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1
Quiz 1 (Notes, books, and calculators are not authorized)
Show all your work in the blank space you are given on the exam sheet. Always justify your answer. Answers with
no justification will not be graded.
R sin(x)
Question 1: Let φ(x, y) = 0
log(1 + y 2 + z 2 )dz. Given α, β, γ ∈ R, compute ∂x φ(α, α + β) and ∂y φ(α − β, γ).
(Do not try to simplify the results).
Recalling the fundamental theorem of calculus
Z
t
f (z)dz
∂t
= f (t),
0
we apply the chain rule repeatedly
∂x φ(x, y) = log(1 + y 2 + sin(x)2 ) cos(x)
This means that
∂x φ(α, α + β) = log(1 + (α + β)2 + sin(α)2 ) cos(α)
Recalling that
Z
v
∂t
Z
v
f (s, t)ds =
∂t f (s, t)ds,
u
u
we apply the chain rule repeatedly
sin(x)
Z
∂y φ(x, y) =
0
This means that
Z
2y
dz
1 + y2 + z2
sin(α−β)
∂y φ(α − β, γ) =
0
2γ
dz.
1 + γ2 + z2
Question 2: Consider the heat equation ∂t T − k∂xx T = f (x), x ∈ [a, b], t > 0, with f (x) = 0, where k > 0.
Compute the steady state solution (i.e., ∂t T = 0) assuming the boundary conditions: −k∂n T (a) = 1, T (b) = 0 (∂n
is the normal derivative).
At steady state, T does not depend on t and we have ∂xx T (x) = 0, which implies ∂x T (x) = α, and T (x) = β+αx, where
α, β ∈ R. The two constants α and β are determined by the boundary conditions. 1 = −k∂n T (a) = k∂x T (a) = kα
and 0 = T (b) = β + αb. We conclude that α = k1 and β = −αb = − kb . In conclusion
T (x) =
x−b
.
k
2
Quiz 1, September 9, 2014
Question 3: Consider the equation ∂t c(x, t) − ∂x (1 + x2 )∂x c(x, t) = 6x/L2 , where x ∈ [0, L], t > 0, with
RL
2
c(x, 0) = f (x), −∂n c(0, t) = 1, −∂n c(L, t) = 1+L
2 , (∂n is the normal derivative). Compute E(t) := 0 c(ξ, t)dξ.
We integrate the equation with respect to x over [0, L]
L
Z
Z
L
∂t c(ξ, t)dξ −
0
Using that
RL
0
∂t c(ξ, t)dξ = dt
0
RT
0
6
∂ξ (1 + ξ )∂ξ c(ξ, t) dξ = 2
L
2
Z
L
ξdξ.
0
c(ξ, t)dξ together with the fundamental theorem of calculus, we infer that
dt E(t) − (1 + L2 )∂x c(L, t) + ∂x c(0, t) = 3.
The boundary conditions ∂x c(0, t) = −∂n c(0, t) = 1, −∂x c(L, t) = −∂n c(L, t) =
2
1+L2
give
dt E(t) + 2 + 1 = 3.
We now apply the fundamental theorem of calculus with respect to t
Z t
E(t) − E(0) =
∂τ E(τ )dτ = 0.
0
In conclusion
Z
E(t) =
L
f (ξ)dξ,
∀t ≥ 0.
0
Question 4: Let φ = x2 + 2y 2 (a) RCompute ∆φ(x, y). (b) Consider the disk of radius 1 centered at (0, 0) and let
Γ be the boundary of Ω. Compute Γ ∂n φdΓ.
(a) The definition ∆φ = ∂xx φ + ∂yy φ implies that
∆φ = ∂xx φ + ∂yy φ = 2 + 4 = 6.
(b)The definition ∆φ = div(∇φ) and the fundamental theorem of calculus (also known as the divergence theorem) imply
that
Z
Z
Z
Z
Z
∂n φdΓ =
n·∇φdΓ =
div(∇φ)dΩ =
∆φdΩ = 6
dΩ = 6π,
Γ
because the surface of Ω,
Γ
R
Ω
dΩ, is equal to π.
Ω
Ω
Ω