Hur¸sit ¨Onsiper
Office hours : Mon. 10:40-11:30, Thurs. 10:40-11:30. Schedule : Mon. 13:40-15:30 (M-13)
Wed. 13:40-15:30 (M-13) Assistant: Burak Yıldız
Office : Z-47
Office hours : Thurs. 10:40-12:30 Course outline :
Topology ofRn.
Functions of several variables; limits and continuity. Partial derivatives, directional derivatives, gradients. Differentials and the tangent plane: the fundamental lemma, approximations. The mean value theorem, implicit and the inverse function theo-rems. Extreme values.
Introduction to vector differential calculus: the gradient, divergence and curl. Curvi-linear coordinates.
Prerequisite: MATH 154. References :
1. T.M.Apostol, Mathematical Analysis, Addison-Wesley (1974).
2. W.Rudin, Principles of Mathematical Analysis, McGraw Hill (1976). 3. V.A.Zorich, Mathematical Analysis I, Universitext, Springer-Verlag (2004). Grading :
Midterm 1 Nov.3, 2014 30 % Midterm 2 Dec.8, 2014 30 %
1. Find the sup and inf of the following sets. a) S ={1 2p + 1 3q + 1 5r :p, q, r∈Z+}. b) S ={x∈R:x2+ 3x+ 1 <0}.
2. Let X ⊂Rn. Define Int(S) to be the set of all interior points of S. Show that Int(S) is an open set.
3. Determine Int(S), Ext(S) and Bd(S) for the following sets. a) S ={(x, y)∈R2 :x−y2+ 1≤0}.
b) S =Z∪ {1
n :n≥1} ⊂R. c) S =Q×Q⊂R2.
4. Show that{(1/n,2/n) :n≥2}is an open cover of the interval (0,1) and that this cover has no finite subcover.
5. Define the closure E of a set E ⊂Rn to be the smallest closed set containing E.
a) Show thatE exists and in fact E =E∪ Bd(E). b) Determine the closure of the following sets :
Q⊂R, Q×Q⊂R2, Z×Q⊂R2, {(x, y, z)∈R3 : 1/2< x2+y2+z2 <1}. 6. True or false ? (Prove the statement or give a counter example).
a) LetA, B ⊂R be bounded sets and let A.B be the set {xy:x∈A, y∈B}. Thensup(A.B) =sup(A).sup(B).
b) Let Ei ⊂Rn, i = 1, .., m be connected sets such that∩Ei 6=∅. Then ∪Ei is connected.
c) Let E, F be sets in Rn such that E ⊂F ⊂E. Then, if E is connected, so isF.
1. Define f : [0,1]→R by f(x) =
x for x∈Q
1−x for x6∈Q. Show that a)f([0,1]) = [0,1], and
b)f is continuous only atx= 1/2.
2. Let f :R2 →R2 be given by f(x, y) = ([|x|], y−[|y|]).
a) Determine the range of f.
b) Find the largest setU ⊂R2 in whichf is continuous.
3. Let f :R2− {(0,0)} →
R be defined by f(x, y) = x
2−y2
x2+y2.
Compute lim(x,y)→(0,0)f(x, y), whenever it exists, along the following curves :
(i)y=ax, (ii) y=ax2, (iii) y2 =ax. Does lim(x,y)→(0,0)f(x, y) exist ? Why ?
4. Define f :R2 → R by f(x, y) = sin(xy)/x if x6= 0 y if x= 0.
a) Computelimx→0(limy→0f(x, y)), limy→0(limx→0f(x, y)).
b) Doeslim(x,y)→(0,0)f(x, y) exist ?
5. For a function F : E ⊂ Rm →
Rn, we define the graph of F to be the set Gr(F) = {(p, F(p))∈Rm×
Rn :p∈E}.
Show that if E is a compact set, then F is continuous on E if and only if Gr(F) is compact.
6. Let Y ⊂ R be a bounded set and let f : Y → R be a uniformly continuous function. Show that f is a bounded function (ie. the image set f(Y) is a bounded set).
1. Consider the function f(x, y) = x2y2 x4+y4 if (x, y)6= (0,0) 0 at (0,0). a) Find ∂f ∂x, ∂f ∂y at all p∈R 2.
b) Show thatf does not have a differential at (0,0). c) Find allp∈R2 wheref is differentiable.
2. Consider f(x, y) = x3 x2+y2 if (x, y)6= (0,0) 0 at (0,0). Show that a)f is continuous in R2.
b) If γ : [0,1]→R2 is a differentiable curve, then the composite map
f◦γ : [0,1]→R is differentiable. c) f is not differentiable at (0,0). 3. Can you find a function F : R2 →
R2 for which the differential dF is given by the matrix 3x2y x3 x x ?
4. Let U ⊂ R2 be an open set and f :U ⊂
R2 → R be a function with partial derivatives ∂f
∂x, ∂f
∂y defined and bounded everywhere inU. Show thatf is a continuous function.
5. Using Taylor’s Theorem, expand f(x, y) = x3+y3 +xy in powers of
(x−1), (y−2).
6. For each of the following functions determine all points, if any, where the function has a local differentiable inverse and write this inverse function. a)F(x, y) = (x+y,2xy2).
1. For each of the following pairs of functionsf1(x, y), f2(x, y), determine if there
is a functional relation Φ(f1, f2) = 0. In case it exists, find the relation.
(i)f1(x, y) =x2, f2(x, y) = y/x.
(ii) f1(x, y) = x2+ 2xy+y2, f2(x, y) = 2x+ 2y.
(iii) f1(x, y) = xcos(y), f2(x, y) = xsin(y).
2. For each pair of functions given in question 1, determine the largest connected regionsU, V ⊂R2, if any, between which the transformation
T = (f1, f2) :U →V defines a coordinate transformation.
3. Consider the plane curvexy+y2−ex = 0. Can you describe this curve in the formx=g(y) in some neighborhood of (0,-1) ? Why ?
Determine the points p0 = (a, b) on the curve, if any, where the curve can not
be written in the formx=g(y).
4. Consider the surface S ⊂R4 given by the equations
u1+ sin(u2) +eu3u4 = 0,
u1u3+u4 = 0.
Find all pointsp∈S around which we can writeS in the form
φ∇ ◦F +F ◦ ∇φ.
2. Let F, G:R3 →R3 be two vector fields. Prove the following identities.
∇ ◦(F ×G) = (∇ ×F)◦G−F ◦(∇ ×G). ∇ ×(∇ ×F) =∇(∇ ◦F)− ∇2F.
3. Let f : R → R be continuously differentiable. Define Φ : R3 →
R3 by Φ(p) =f(|p|). Show that there exists a scalar fieldλsuch that∇Φ(p) = λ(p)p if p6=0.
4. Find an example of a scalar field φ and a vector field F neither of which is constant, for which∇ ◦(φF) is identically equal to φ∇ ◦F.
5. Find a vector field F :R3 →R3 such that ∇ ×F =y~i+x~j.
6. Consider the family of curves xy =c, c > 0 in
Ω = {(x, y)∈R2 :x >0, y >0}.
a) Find the orthogonal trajectoriesv(x, y) =d of this family.
b) Write the operator ∇ in Ω, in the orthogonal coordinate system thus ob-tained.
a) Determine Int(X).
b) Determine all limit points of X. c) Is X open? Closed ?
d) Is the closureX of X compact ?
Can you obtain a connected set by adding onlyone point to X ? How ? 2. True or false ? (Explain why the statement is true, or give a counter example.)
a) LetX ⊂ R be a compact and connected set. Then R−X consists of two disjoint open nonempty sets.
b) If f : Rn →
R is a continuous function, then for any compact set Y ⊂ R the setf−1(Y) is compact.
c) Let D ⊂ Rn be compact and f : D →
R be continuous. Let {pn} be a sequence inDsuch thatlimn→∞f(pn) = 3. Show that there existsp∈Dsuch that f(p) = 3.
d) One can find real valued functions which are continuous on
S={(x, y, z) :x2+y2+z2 ≤4}, but not uniformly continuous onS. e) If f(x)>0 and is uniformly continuous inR, then g(x) = 1
1 +f(x) is also uniformly continuous on R.
3. True or false ? Prove the statement or give a counter example. a) Suppose limx→0(limy→0f(x, y)) =L=limy→0(limx→0f(x, y)).
Thenlim(x,y)→(0,0)f(x, y) =L.
b) Let f : R2 → R be such that f(x,0) and f(0, y) are continuous at (0,0). Then f is continuous at (0,0).
c) Iff :R2 →
R has directional derivatives in all directions at a pointp∈R2, then f is differentiable atp.
4. a) Show that the function f(x, y) =
ysin(x/y) if y6= 0
5. Consider f :R2 → R, f(x, y) = x2sin(y) +y2sin(x) x2+y2 if (x, y)6= (0,0) 0 at (0,0).
a) Show thatf is continuous in R2.
b) Compute ∂f
∂x(0,0) and ∂f ∂y(0,0).
c) Show thatf is not differentiable at (0,0). 6. Consider the function f(x, y) = x2ye−(x2+y2).
a) Find and classify all critical points of f. b) Show thatlimkpk→∞f(p) = 0.
c) Show thatf has an absolute maximum and an absolute minimum. (Hint : Use part (b)).
7. Show that in a neighborhood of p= (1,−1,2), the curve of intersection of the surfaces x2(y2+z2) = 5 and (x−z)2 +y2 = 2 can be described in the form y=g(x), z =h(x).
8. Find the maximum and the minimum values of f(x, y, z) = xy+ 2z on the intersection of the surfaces x+y+z = 0 and x2+y2 +z2 = 24.
9. Suppose that F : R2 →
R3 has continuous first order partial derivatives and that F(1,1) = (2,2,1). Let dF at(1,1) be given by the matrix
2 0 1 2 0 0 . LetG:R3 →
R2 be the functionG(u, v, w) = (ewsin(u), ewcos(v)). a) Show thatG◦F has a local differentiable inverse around (1,1).
b) Show that forp0 ∈G−1((1,1)), F ◦G does not have an inverse around p0.
10. True or false ? Why ?
a) LetS ⊂R3 be the surface given by the equationz2−xey = 1. There exists no point onS where the tangent plane is parallel to the xy-plane.
c) The functionf(x, y) = |xy| is not differentiable at (0,0). d) If w=F(xz, yz), then we have x∂w
∂x +y ∂w ∂y =z ∂w ∂z. 11. True or false ?
a) E ⊂ Rn is such that any set F ⊂
Rn which contains E properly, is connected. ThenE is connected.
b) Letf :Rn →
Rm,g :Rm →RN be two functions such thatg and g◦f are differentiable everywhere. Then f is also differentiable.
c) The function F : R2 →
R2, F(x, y) = (xcos(y), xsin(y)) is differentiable but does not have a differentiable inverse in the upper half-plane.
d) Suppose that f :R2 →
R has continuous first order partial derivatives and that each point of a dense subset of R2 is a critical point for f. Then f is constant.
12. Suppose that the functions u : R2 → R, v : R2 → R have continuous first order partial derivatives everywhere.
a) Show that the function F(x, y) = exp((u(x, y) +v(x, y)) has a maximum and a minimum value in every disk D(r) ={(x, y) :x2+y2 ≤r2}.
b) Suppose that the function G(x, y) = (u(x, y), v(x, y)) has a local differen-tiable inverse around each point inR2. Then the points where F(x, y) attains
its maximum and minimum values inD(r) all lie on the boundaryx2+y2 =r2. c) Is the converse of the statement in (b) true ? Why ?