PHYS 216 Final, Fall 2018

vecember 15, 4U16, 4 hours Solve only 10 Problems out of 13 => ~ ~ . ~ . oo . Given the vector V = r? (cos Or + cos $8 — cos @ sin $9) . Find V -V and evaluate the volume integral / V - Vdv where V is the volume of the upper hemisphere of radius R centered at Vv the origin. Check the divergence theorem by comparing with the surface integral fv -d@ Ss where S is the boundary of V consisting of the upper hemisphere and the flat disk of radius => = => Rin zy plane. Hint: To find V -da@ on flat disk use F = cos@k +sin9 (cos gi +sin e3) * a => => => > i => = 6=-sindk +cos6 (cos pi +sindj),6=—singi +cos¢]. => = = = " . => . Consider the vector field F = (x+y) i +(x —y) 7. Evaluate the line integral pra where C C is the square of vertices O = (0,0), A= (1,—-1), B = (2,0), C = (1,1). Is the vector F conservative. Hint: Find equations of the lines OA, AB, BC, CA. . Let {€1, €, €’3} be a basis of a three dimensional vector space V3 and T’ an operator that transforms this basis to the vectors { 4, €5, €4} where €, = T(€i1) = €1+ @3, 5 =T (eo) = 2614 €o, €45=T(€3) =3e2+ €3. Show that the matrix representation 3 12.0 of the operator T where Te’; = Deer isT =] 0 1 8 |. Is T invertible? Find the j=l IES) et inverse of T by first finding the adjoint elements. . Find the eigenvalues and normalized eigenvectors of the matrix ee ne ( cos@ sin ée ) sinfe’? —cosé Express your answer in terms of sin § and cos g. ab ed matrix. Find the most general solution for a,b,c,d satisfying the orthogonality condition. Assume det A = 1 then express your answer in terms of trigonometric functions. Hint: Let a=cos6, b=cosé’. ) ber A ( ) be an orthogonal matrix satisfying ATA = I where I is the identity }. Given the function f (x) = {a—|x|,— a < x < a} of period 2a. Is f (x) even or odd? Find co the Fourier series for f (z). Use the series to show that Dee = ue k=0 _ -a<r< . Given the function f (x) = { t ile edn” ° Find the Fourier transform F (w) of f (x). Is F (w) even or odd. . Is the function f (a) even or odd? . Verify the delta function expansion 6(+a)= (1 =p, (2) n= 10. 1. Le 13. ai 2 Hint: Use the generating function g (#,t) = (1—2xt +t”)? to find the value of P, (—1). . Use the recursion relation for Hermite polynomials Doi, (z) = Aye (x) +2nHAy-1 (x) co 2 and orthogonality condition / e—* Ay (2) Hm (x) d& = Onms/72"n! to evaluate the integral / we Hy (2) Hn (2) dx = /72”n! (» + i) Find the analytic function f(z) = u(z,y) + iv (x,y) where u(a,y) = 2° — 3ay?. Derive the Laurent series for the function f(z) = Fa around the point z) = —2 for |z+2| < 2. Verify the values of aj and a_; in the expansion f(z) = 5 Gn (2 — 29)” n=—00 by comparing with the formula a, = Hh pede LO. Hint: Write z = (z2+2)-2= —2 (1 — 44) and to find the series a differentiate the series 7+... Prove that on Sa a Ta ne 0 by evaluating an appropriate contour integral. Hint: let C be the unit circle and z = e””. Find the residues of the function z i) = aay about the isolated singular points. Use theorem of residues to deduce the value of the integral pach (z) where C is the circle of center at the origin and radius equal to 4. Cc