PILOT
Calculus III
Fall 2015
Problem Set 5
1. Show that Newton’s potential V = −GmM/r satisfies Laplace’s equation
𝜕2𝑉 𝜕2𝑉 𝜕2𝑉
+
+
= 0 for (𝑥, 𝑦, 𝑧) ≠ (0,0,0).
𝜕𝑥 2 𝜕𝑦 2 𝜕𝑧 2
2. Show that the rectangular parallelepiped with fixed surface area and maximum volume is a cube.
3. A function 𝑢 = 𝑓(𝑥, 𝑦)with continuous second partial derivatives satisfying Laplace’s equation
𝜕2𝑢 𝜕2𝑢
+
=0
𝜕𝑥 2 𝜕𝑦 2
is called a harmonic function. Show that the function 𝑢(𝑥, 𝑦) = 𝑥 3 − 3𝑥𝑦 2 is harmonic.
4. Determine the second-order Taylor formula for the given function about the given point (x0, y0).
1
(𝑥0 , 𝑦0 ) = (0,0)
a. ) 𝑓(𝑥, 𝑦) = 2
,
𝑥 − 𝑦2 + 1
(𝑥0 , 𝑦0 ) = (0,0)
b. ) 𝑓(𝑥, 𝑦) = 𝑒 𝑥−𝑦 ,
(𝑥0 , 𝑦0 ) = (0,0)
c. ) 𝑓(𝑥, 𝑦) = sin(𝑥 + 𝑦) + cos(𝑥 − 𝑦) ,
5. Challenge if you want to try it
A function 𝑓: ℝ → ℝ is called an analytic function provided
𝑓 𝑘 (𝑥) 𝑘
𝑓(𝑥 + ℎ) = 𝑓(𝑥) + 𝑓 ′ (𝑥)ℎ + ⋯ +
ℎ +⋯
𝑘!
[In other words, the series on the right hand side of the equation converges and equals f (x + h).] Suppose f
satisfies the following condition: On any closed interval [a, b] there is a constant M such that for all 𝑘 =
1, 2, 3, … , |𝑓 (𝑘) (𝑥)| ≤ 𝑀𝑘 for all 𝑥 ∈ [𝑎, 𝑏]. Prove that f is analytic.
6. Let u be a C2 function on D (the unit disc) which is “strictly subharmonic”; that is, the following inequality holds:
𝜕2𝑢
𝜕2𝑢
∇2 𝑢 = ( 2 ) + ( 2 ) > 0.
𝜕𝑥
𝜕𝑦
Show that u cannot have a maximum point in 𝐷\𝜕𝐷 (the set of points in D, but not in ∂D.
7. a.) A path is said to be parameterized by arc length or, what is the same thing, to have unit speed is ‖𝐜′(𝑠)‖ = 1.
For a path parameterized by arc length on [a,b], show that l(c) = b – a.
b.) The curvature at a point c(s) on s path is defined by 𝑘 = ‖𝐓′(𝑠)‖ when the path is parameterized by arc
length. Show that 𝑘 = ‖𝐜′′(𝑠)‖.
c.) If c is given in terms of some other parameter t and 𝐜′(𝑡) is never 0, show that 𝑘 = ‖𝐜′(𝑡) × 𝐜′′(𝑡)‖/‖𝒄′(𝑡)‖𝟑 .