PHAS0009 - Mathematical Methods II 2022
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PHAS0009 - Mathematical Methods II
2022
1. (a) Consider the scalar field 4(r) = q r3 with q a positive constant and r = ^x2 + y2 + z2 . Evaluate a surface integral to obtain the flux of the vector field G(r) = 史4 through the surface of a sphere of radius R centred at the origin. (You can use the vectorial surface area element dS of a sphere without derivation.)
(b) Calculate the flux by rewriting the surface integral of part (a)
as a volume integral, using Gauss’s theorem.
(c) Show that the line integral
I = F(r).dS
of the vector field
F(r) = 3y x + z y 一 2x z
along the closed triangular contour C from A(1, 0, 0) to B(0, 1, 0) to C(0, 0, c) (c > 0) and then back to A is equal to (c 一 3)/2.
(d) i. Obtain the result of question (c) by using Stokes’s
theorem and evaluating the resulting surface integral.
ii. Give a geometrical interpretation of why the flux vanishes for c = 3.
2. (a) Determine the general solutions of the following ordinary differential equations (ODEs),
i. = x2 一3x(y) 一 2 ii. x + 3y = x3 .
(b) Consider a damped harmonic oscillator with eigen-frequency o0 > 0 and damping y = 2o0 . The equation of motion for the displacement x as a function of time t is given by the
second-order ODE
dt2 + y dt + o0(2)x = 0.
Show that the solution that satisfies the initial conditions
x(0) = 0 and x\ (0) = v0 0 is given by
x(t) = v0 te–o0 t .
(c) The same damped oscillator is now subject to a periodic driving force, resulting in the modified equation of motion
d2x dx
Show that after a long time the system oscillates with
amplitude f0 /(o2 + o0(2)). (You might use that
仪 sin(ot) 一 β cos(ot) = ^仪2 + β2 sin(ot 一 4) with
4 = arctan(β/仪).)
3. Consider the matrix
╱ 1 0
0一b 、.
.
with real numbers a, b > 0. This matrix shall have unit
determinant, i.e. det M = 1.
(a) Show that for M to have a determinant of 1, the following
condition must hold:
a2 + b2 = 1 .
(b) Calculate the nine cofactors of M, then use these to assemble the inverse, M– 1 .
To which group of special matrices does M belong, given the form of its inverse?
(c) Show that the eigenvalues of M are 1, a + ib, and a 一 ib. Detail your calculation and describe the key steps with a brief bullet point.
(e) Calculate the square root of M. Write the complex
eigenvalues of M in polar form and make use of Euler’s formula, eiα = cos α + i sin α when simplifying the expression for ^M.
(f) How can the two linear operations represented by M and by
its square root be interpreted geometrically?
2022-08-16