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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