All questions may be attempted but only marks obtained on the best four solutions will
count.
The use of an electronic calculator is not permitted in this examination.
1. (a) State, without proof, the chain rule for a function f (x(t), y(t)).
(b) Let f (x, y) = exy and x(t) = sin(t) + cos(t), y(t) = sin(t) − cos(t). Use the
chain rule to compute
df
(x(t), y(t)).
dt
Give the answer both as a function of x, y and as a function of t.
(c) Considering the function f (x, y) from part (b), find the tangent plane of f at
the point (−1, 2).
2. (a) Define the Jacobian
∂(x, y)
.
∂(u, v)
(b) Using the definition from part (a), determine the Jacobian for the coordinate
transformation
x(u, v) = u + v and y(u, v) = u − v.
(c) Let R := {(x, y) ∈ R2 : 0 6 x + y 6 1, 0 6 x − y 6 1} Using the coordinate
transformation from part (b), or otherwise find
ZZ
exy (x + y)(x − y) dxdy.
R
3. (a) State the Divergence theorem.
(b) Let S be the surface of the closed box defined by 0 6 x 6 2, 0 6 y 6 1/2, and
0 6 z 6 1 Let a be a positive constant and consider the vector field
F(x, y, z) = (a2 + 1)xi + x2 12 cos3 (az)j − 2az 2 k.
Find the exact value of a so that the net flux of F over S is 0.
(c) The vector field F in part (b) is replaced by F+G, where G(x, y, z) : g1 (y, z)i+
g2 (x, z)j + g3 (x, y)k with smooth functions g1 , g2 , g3 : R2 7→ R. How does the
value of a change if the net flux over S must remain zero?
MATH1402
PLEASE TURN OVER
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4. (a) State Stokes’ Theorem.
(b) Let S be the surface defined by
S := {(x, y, z) ∈ R3 : x2 + y 2 6 1 and z = 1 − x2 − y 2 }.
Find the unit normal vector field of S.
(c) Verify Stokes’ Theorem for the vectorfield
F := −yi + xj + z 2 k
and the surface S introduced in part (b).
5. (a) Let f : R3 7→ R. Define the Hessian matrix of f .
(b) Consider the function f (x, y) := sin(xy). Compute the Hessian of f (x, y) in
the point x0 = ( π2 , −1) and find its eigenvalues.
(c) Consider the function f (x, y) = x4 + x3 − y 2 + y 4 . Find the critical points of f
and identify each point as either a local minimum, a local maximum or a saddle
point. For the local extrema also state if the critical points are degenerate.
MATH1402
CONTINUED
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6. Let Ω be a simply connected subset of R3 with smooth boundary ∂Ω. Assume that
the vector field u describes the velocity of a fluid in Ω, such that u · n = h on ∂Ω,
for some function h, and with n defined as the outward pointing unit normal on ∂Ω.
(a) If there holds ∇ · u = 0 and ∇ × u = 0 everywhere in Ω, show that there exists
a function φ : R3 → R, from which we may deduce u, with ∇φ · n = h on ∂Ω
and such that
∇2 φ = 0.
(1)
What is the relation between φ and u?
(b) Let f : R3 7→ R and g : R3 7→ R be two smooth functions. Show the identity
∇ · (f ∇g) = f ∇2 g + ∇f · ∇g.
(c) Prove the First Green’s identity:
ZZZ
ZZ
∂g
2
dS.
(f ∇ g + ∇f · ∇g) dxdydz =
f
V
S ∂n
Here f (x, y, z), g(x, y, z) are smooth functions in a bounded domain V ⊂ R3 ,
S is a smooth surface forming the boundary of V and n is an outward pointing
normal.
(d) Show that any solution to (1) satisfies
ZZ
ZZZ
∇φ · ∇w dxdydz =
hw dS
(2)
∂Ω
Ω
for all differentiable functions w : Ω 7→ R. Show RR
that if φ satisfies (2) then
so does φ̃ = φ + α for any α ∈ R. Show that if ∂Ω h dS > 0 no function
φ(x, y, z) can satisfy (2).
MATH1402
END OF PAPER
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