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Besides some of the tricky points we discussed above, you are of course expected to know all the basics of differential calculus as well as the properties of the special functions we encountered. The following set of problems is by no means an exhaustive test of your background. It should however suffice to give you an idea of where you stand. If you find any weak areas while doing them, you should strengthen up those areas by going to a book devoted to calculus.
Problem 1.6.1. Expand the function f(x) = sinx/(coshx + 2) in a Taylor series around the origin going up to x3 . Calculate f(.l) from this series and compare to the exact answer obtained by using a calculator.
-10 10
Figure 1.7. Plot of the function S(x).
Problem 1.6.2. Find the derivatives of the following functions: (i) sin(x3 + 2), (ii) sin(cos(2x~). (iii) tan3 x, (iv) ln(coshx), (v) tan-1 x, (vi) tanh-1 x, (vii) cosh2 x - sinh x , (viii) sin xI ( 1 + cos x ).
Problem 1.6.3. A bank compounds interest continually at a rate of6% per annum.
What will a hundred dollars be worth after 2 years? Use an approximate evaluation of e"' to order x2•
Problem 1.6.4. According to the Theory of Relativity. if an event occurs at a space-time point (x, t) according to an observer, another moving relative to him at speed v (measured in units in which the velocity of light c = 1) will ascribe to it the coordinates
x' = ~ x- vt (1.6.1)
t' t- vx
(1.6.2)
= ~ã
VerijY that 8, the space-time interval is same for both: 82 = t2 - x2 = t'2 - x'2 =
s'2• Show that if we parametrize the transformation terms of the rapidity(}, x' = xcoshB-tsinhB
t' = t cosh(} - x sinh(}
(1.6.3) ( 1.6.4) the space-time interval will be automatically invariant under this transformation thanks to an identity satisfied by hyperbolic functions. Relate tanh(} to the velocity.
Suppose a third observer moves relative to the second at a speed v', that is, with
rapidity 9'. Relate his coordinates (x", t") to (x, t) going via (x'.t'). Show that the rapidity parameter 9" = 6' + 9 in obvious notation. (You will need to derive a formula for tanh( A+ B).) Thus it is the rapidity, and not velocity that really obeys a simple addition rule. Show that if v and v' are small (in units of c), that this reduces to the daily life rule for addition of velocities. (Use the Taylor series for tanh9.) This is an example of how hyperbolic functions arise naturally in
mathematical physics.
Problem 1.6.5. A magnetic moment 1-1 in a magnetic field h has energy E± = T#Jh
when it is parallel (antiparallel) to the field. Its lowest energy state is when it is aligned with h. However at any finite temperature, it has a nonzero probabilities for being parallel or antiparallel given by P (par) I P ( antipar) = exp[-E+/T]/ exp[-E-/T] where T is the absolute temperature. Using the fact that the total probability must add up to 1. evaluate the absolute probabilities for the two orientations. Using this show that the average magnetic moment along the field h is m = 1-1 tanh(~Jh /T) Sketch this as a function of temperature at fued h. Notice that if h = 0, m vanishes since the moment points up and down with equal probability. Thus h is the cause of a nonzero m. Calculate the susceptibility,
';;: lh=O as a function ofT.
Problem 1.6.6. Consider the previous problem in a more genera/light. According to the laws of Statistical Mechanics if a system can be in one of n states labeled by an index i, with energies Ei, then at temperature T the system will be in state i with a relative probability p(i) = e-IJE, where /3 = 1/T. Introduce the partition function Z = Ei e -tJE,. First write an expression for P ( i ), the absolute probability (which must add up to 1). Next write a formula for (V), the mean value of a variable V that takes the value Vi in state i, i.e., (V) is the average over all allowed values, duly weighted by the probabilities. Show that < E > = - ~.
Give an explicit formula for Z for the previous problem. Show that ~ gives the mean moment along h. Use the formula for Z. evaluate this derivative and verify that it agrees with the result you got in the last problem.
Problem 1.6.7. A wire of length L is used to fence a rectangular piece of land.
For a rectangle of general aspect ratio compute the area of the rectangle. Use the rule for finding the maximum of a function to find the shape that gives the largest area. Find this area.
Problem 1.6.8. Sketch and locate the maxima and minima of f(x) = (x2 - 5x +
6)e-"'.
Problem 1.6.9. Find the first and second derivatives of f(x) = ez/(l-z) at the origin.
Problem 1.6.10. Imagine a life guard situated a distance dt from the water. He sees a swimmer in distress a distance L to his left and distance d2 from the shore.
Given that his speed on land and water are v1 and v2 respectively, with v1 > v2,
what trajectory will get him to the swimmer in the least time? Does he rush towards the victim in a straight line joining them, does he first run on land until he is in front qf the victim and then swim, does he head for the water first and then swim over. or does he do something else? Pick some trajectory composed of two straight line segments in each medium (why) and show that for the least time ::: :! = ~
where the angles ei are the angles of the segments with respect to the normal to the shoreline.
This problem has an analog in optics. If light is emitted at a point in a medium where its velocity is v1 and a"ives at a point in an aqjacent medium where its velocity is v2, the route it takes is arrived at in the same fashion since light takes the path of least time. The above equation is called Snell's Law.
Problem 1.6.11. The volume of a sphere is V(R) = 41rt. What is the rate of change of the volume with re.vpect to R? Does it make sense?
Problem 1.6.12. (Implicit Dlfferendadon). You know how to find the derivative dy/dx when y(x) is given. Suppose instead I tell you that y and x are related by an equation, say x2 + y2 = R2 and ask you to find the derivative at each point.
There are two ways. The first is to solve for y as a function of x and then let your spinal column take over, i.e., by changing x infinitesimally and computing the corresponding change in y given by the functional relation. The second is to imagine changing x and y infinitesimally while preserving the constraining relation (a circle in our example). The latter condition allows us to relate the if!finitesimals f:J.x and f:J.y and allows us to compute their ratio in the usual limit. Show that the derivative computed this way agrees with the .first method.
Find the slope at the point (2, 3) on the ellipse 3x2 + 4y2 = 48 using implicit differentiation.
Problem 1.6.13. Find the stationary points of f(x) = x3 - 3x + 2 and classify them as maxima or minima.
f
----;---~---x
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Figure 1.8. The meaning of the differentials d/ and d:J:.
1. 7. Differentials
Consider a function f(x) shown in Fig. 1.8.
If we change x by Ax at the point xo, we write the change in f as df I
A/ = dx "'o Ax + ... (1.7.1)
where the dots stand for tenns of order (Ax)2 and beyond. We expect the latter to be relatively insignificant as Ax -+ 0. Let us now introduce the differentials df and dx such that
df = dx df I .., dx (1.7.2)
0
with no approximation or requirement that either differential be small. What this means is that df is the change the function would suffer upon changing x by dx,
if we moved along the tangent to the function at the point xo as in Fig. 1.8.
Note that we always have the option of taking dx vanishingly small, in which case df, which is the change in f to first order in dx, becomes a better and better approximation to A/, the actual change in f (and not just along its tangent at xo).
This is always how we will use the differentials in this book, although the concept has many other uses. Thus when you run into an equation involving differentials you should say: "I see he is working to first order in the change dx." The advantage of using df will then be that I don't have to keep saying to "to first order" or use the string of dots.