MATH 282 Field Theory, Partial Differential Equations and Methods of Solution 2017
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH 282
2017 JANUARY EXAMINATIONS
Field Theory, Partial Differential Equations and Methods of Solution
1. Evaluate the following volume integrals:
(a)
╱x2 + y2 + z2 、dV,
尸4
where τ← is the cube 0 < x < 2, 一1 < y < 1, 0 < z < 2 .
[5 marks] (b)
x2 z dV,
尸乙
where τ) is the region defined by x > 0, y > 0, z > 0, x + y + z < 2 .
[6 marks] (c)
╱xy + z2 、dV,
尸d
where τ| is the region defined by x > 0, y > 0, x2 + y2 + z2 < 16 .
[7 marks] (d)
x2 z dV,
尸X
where τ4 is the region defined by x2 + y2 < 4, 0 < z < 4 .
[7 marks]
2. Throughout this question v is the vector field v = (3x2 + yz, xz 一 z, xy 一 y) .
(i) Evaluate the line integral
v ? dr
√ 1
along the parametric path √ 1 given by
(x, y, z) = (s, s2 , s3 ), 0 ≤ s ≤ 1 .
[6 marks]
(ii) Use the definitions of curl and gradient to show that
V × Vφ = 0
for any smooth scalar function φ .
[4 marks] (iii) Show that the vector field v, defined in part (i), is irrotational.
[4 marks] (iv) Find a scalar field φ such that v = Vφ .
[6 marks]
(v) Evaluate the line integral
v ? dr
√2
along √2 , the straight line path from (0, 0, 0) to (1, 1, 1). Comment briefly on how your result is related to the answer to (i) and the answer to (iv).
[5 marks]
3. (i) Consider the triangle defined by the vertices
(1, 0, 1), (4, 0, 1), and (4, 1, 0) .
Find this triangle’s area and n, the unit normal to the plane of the triangle.
[6 marks]
(ii) Let
u = (一xy2 , xz, 1 + x + z) .
Calculate
u ? dr ,
√
the integral of u around the triangle defined in part (i).
[10 marks]
(iii) Calculate V × u, and verify that the surface integral over the triangle agrees with Stokes’ theorem.
[9 marks]
4. The displacement of an elastic band, u(x, t), obeys the wave equation
∂2u 2 ∂2u
∂t2 ∂x2
where c is a constant. The ends of the elastic are fixed, giving the boundary conditions
u(0, t) = 0, u(L, t) = 0 .
(i) Use separation of variables to show that the general solution of the given Partial Differential Equation is
u(x, t) = 素 ┌A素 sin ╱ 、cos ╱ 、 + B素 sin ╱ 、sin ╱ 、┐
where all the A素 and B素 are constant.
[10 marks]
(ii) At t = 0 the elastic band has an initial position and velocity given by
u(x, 0) ∂u(x, 0)
∂t
= a x(L 一 x) ,
= 0 .
Use the results of part (i) to find an expression for u(x, t) at later times.
[15 marks]
5. A superconducting sphere of radius A is located at the origin. The mag- netic field for r > A (outside the sphere) is given by
B = β ╱ 1 + 一 2x2 ) , 一 , 一 、
where β is a positive constant and r = 尸x2 + y2 + z2 is the distance from the origin. For r < A (inside the superconducting sphere) the magnetic field is zero.
(i) Calculate the field strength |B| and B一 , the component of B normal to the surface of the sphere at the point r = (A cos θ, A sin θ, 0), where θ can take any value. [5 marks]
(ii) Derive the identities
V ? ╱ 、 = 一 ,
V × ╱ 、 = 一
where v is a vector field and f a scalar field. [5 marks]
(iii) Calculate V ? B and V × B outside the sphere, and verify that B satisfies both vacuum Maxwell’s equations.
[8 marks]
(iv) Calculate B ? dr around the semi-circular path shown above. ρ, the radius of the semicircle, is greater than A, the radius of the sphere. Use your result to find the total current circulating around the sphere.
[7 marks]
6. (i) The electric potential in a region is given by φ(x, y, z) = 2x + y 一 z + xy + x2 + yz .
Calculate the resulting electric field and the charge density.
[6 marks]
(ii) By calculating surface integrals find the electric flux out of the hemi- sphere defined by
z > 0, x2 + y2 + z2 < 1 .
How much flux leaves through the flat base of the hemisphere, and how much through the curved upper surface?
[10 marks]
(iii) By calculating a volume integral find the total charge inside the hemi- spherical region defined in part (ii). Comment on your result.
[9 marks]
2022-01-26