Tentamen i GRUNDLÄGGANDE MATEMATISK FYSIK

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Karlstads Universitet Fysik

Tentamen i

G

RUNDL ¨

AGGANDE

M

ATEMATISK

F

YSIK

[ VT 2008, FYGB05]

Datum: 2008 - 03 - 26 Tid: 8.15 – 13.15 L¨arare: J¨urgen Fuchs

c/o Carl Stigner Tel: 054 - 700 1815

Total po¨ang: 28 Godk¨and: 50 % V¨al godk¨and: 75 %

Tentan best˚ar av 2 delar som inl¨amnas separat : Del 1: 5 p.

Del 2: 23 p.

Hj¨alpmedel:

Del 1 & 2: Ordbok engelska ←→svenska Del 2 (efter del 1 har

inl¨amnats) dessutom : Ett handskrivet A4 ark med valfritt inneh˚all (skrivet p˚a ena sidan,

ej maskinskriven eller maskinkopierad) –inl¨amnas tillsammans med tentan –

Endast en uppgift per sida. Svaren m˚aste vara v¨al motiverade.

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Problem 1 – Basics: Curvilinear coordinates – 1 p.

1 p. Separately for Cartesian coordinates, cylindrical coordinates, and spherical

co-ordinates, describe in words and/or draw the following:

• the infinitesimal volume element dV ;

• for each i∈ {1,2,3}, a surface S(i) on which the ith coordinate has a

constant value;

• the vector-valued infinitesimal surface element d~S(i) on each of these

three surfaces.

Problem 2 – Basics: Vector algebra – 2 p.

a Describe in words and/or draw the following: 1 p.

• the geometric meaning of the gradient ∇~f(~r) of a scalar function f(~r) ;

• the geometric meaning of the divergence ∇·~ F~(~r) of a vector field F~(~r) .

b Give examples (describing them in words and/or using drawings) 1 p. for the following types of fields:

• a vector field F~1(~r) which satisfies ∇·~ F~1(~r) = 0 , and a vector field F~2(~r) which satisfies ∇·~ F~2(~r)6= 0 ;

• a vector field F~3(~r) which satisfies ∇×~ F~3(~r) = 0 , and a vector field F~4(~r) which satisfies ∇×~ F~4(~r)6= 0 .

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Problem 3 – Basics: Series – 1 p. 1 p. Give an example of • a series ∞ X n=0 an which is divergent; • a series ∞ X n=0

an which is absolutely convergent;

• a series ∞ X

n=0

an which is convergent, but not absolutely convergent;

• a power series ∞ X

n=0

anxn which converges for all x∈R;

• a power series ∞ X

n=0

anxn having a finite (but non-zero) radius of convergence.

Problem 4 – Basics: Differential equations – 1 p.

1 p.

• Which differential equations have the property that every linear combination of solutions is again a solution?

• How is the general solution of an inhomogeneous linear ordinary differential equation related to the general solution of the corresponding homogeneous differential equation?

• What is the difference between anordinary point and a singular point of a linear ordinary differential equation?

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Problem 5 – Vector algebra – 2 p.

2 p. Determine the perpendicular distance between the point (4,7,2) and the

straight line that joins the points (−2,1,−1) and (2,9,3) .

Problem 6 – Curvilinear coordinates – 2 p.

2 p. Compute the three volume integrals

Z V z2dV, Z V (x2 +z2) dV and Z V (x2+y2) dV , over the ball of radius 3 centered at the origin.

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Problem 7 – Vector analysis and integrals – 6 p.

a Compute the value of the line integral 2 p.

Z

C1 ~ F ·d~ℓ

of the vector field

~

F = (2x−5y)~ex−(5x−2y)~ey

along the curve C1 that consists of the part of the circle x2+y2= 2 between

the points (1,1) and (−1,−1) .

b Compute 1 p.

Z

C2 ~ F ·d~ℓ,

where F~ is the same vector field as in part a and C2 is the straight line

from (1,1) to (−1,−1) .

c Obtain the curl of F~ and show that F~ can be written as the gradient of a 1 p. scalar function.

Discuss how this is related to the results of parts a and b .

d Use the divergence theorem to rewrite the surface integral 2 p. I

S

~ F·d~s,

as a volume integral, where F~ is the vector field

~

F = (4x23y2)~e

x+ (9y−8x y)~ey+ (−5z+ 3x y)~ez

and S is the entire surface (i.e., including top and bottom pieces) of the full cylinder given by 0≤x2+y2 ≤9 and −7≤z ≤7 .

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Problem 8 – Power series – 2 p.

a For which values of x∈R does the power series 1 p.

f(x) = ∞ X n=0 5x−1 7 n converge?

b For which values of x∈R does the power series 1 p.

g(x) = ∞ X n=1 1 x2+n x+n2 converge?

Problem 9 – Fourier series – 2 p.

2 p. Determine the Fourier series for the function f that has period 2a and in the

range −a≤x < a is given by

f(x) = (

x for −a≤x <0,

0 for 0 ≤x < a .

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Problem 10 – Fourier transform – 2 p.

2 p. Determine the Fourier transform of the function

f(x) =

d for −2≤x≤2,

0 else (with d constant).

Write down an expression for the inverse transformation and show that it can be used to obtain the value of the definite integral

I = Z ∞ 0 sin(t) cos(t) t dt. Problem 11 – Matrices – 3 p.

a Find the eigenvalues and eigenvectors of the matrix 1 p.

M = 1 −1

−2 0

! .

b Use the result of part a to bring M to diagonal form, 1 p. and compute the exponentiated matrix eM.

c Find the eigenvalues and eigenvectors of the matrix 1 p.

N =    1 2 −1 0 2 −1 0 5 −4   .

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Problem 12 – Differential equations – 4 p.

a Using the Frobenius method, find two independent series solutions of the dif- 2 p. ferential equation

2x2f′′

(x) + 3x f′

(x)−(1 +x)f(x) = 0 around its regular singular point x= 0 .

(If you cannot solve the recursion relation, work out the first few coefficients numerically.)

b Determine the eigenvalues of the three-dimensional Laplace operator for the 2 p. situation that the eigenfunctions are required to vanish at the surface of the

cuboid given by −π ≤x≤π, −2π≤y ≤2π and −3π ≤z ≤3π

Figure

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