Tentamen i GRUNDLÄGGANDE MATEMATISK FYSIK

Transcription

1 Karlstads Universitet Fysik Tentamen i GRUNDLÄGGANDE MATEMATISK FYSIK [ VT 2008, FYGB05] Datum: Tid: Lärare: Jürgen Fuchs c/o Carl Stigner Tel: Total poäng: 28 Godkänd: 50 % Väl godkänd: 75 % Tentan består av 2 delar som inlämnas separat: Del 1: 5 p. Del 2: 23 p. Hjälpmedel: Del 1 & 2: Ordbok engelska svenska Del 2 (efter del 1 har inlämnats) dessutom: Ett handskrivet A4 ark med valfritt innehåll (skrivet på ena sidan, ej maskinskriven eller maskinkopierad) inlämnas tillsammans med tentan Endast en uppgift per sida. Svaren måste vara väl motiverade. FYGB05 Tentamen

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3 Problem 1 Basics: Curvilinear coordinates Separately for Cartesian coordinates, cylindrical coordinates, and spherical coordinates, 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 ds(i) on each of these three surfaces. Problem 2 Basics: Vector algebra a Describe in words and/or draw the following: 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) 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) 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) 0. FYGB05 Tentamen

4 Problem 3 Basics: Series Give an example of a series a n which is divergent; a series a series a n which is absolutely convergent; a n which is convergent, but not absolutely convergent; a power series a n x n which converges for all x R; a power series a n x n having a finite (but non-zero) radius of convergence. Problem 4 Basics: Differential equations 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 an ordinary point and a singular point of a linear ordinary differential equation? What is the difference between a regular singular point and an irregular singular point of a linear ordinary differential equation? FYGB05 Tentamen

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6 Problem 5 Vector algebra 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 Compute the three volume integrals z 2 dv, (x 2 + z 2 ) dv and V V over the ball of radius 3 centered at the origin. V (x 2 + y 2 ) dv, FYGB05 Tentamen

7 Problem 7 Vector analysis and integrals 6 p. a Compute the value of the line integral F dl C 1 of the vector field F = (2 x 5 y) e x (5 x 2 y) e y along the curve C 1 that consists of the part of the circle x 2 +y 2 = 2 between the points (1, 1) and ( 1, 1). b Compute C 2 F dl, where F is the same vector field as in part a and C 2 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 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 F ds, S as a volume integral, where F is the vector field F = (4 x 2 3 y 2 ) e x + (9 y 8 xy) e y + ( 5 z + 3 xy) e z and S is the entire surface (i.e., including top and bottom pieces) of the full cylinder given by 0 x 2 +y 2 9 and 7 z 7. Compute the value of the resulting volume integral. FYGB05 Tentamen

8 Problem 8 Power series a For which values of x R does the power series ( 5 x 1 ) n f(x) = 7 converge? b For which values of x R does the power series 1 g(x) = x 2 + n x + n 2 converge? n=1 Problem 9 Fourier series 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. For which values of x R does this Fourier series converge to f(x)? FYGB05 Tentamen

9 Problem 10 Fourier transform Determine the Fourier transform of the function { d for 2 x 2, f(x) = 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 = 0 sin(t) cos(t) t dt. Problem 11 Matrices 3 p. a Find the eigenvalues and eigenvectors of the matrix ( ) 1 1 M =. 2 0 b Use the result of part a to bring M to diagonal form, and compute the exponentiated matrix e M. c Find the eigenvalues and eigenvectors of the matrix N = FYGB05 Tentamen

10 Problem 12 Differential equations 4 p. a Using the Frobenius method, find two independent series solutions of the differential equation 2 x 2 f (x) + 3 xf (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 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 ( particle in a box -problem). FYGB05 Tentamen