PHYS 216 Final, Fall 2012

Physics 216, Final Exam January 10, 2013, 2 hours Solve ten problems out of twelve 1. Check the divergence theorem for the vector −→ V = r2 cos θ�r + r2 cos φ�θ − r2 cos θ sin φ�φ using as your volume a hemisphere of radius R. Take the flat side in the x − y plane. 2. Expand the function f (x) = x2, 0 < x < 2π in a Fourier series if the period is 2π. 3. Find the eigenvalues and normalized eigenvectors of the matrix A = � cos θ sin θe−iϕ sin θeiϕ − cos θ � 4. Let A = � a b c d � be an orthogonal matrix satisfying AT A = I where I is the identity matrix. Find the most general solution for a, b, c, d satisfying the orthogonality condition. Compare with the rotation matrix in two dimensions. 5. Let {e1, e2, e3} be a basis of a three dimensional vector space V3 and T an operator that transforms this basis to the vectors {e′ 1, e′ 2, e′ 3} where e′ 1 = T (e1) = e1 + e3, e′ 2 = T (e2) = 2e1 + e2, e′ 3 = T (e3) = 3e2 + e3. Find the matrix representation of the operator T and determine whether the vectors {e′ 1, e′ 2, e′ 3} are independent. 6. Find the Fourier transform of the function f (x) = � 1 − |x| , −a ≤ x ≤ a 0, otherwise . 7. Verify the delta function expansion δ (1 + x) = ∞ � n=0 (−1)n 2n + 1 2 Pn (x) 8. Find the analytic function f (z) = u (x, y) + iv (x, y) where u (x, y) = x3 − 3xy2. 9. Let {Qn, n = 0, · · · , ∞} be an orthogonal set of Sturm-Liouville polynomials with respect to the weight function w (x) ∞ � −∞ w (x) Qn (x) Qm (x) dx = 0, n ̸= m where Qn (x) satisfies the differential equation 1 w d dx � wα (x) dQn dx � + λnQn = 0. Show that � Q′ n = dQn dx , n = 1, · · · , ∞ � satisfy the property ∞ � −∞ w (x) α (x) Q′ n (x) Q′ m (x) dx = 0, n ̸= m Hint: Integrate by parts only on one of the dQn dx and use the defining equation for Qn. 1 10. Expand f (z) = z3 − 2z2 + 1 (z − 3)2 (z2 + 3) in a Laurent series about z = 3 and z = ±i √ 3. 11. Using complex integration over the unit circle show that 2π � 0 dθ a ± b cos θ = 2π (a2 − b2) 1 2 , a > |b| 12. Find the residues of the function f (z) = ez z2 (z2 + 9) about the isolated singular points. 2