24102018Dr Jonas HirschFlorian Oschmann

Mathematics 3Mock Exam

Exercise 1

Cross the statements which are true

1 Let AS sub Rn be arbitrary then

|S| ge |S capA|+ |S A|

Let AS sub Rn be measurabel then

|S| ge |S capA|+ |S A|

Let S sub Rn be arbitrary A sub Rn be measurable then

|S| ge |S capA|+ |S A|

2 The following set is a C1 regular hypersurface

M1 = part[0 1]n = x = (x1 xn) isin Rn xi isin 0 1 for some i

M2 = (x f(x)) x isin Rn for f isin C1(RnR)

M3 = (cos(θ) + 1 sin(θ)) θ isin [0 2π] cup (cos(θ)minus 1 sin(θ)) θ isin [0 2π]

3 Let f isin L1(Rn) be given but otherwise arbitrary then we have

supξisinRn |f(ξ)| lt infin

983125Rn |f |2 lt infin

983125Rn gf =

983125Rn gf for every g isin L1(Rn)

4

Let X isin C1(B1R3) with div(X) = 0 then X = nablaf for some f isin C1(B1R) Let X isin C1(B1R3) with div(X) = 0 then X = curlY for some Y isin C1(B1R3)

Let f isin C1(B1R) then curl(nablaf) = 0

Exercise 2

Let f isin L1(Rn) Show that for p ge 1 one has

983133

Rn

|f |p = p

983133 infin

0tpminus1|x f(x) ge t| dt

Hint As on the exercise sheet write |f(x)|p as an certain integral and apply Fubinistheorem Argue shortly why this is allowed

1

Exercise 3

What is the volume of the enclosed ellipsoid

x2

a2+

y2

b2+

z2

c2= 1

Exercise 4

Suppose u isin C2(B1) u(x) = 0 if |x| = 1 satisfies

∆u = λu in B1 sub R3

Show that 983133

B1

|nablau|2 + λu2 = 0

and argue why λ lt 0 if u is not constant 0

Exercise 5

Consider the set

H = f isin L2((minusππ)) f(minusx) = minusf(x) for ae x

We endow it with inner product (f g) = 12π

983125 πminusπ f g

1 Show that H is a Hilbert space

2 Show that the familyradic2 sin(nx) with n isin N form an orthonormal basis

2

Exercise 3

What is the volume of the enclosed ellipsoid

x2

a2+

y2

b2+

z2

c2= 1

Exercise 4

Suppose u isin C2(B1) u(x) = 0 if |x| = 1 satisfies

∆u = λu in B1 sub R3

Show that 983133

B1

|nablau|2 + λu2 = 0

and argue why λ lt 0 if u is not constant 0

Exercise 5

Consider the set

H = f isin L2((minusππ)) f(minusx) = minusf(x) for ae x

We endow it with inner product (f g) = 12π

983125 πminusπ f g

1 Show that H is a Hilbert space

2 Show that the familyradic2 sin(nx) with n isin N form an orthonormal basis

2