8—Multivariable Calculus 2458.27 Compute the area of an ellipse having semi-major and semi-minor axesaandb. Compare your result to
that of Eq. ( 22 ). Ans:πab
8.28 Two equal point chargesq are placed at z =±a. The origin is a point of equilibrium,E~ = 0there.
Compute the potential near the origin, writingV in terms of powers ofx,y, andznear there, carrying the powers
high enough to describe the nature of the equilibrium point. IsV maximum, minimum, or saddle point there?
(b) Write your result forV near the origin in spherical coordinates also.
8.29 When currentI flows through a resistanceRthe heat produced isI^2 R. Two terminals are connected
in parallel by three resistors having resistance R 1 , R 2 , and R 3. Given that the total current is divided as
I = I 1 +I 2 +I 3 , show that the condition that the total heat generated is a minimum leads to the relation
I 1 R 1 =I 2 R 2 =I 3 R 3. You can easily do problem 8 by eliminating a coordinate the doing a derivative. Here it’s
starting to get sufficiently complex that you should use Lagrange multipliers. Doesλhave any significance this
time?
8.30 Given a right circular cylinder of volumeV, what radius and height will provide the minimum total area for
the cylinder. Ans:r= (V/ 2 π)^1 /^3 ,h= 2r
8.31 Sometimes the derivative isn’t zero at a maximum or a minimum. Also, there are two types of maxima and
minima; local and global. The former is one that is max or min in the immediate neighborhood of a point and
the latter is biggest or smallest over the entire domain of the function. Examine these functions for maxima and
minima both inside the domains and on the boundary.
|x|, (− 1 ≤x≤+2)
T 0(
x^2 −y^2)
/a^2 , (−a≤x≤a, −a≤y≤a)
V 0 (r^2 /R^2 )P 2 (cosθ), (r≤R, 3 dimensions)8.32 In Eq. ( 26 ) it is more common to specifyN andβ= 1/kT, the Lagrange multiplier, than it is to specify
NandE, the total energy. Pick three energies,E`, to be 1, 2, and 3 electron volts. What is the average energy,
E/N, asβ→∞(T→ 0 )?
(b) What is the average energy asβ→ 0?
(c) What aren 1 ,n 2 , andn 3 in these two cases?