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Multivariable calculus, 2008-10-31.
Per-Sverre Svendsen, Tel.035 - 167 615/0709 - 398 526.
Minimum requirements: Grade 5: 27p, 4: 21p, 3: 15p.You are
allowed to use one unmarked, ordinary (non-mathematical) dictionary
of your choice.No additional written material is allowed. Also, no
calculators or any other electronic equipment.
1. The function f (x, y) = ex2+2x+y2 is
defined on the disk D = {(x, y)
R2| x2 + 2x + y2 ≤ 0}.
(a) Calculate the maximum directional derivative
of f (x, y) at the point (−12 , 12).
(2p)(b) Calculate the absolute minimum and maximum values
of f (x, y) (on D). (3p)
(c) Calculate
ZZ D
f (x, y) dx dy. (3p)
2. Find and classify all critical points of the function
g(x, y) = x2y + y2 + 2xy. (3p)
3. Calculate
ZZ A
r x
y dx dy, where A is bounded by the curves
x2y = 1, y = 1 and x = 4.
(3p)
4. Calculate
ZZ D
(x2 − y2) ln(x + y) dx dy , D
= {(x, y) R2| 0 ≤ x− y ≤ 1, 1
≤ x + y ≤ 2}. (3p)
5. Calculate
ZZZ B
1p x2 + y2 + z2
dx dy dz, where B is the solid sphere x2
+ y2 + z2 ≤ 3. (3p)
6. Find the absolute minimum and maximum values
of f (x, y) = x2 + y2
subject to the constraint 5x2 − 6xy + 5y2 = 4.
(5p)
7. Calculate
ZZZ K
zp
x2 + y2 dx dy dz , K =
{(x,y,z) R3 |p
x2 + y2 ≤ z ≤ 2− x2 − y2 }. (5p)
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Taylor Series
Taylor’s formula for a function f : R→
R
f (a + h) = f (a) +
f 0(a) h + f 00(a)
2 h2 + · · · =
∞Xn=0
f (n)(a)
n! hn
Table of particular expansions (a = 0, h → x)
1. 1
1− x =∞Xk=0
xk = 1 + x + x2 + x3 + · · ·
(−1 < x
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Double Integrals
General substitution
Assume a one-to-one mapping between a region D in
the xy-plane and a region Duv in the
uv-plane
x = x(u, v)
y = y(u, v)⇔
u = u(x, y)
v = v(x, y)
Then
ZZ D
f (x, y) dxdy =
ZZ Duv
f (x(u, v), y(u, v))∂ (x, y)∂ (u, v)
dudv,
with ∂ (x, y)
∂ (u, v) =
xu xv
yu yv
6= 0.
Polar coordinates
x = r cos θ
y = r sin θ
∂ (x, y)
∂ (r,θ) = r,
ZZ D
f (x, y) dxdy =
ZZ Drθ
f (r cos θ, r sin θ) rdr dθ
Triple Integrals
General substitution
As above assume a one-to-one mapping between points (x,y,z) in
∆ and (u,v,w) in ∆uvw.
ZZ ∆
f (x,y,z) dxdydz =
ZZ ∆uvw
f (x(u,v,w), y(u,v,w), z(u,v,w))
∂ (x,y,z)∂ (u,v,w)
dudvdw,
with ∂ (x,y,z)
∂ (u,v,w) 6= 0
Spherical coordinates
x = ρ sinφ cos θ
y = ρ sinφ sin θ
z = ρ cosφ
∂ (x,y,z)
∂ (ρ, φ, θ) = ρ2 sinφ
ZZZ ∆
f (x,y,z) dxdydz =
ZZZ ∆ρφθ
f (ρ sinφ cos θ, ρ sinφ sin θ, ρ cosφ) ρ2
sinφ dρ dφ dθ
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Line Integrals
Line integral with respect to arc length
Given a parametrized curve C : r(t) =
(x(t), y(t), z(t)), a ≤ t ≤ b and a function
f : R3 → R.
Z C
f (x,y,z) ds =
b
Z a
f (x(t), y(t), z(t))p
(x0
(t))2
+ (y0
(t))2
+ (z0
(t))2
dt, (ds = d|r
|)
Surface Integrals
General parametrized surface
S : r = r(u, v) = (x(u, v), y(u,
v), z(u, v)), (u, v) D.
ZZ S
f (x,y,z) dS =
ZZ D
f (r(u, v))∂ r∂ u × ∂ r
∂ v
dudv
Function graph z = h(x, y)
S : r = (x,y,z) = (x,y,h(x, y)),
(x, y) S xy.ZZ S
f (x,y,z) dS =
ZZ S xy
f (x,y,h(x, y))q
1 + h2x + h2y dxdy
