Problem Set # 5 / MATH 20400

Problem Set # 5 / MATH 20400
Due: Friday, February 13, 2015, at 13:30 CST
1. Let U ⊆ Rn be open and let f : U → Rm be differentiable on the subset V of U . Fix a ∈ V and
x ∈ Rn . Also fix i and j such that 1 ≤ i ≤ m and 1 ≤ j ≤ n. Finally, fix a multi-index α on Rn .
Describe precisely what each of the following objects is (for example, is it a number, a vector, a map, a
matrix? If it is a vector or a matrix, what are its dimensions? If it is a map, what are its domain and
codomain?). In the parts that involve α, you may assume V = U and f ∈ C |α| (U, Rm ). The purpose
of this problem is to fix in your mind precisely what the different objects are that we have discussed
in class.
(a) f , fi
(b) Df (a) (x), Dfi (a) (x), Df (a), Dfi (a), Df , Dfi
(c) f 0 (a), fi0 (a), f 0 , fi0
(d) Dj f (a), Dj fi (a), Dj f , Dj fi
(e) Dα f (a), Dα fi (a), Dα f , Dα fi
(f) ∇fi (a), ∇fi
2. Let g ∈ C 1 (Rn , Rp ) and let f : Rp → Rm be differentiable at g (a) ∈ Rp . Define F : Rn → Rm by
F := f ◦ g. Prove that, for 1 ≤ i ≤ m and 1 ≤ j ≤ n
Dj Fi (a) =
p
X
D` fi [g (a)] Dj g` (a) .
`=1
Remark: This is an extremely important and useful corollary of the chain rule. In fact, in many
books, this is called “the chain rule”. This says, for example, that if f = f (x, y) and x = x (u, v) and
y = y (u, v), then, if we define F (u, v) by F (u, v) = f (x (u, v) , y (u, v)), we have
D1 F (u, v) = D1 f [x (u, v) , y (u, v)] D1 x (u, v) + D2 f [x (u, v) , y (u, v)] D1 y (u, v)
and
D2 F (u, v) = D1 f [x (u, v) , y (u, v)] D2 x (u, v) + D2 f [x (u, v) , y (u, v)] D2 y (u, v) .
These are written in various other ways in different textbooks. One way would be
Du F (u, v) = Dx f (x, y) Du x (u, v)+Dy F (x, y) Du y (u, v)
and
Dv F (u, v) = Dx f (x, y) Dv x (u, v)+Dy F (x, y) Dv y (u, v) .
Another would be
∂f
∂x
∂f
∂y
∂F
(u, v) =
(x, y)
(u, v)+
(x, y)
(u, v)
∂u
∂x
∂u
∂y
∂u
and
∂F
∂f
∂x
∂f
∂y
(u, v) =
(x, y)
(u, v)+
(x, y)
(u, v) .
∂v
∂x
∂v
∂y
∂v
A particularly compact way to denote this would be to write
fu = fx xu + fy yu
and fv = fx xv + fy yv .
All of these different formulas are just different notations for the same idea. The point is that the result
of this problem gives you an easy way to compute partial derivatives of a function whose arguments
themselves are functions without having to compute the full chain-rule derivative of the given function.
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3. The proof of Theorem 4.5.1 of your textbook contains an error. Specifically, one of the equations that
is written down is not correct. It can be corrected if an additional hypothesis is added to the statement
of the theorem. Find and correct the error.
4. Define f : R2 → R by
f (x, y) :=
xy (x − y) (x + y) x2 + y 2
0,
−1
,
if (x, y) 6= (0, 0) ,
if (x, y) = (0, 0) .
Prove that D12 f (0, 0) 6= D21 f (0, 0). Does this contradict anything we have proven in class? Explain.
5. In each of the following, an f : R2 → R is given. Write down the Taylor polynomials of degrees 0,
1, and 2,
called T0 , T1 , and T2 in class, at the point x0 = (0, 0). Use them to estimate the value of
1 1
,5 .
f 10
(a)
(b)
(c)
(d)
(e)
f (x, y) = sin (xy).
f (x, y) = ex+y .
f (x, y) = cos x2 + y .
f (x, y) = ex sin y.
f (x, y) = x + xy + 2y 2 .
6. Suppose f ∈ C 2 R2 is given, and define g : R2 → R by g (r, θ) := f (x, y), where x = r cos θ and
y = r sin θ. Prove that the following identities hold for all (r, θ) ∈ (0, +∞) × (−π, π]:
(a) D1 f (x, y) = D1 g (r, θ) cos θ − r−1 D2 g (r, θ) sin θ.
(b) D2 f (x, y) = D1 g (r, θ) sin θ + r−1 D2 g (r, θ) cos θ.
(c) D12 g (r, θ) + r−1 D1 g (r, θ) + r−2 D22 g (r, θ) = D12 f (x, y) + D22 f (x, y).
7. Suppose f ∈ C 2 (Rn ) is radial and f = g ◦ r as in Problem Set #4. Prove that if x 6= 0, then
−1
00
g (kxk) + kxk
0
(n − 1) g (kxk) =
n
X
Dj2 f (x) .
j=1
8. In class, we proved Taylor’s theorem for real-valued functions. Explain how the proof should be
modified if we want it to hold for f ∈ C N +1 (U, Rm ) instead of just f ∈ C N +1 (U ).
9. In each of the following, a smooth hypersurface S in R3 is given and a point (a, b, c) ∈ S is given. Find
the equation for the tangent hyperplane to S at (a, b, c).
(a) S = (x, y, z) ∈ R3 : x2 + y 2 + z 2 = 49 ; (a, b, c) = (6, 2, 3).
(b) S = (x, y, z) ∈ R3 : xy + yz + xz − 1 = 0 ; (a, b, c) = (1, 1, 0).
(c) S = (x, y, z) ∈ R3 : x2 y 2 + xz − 2y 3 = 10 ; (a, b, c) = (2, 1, 4).
(d) S = (x, y, z) ∈ R3 : sin (xy) + sin (yz) + sin (xz) = 1 ; (a, b, c) = 1, 21 π, 0 .
(e) S = f −1 ({0}), where f (x, y, z) = x − e2y−z ; (a, b, c) = (1, 1, 2).
10. For each given f , find the set of real numbers k such that S := f −1 ({k}) defines a smooth hypersurface.
For those k, also find the equation for tangent hyperplane to S at a ∈ S.
(a) f : R3 → R defined by
f (x, y, z) :=
(b) f : R4 → R defined by
x−1 + y −1 + z −1 , x 6= 0, y 6= 0, and z 6= 0,
0,
x = 0 or y = 0 or z = 0.
f (x, y, z, w) := x2 sin y + y 2 sin z − zwexy .
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