a slope of -1/2 (such as y=-(1/2)x) intersect at 90 degrees. Use the CRE for analyticity and the results from (a) to show that this condition is satisfied.

(c) For f(z) = z², u(x,y) = x²-y², v(x,y)=2xy. So the curves u=constant and v=constant are hyperbolas (of different axis orientations).

Problem 2:

(a) You can always use z^a = e^{a ln z}. When you write z in the polar form, allowing adding any integer multiple of 2π to the polar angle θ, you will get all possible values of z^a.

(b) Again, write 1 = e^{2nπ i} and raise this expression to the 1/4 power.

Prolem 3:

By locating and naming the singularities, I mean showing where each singularity is and what type of singularity it is (i.e., a pole (of what order?), essential singularity, or branch point). If it is a simple pole (pole of first order), then also calculate the residue there. Simple poles are singularities of the form C/(z-z₀) where C is either a constant or something that approach a non-zero, finite constant as z approaches z₀. In this case that limit of C is the residue.

(a) Factor the denominator in to the product of 4 terms of the form (z-z_i), where z_i is the i-th root of z⁴=-1.

(b) tan(z) = sin(z)/cos(z). Since neither sin(z) nor cos(z) has singularities, the only singularities of tan(z) are where the denominator cos(z) = 0. For what values of z is cos(z)=0? You may look at Homework 6, Problem 2(b). In order to obtain the residues at each simple pole, you can replace cos(z) near a pole at, say, π/2, by the first non-vanishing term of its Taylor series expansion about z=π/2.

(c) There are some poles and also one other type of singularity that comes from the numerator. Because of this last singularity, the residue from a simple pole is triply-valued.

Problem 4 (6.1.12):

This is a very important problem. Analytic functions of a complex variable are so special that knowing just its real (or imaginary) part allows you to reconstruct the whole function (up to a constant). For example, if u(x,y) is given, then CRE tells you what v_x and v_y. Each of these can be integrated to get (parts of) v(x,y). For example, integrating v_x with respect to x (considering y to be a constant) leads to v(x,y) up to an integratino 'constant', which in this case is an unknown function of y (because y is held fixed like a constant). Analogous procedure on v_y will also lead to v(x,y) but up to an unknown function of x. Compare the two expressions and use the fact that they have to be the same because they are both expressions of the same function v(x,y).

Exam II date changed to Nov. 3 Tue 14Oct2008 2:07PM

Due to request from many of you who wanted to track the election night media, I have just changed the date of our Exam II to Monday, Nov. 3. It will still be from 8 - 10 pm in MSEE room B012. If any of you have a legitimate conflict or if you just have to have the exam on Nov. 4 as previously scheduled, please contact me so that we can discuss it. Thanks.

Mr. Svyatkovskiy will hold his office hours on Wednesday, Oct. 15 from 10:30 am - 12:30. Mon 13Oct2008 11:43AM Mr. Svyatkovskiy will hold his office hours on Wednesday, Oct. 15, 10:30 am - 12:30. If you have questions about homework 6 (or other related material) and didn't make Prof. Nakanishi's office hour on Monday, please do use his office hours.

Help for Homework 6 Sun 12Oct2008 11:04AM

Problem 1:

Problem 2:

From the given identity e^iz = cos z + i sin z and another one you get by replacing z by -z, you can write cos z = (e^{iz+e^-iz)/2 and sin z = (e^iz-e^-iz)/(2i). These relations are valid for a complex z as well as for real z because they are valid term by term in their Taylor series expansions. [We will later talk about how the convergence of these series is enlarged when the variables are extended to the complex plane.]}^{If you write, e.g., sin(x+iy) = (e^i(x+iy)-e^-i(x+iy))/(2i) and expand e^ix and e^-ix in terms of cos x and sin x, the righthand side of (a) should just fall out.
Problem 3:
You just need to calculate the various partial derivatives of the real and imaginary parts of the given functions and find out whether CRE are satisfied or not. If they are, then you know the function should be expressible as f(z). In that case, take a guess at what this form should be and check to make sure that your guess is indeed correct. (It shouldn't be hard to make a good guess.)
Problem 4 (5.3.5):
In this problem, calculate the sum of the n terms of the geometric sequence (5.3.30). (It is a geometric sequence where the k-th term is of the form a₁r^k-1 where a₁ = e^iθ and r = e^2iθ.)
Problem 5 (6.1.1):
The problem has 3 parts. First, investigate the limits when you approach the origin along x and y axes. Second, write x = r cos θ, y = r sin θ, and take the limit of r → 0 at fixed θ. Lastly, do the same for approaching any point other than the origin.
Problem 6 (6.1.8):
As I explained in class, work this problem in 3 parts. First, for f(r,θ), df = (∂f/∂r)dr + (∂f/∂θ)dθ. Apply this to f(z) = z = r e^iθ. Second, "interpret" the two factors of dz, that is, (dr + ir dθ) and e^iθ. Use geometry to show that the first factor is the variation dz when r is changed by dr and θ (=0) is changed by dθ. Interpret the second factor in terms of rotation. Lastly, use the same approach as we did in class to derive the CRE in polar coordinates. That is, calculate (∂f/∂r) and (∂f/∂θ) in terms of (df/dz). Then relate the two partial derivatives to each other using the fact that (df/dz) is common in the two equations. Finally, write the relationship between the two partial derivatives in terms of the real part u(r,θ) and imaginary part v(r,θ) of f(r,θ).}

Homework 5 Mon 29Sep2008 11:49AM As you know, I spent most of the last lecture on Friday discussing the current homework (Homework 5). However, there seem to be many questions, particularly about 5(a).

5 (a): So here is what I said about that problem:

If you apply the ratio test to the series a₁+a₂+...+a_n+..., you get the lim_n→∞ a_n+1/a_n = 1. This means that the ratio test is inconclusive. Therefore, you would have to do something else. In this case, the first thing to do is actually to find out whether a_n converges to zero as n goes to infinity. If it doesn's, then there is no hope of convergence. To consider this, my suggestion is to look at ln a_n instead of a_n itself (because of the exponent n). Thus, ln a_n = n ln (1-1/n). You can write this a ln(1-1/n) / (1/n). Then look at the limit of this quantity as n → ∞. (Use L'Hospital's Rule for this.) From this, you will know what lim_n→∞ a_n is equal to. This should tell you whether this series converges or not.

4 (a),(b): I forgot to state that a>0. If a<0, these integrals would be divergent; so we had better have a>0. I am sorry for the omission.

Exam I coming up Wed 24Sep2008 4:23PM Exam I will be given on Tuesday, Sept. 30, 8 - 10 pm in MSEE, rm. B012. It will cover Chapter 1, 2, 4, and Chapter 5 up to and including 5.3 on p.98. (We skip Chap.3 until later in the course.) There will be 5 problems (with some of them having multiple small parts). You can bring a scientific calculator if you wish (but NOT graphing or programmable one), and in any case a calculator is not likely to be of much use.

No crib sheets or other material is allowed at the exam. You would need to know some fundamental formulas such as the general Taylor series expansion (around the origin or another point) and basic identities such as cos²θ + sin²θ = 1, among others. If in doubt about whether you need to know some formula, just ask beforehand.

Correction from today's lecture: Today I wrote 1+z+z²+...+zⁿ = (1-zⁿ)/(1-z). This is unfortunately slightly wrong. The last term on the left needs to be z^n-1 - or the power of z in the numerator of the righ side needs to be n+1. (Either would correct the error.) Corresponding changes need to be made to the ensuing equations that I wrote as well. I am sorry for overlooking this obvious error. I will talk about it on Friday to make sure that everyone corrects it in their notes.

Help for Homework 4 Thu 18Sep2008 11:44AM

(This may be further added to later on.)

Problem 1:

Just remember that the Taylor series for (1+x)^p = 1 + px + p(p-1) x²/2 + ... and apply this to the problem at hand where there are two terms with p=-2 and x = d/(2x) or -d/(2x). What we did in class had p=-1 because it was for the potential. For the electric field, p=-2.

Problem 2:

Generalize what we did in class for the integral of u/(e^u-1) (or x/(e^x-1) in the notation used in class) to u^x-1/(e^u-1). To do this, follow the same steps as in class, and write
u^x-1/(e^u-1) = (u^x-1/e^u) [(1/(1-e^-u)] = (u^x-1/e^u) (1+e^-u+e^-2u+e^-3u+...)
Write the integral of this sum as the sum of integrals and use a change of variables where, say, v = nu. Then compare the integral you get with (2.1.39) on p.43.

Note: My powerpoint slide for Lecture 10 had misprints where the factors of Γ(n) were omitted inadvertently from the general Bose-Einstein and Fermi-Dirac integrals. I am sorry for the error; the corrected version has been uploaded. (However, in this problem, you are asked to actually produce the integral form of Γ(x); so I hope you realized my earlier error in the powerpoint while doing this problem.)

Problem 3:

You can use the known Taylor series of the functions given in parentheses. For (a), tan^-1 x = x - x³/3 + x⁵/5 - x⁷/7 + ... for |x| < 1. [This can be derived (though you need not do it for this problem), by noting (tan^-1)' = 1/(x²+1).]

Problem 4 (4.2.4):

(i) You can just compare with known series.

(ii) Ratio test will work.

(iii) Integral test

(iv) Telescopic!

(v) Look at a_n as n → ∞.

(vi) a_n = 1/(n n^1/n). Note that n^1/n → 1 as n → ∞. Therefore, if you pick any value, say, 2, n^1/n < 2 for any n that is large enough (i.e., for n > n₀). Use this fact and find a lower bound to the original series that is itself divergent. [n^1/n → 1 because n^1/n = e^{(1/n)ln n} and the exponent → 0 as n → ∞.]

Some hints on Homework 3 (may be added to later) Sun 14Sep2008 12:00AM Problems 2(a), (b), and 4:

Trigonometric and hyperbolic identities like below come in handy in these problems:
tan²x + 1 = 1/cos²x, 1/tan²x + 1 = 1/sin²x,
(tan x)' = 1/cos²x, (1/tan x)' = -1/sin²x,
cosh²x - sinsh²x = 1, (cosh x)' = sinsh x, etc.

Problem 3(c):

You can use integration by parts a couple of times to come up with the same expression as part of the right-hand side as you started from on the left-hand side. Then solve for that.

Problem 6 (2.2.10):

This problem has 4 parts. (a) is to define the following integral:
I(a) ≡ ∫₀¹ (t^a-1)/ln t dt
and evaluate I'(a). Just as we discussed in class, the derivative of the integral I(a) can be evaluated as the integral of the derivative of (t^a-1)/ln t. Note that this is a surprisingly easy integral to do if you know what the derivative of (t^a-1) is equal to.

Part (b) is to go from I'(a) to I(a) by integrating the result you got in part (a). This will involve an undetermined constant of integration.

In (c), you will determine the constant of integration in (b) by using the value that I(0) will have to assume.

(d) Once you have I(a), just evaluate I(1) by substituting a=1.

Typo in Homework 3, Problem 6 (2.2.10) Thu 11Sep2008 3:41PM Homework 3 Problem 6 (2.2.10) in the book has two typos. The first one is that the integral shown should have had a dt at the end. Secondly, where it says "Think of the t in t-1 as the a=1 limit of t^a," the expression "t^a" should have read "t^a-1".

Help for Homework 2 Sun 07Sep2008 9:23AM Problem 1

(a) As I worked out in class, you need to use the trigonometric identities:
a sin x + b cos x = (a²+b²)^1/2 sin (x+y)
a cos x - b sin x = (a²+b²)^1/2 cos (x+y)
where the angle y is given implicitly by cos y = a/(a²+b²)^1/2 and sin y = b/(a²+b²)^1/2

Problem 3 (1.6.5)

This problem is in the Ising class, that is the magnetic moment μ can only point parallel (up) or anti-parallel (down) to the external magnetic field h. It has 4 parts:

(a) You are given the ratio of the probabilities P_up and P_down and the fact that they add up to 1. Then you are to calculate the P_up and P_down individually. Do just like you would any problem where there are two unknowns and you know their ratio and sum.

(b) The average magnetic moment m is given by
m = μ P_up + (-μ) P_down
just like the average (expected) value when you know values of all possible cases (outcomes) and their probabilities of occurrences.

Office Hours are set for the semester Wed 03Sep2008 5:45PM Office hours for the rest of the semester have now been set:

Nakanishi: Thursday and Monday, 10:30 - 11:30 am.

Grader: Friday, 10:30am - 12:30 pm
(Mr. Lim in Rm.19 and Mr. Svyatkovskiy in 19B alternate each week. This week, it is Mr. Svyatkovskiy. Next week it will be Mr. Lim.)

Help for Homework 1 Tue 02Sep2008 11:00AM Problem 5 (1.4.1): If you don't know where to start on this problem, read p.8-11, particularly p.10. The main part asks you to derive (1.4.18) and (1.4.20) from the general [removed]1.3.16), using your knowledge of the derivatives of cos x and sin x evaluated at x=0. The second part is asking for the radius of convergence of the Taylor series for cos x and sin x. Use the ratio test just as they do on p.10-11 for the exp(x) function.

Problem 6 (1.5.1): We didn't cover L'Hospital's rule in class, but this is something you all did (I am sure) in your calculus classes. Write xⁿe^-x as xⁿ/e^x and apply the L'Hospital's rule (p.24 about the middle).

Problem 7 (1.6.4): This problem has 4 parts (though they are not explicitly marked as separate parts). In the third part (starting from the bottom of p.26 to the top 4 lines of p.27), they define θ" and v" as those that relate (x",t") directly to (x,t), while θ' and v' relate (x",t") to (x',t') and θ and v relate (x',t') to (x,t). In this part, it is useful to realize that:

cosh(θ+θ') = cosh θ cosh θ' + sinh θ sinh θ'

sinh(θ+θ') = sinh θ cosh θ' + cosh θ sinh θ'
Don't forget the fourth part where you will consider the limit of small v and v'.

Grader (Mr. Lim) Office Hours on Tuesday, September 2

Mon 01Sep2008 11:35PM Mr. Youbong Lim (grader for Homework #1) will be holding his office hours on Tuesday, September 2, 3 - 5 pm in his office, PHYS Rm. 19 on the ground floor. Don't forget that the first Homework Assignment is due on Wednesday in class.

Announcement 1:

Thu 28Aug2008 10:06AM

Letter grades:

Yes, this course will be taking advantage of the newly available +/- grading system.

Textbook:

All bookstores (Follet's and University) are currently out of the Shankar text. They tell us that you would need to go to one of the stores and ask to reserve a copy in order for them to order more. You may also try on-line shops such as Amazon and Alibris. In the mean time, I will post the actual problems from the book that have been assigned as homework. So I just posted the last 3 problems under Homework Assignments link on the left side of the course home page. All books (including the text) listed on the references section of the syllabus should be available on reserve in the Physics library (2nd floor, PHYS) as well.

Office Hours:

Nakanishi will hold office hours 10:30 - 11:30 am on Thursday, August 28 and Friday, August 29. For the subsequent weeks, we will decide on more permanent office hour times and post them.

Homework:

The homework should be worked out and neatly and legibly written out, step by step, on paper and turned in to me during the class. The first set is due on Wednesday, Sept. 3. Please use 8.5 by 11 pieces of paper and staple all the sheets together. We would appreciate it if you either use sheets with no spiral binding perforation on the side (or those hanging perforation left-overs cut off neatly) because it interferes with stacking and handling.

Last Updated: May 9, 2023 8:31 AM