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Multivariable calculus, 2013-10-30.Per-Sverre Svendsen, Tel.035 - 16 76 15/0709 - 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. Find the following limit or show that it does not exist lim(x,y ) → (0 ,0)

sin(x2 −y2)

x2 + y2. (2p)

2. We are given a point P = (1 , 1, −1) on the surface S : x3 −xy + z2 = 1.Calculate, at the point P , the directional derivative of f (x,y,z ) = x 2 + y2 + 3 z in a direction n

normal to S and having a positive z -component, that is n ·(0, 0, 1) > 0. (3p)

3. Find the points on the curve x 2 + xy + y2 = 2 that are closest to and farthest from the origin. (4p)

4. Find the absolute minimum and maximum values of g(x, y ) = ( xy −1) e2x + y

on the set ∆ = {(x, y ) ∈ R 2 | 2x + y ≤4, x ≥0, y ≥0}. (5p)

5. Calculate D

1y + 1

dx dy , D = {(x, y ) ∈ R 2 | x ≤y ≤1, x ≥0 }. (3p)

6. Calculate the area of the part of the surface z = xy that lies inside the cylinder x 2 + y2 = 1. (3p)

7. Calculate

K

1

x2 + y2 + z2 + 1 dx dy dz ,

where K = {(x,y,z ) ∈ R 3 | x2 + y2 + z2 ≤1, 0 ≤z ≤ x2 + y2}. (5p)

8. Find the center of mass r cm = ( x, y ) of a region A with density f (x, y ) = 1

x2 + y2,

and A = {(x, y ) ∈ R 2 | 1 ≤x2 + y2 ≤2y}. (5p)

Hint: x = 1M A

A

x f (x, y ) dx dy, y = 1M A

A

y f (x, y ) dx dy ,

where M A = Af (x, y ) dx dy represents the

total mass of the region A .

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Reference formulas and equations in Multivariable Calculus

Trigonometry and Logarithms

sin(x ±y) = sin x cos y ±cos x sin y sin x ±sin y = 2 sin x ±y2 cos x

∓y

2cos(x ±y) = cos x cos y

∓sin x sin y cosx −cos y = −2sin x + y

2 sin

x −y2

tan( x ±y) = tan x ±tan y1∓tan x tan y

cosx + cos y = 2cos x + y

2 cos

x −y2

cot( x ±y) = cot x cot y

∓1

±cot x + cot y 2sin x sin y = cos( x −y) −cos(x + y)

sin2x = 2sin x cos x 2cosx cos y = cos( x −y) + cos( x + y)

cos2x = cos 2 x −sin2 x = 2cos 2 x −1 = 1 −2sin2 x 2sin x cos y = sin( x −y) + sin( x + y)

ln x + ln y = ln xy ln x

−ln y = ln

x

yln x a = a ln x (x, y > 0)

Standard limits

limx → 0+

x α loga x = 0 ( a > 1, α > 0) limx →∞

a x

x α = ∞ (a > 1)

limx → 0

sin xx

= 1 limx →∞

x α

loga x = ∞ (a > 1, α > 0)

limx → 0

ln(1 + x)x

= 1 limn →∞

a n

n ! = 0

limx → 0

ex −1

x

= 1

Basic derivatives

f (x ) f (x )

x a ax a − 1

a x a x ln a

ln |x| 1

x

sin x cosx

cos x −sin x

tan x 1 + tan 2 x = 1cos2 x

arcsin x 1

√ 1 −x2

arccos x − 1

√ 1 −x2

arctan x 1

1 + x2

ln x + x2 + α1

√ x2 + α12 x√ x

2+ α +

α2 ln x + √ x

2+ α x

2+ α

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Taylor Series

Taylor’s formula for a function f : R →R

f (a + h) = f (a ) + f (a) h + f (a)

2 h2 + · · · =

∞

n =0

f (n ) (a )n !

hn

Table of particular expansions ( a = 0 , h →x )

1. 11 −x

=∞

k =0

xk = 1 + x + x2 + x3 + · · · (−1 < x < 1)

2. (x + 1) α = 1 + α x + α (α −1)

2 x2 +

α (α −1)(α −2)2 ·3

x3 + · · · (−1 < x < 1)

3. ex =∞

k =0

1k!

x k = 1 + x + 12

x2 + 16

x3 + · · ·

4. sin x =∞

k =1

(−1)k +1

(2k −1)! x2k − 1 = x − 1

6x3 + 1

120x5 −· · ·

5. cos x =∞

k =0

(−1)k

(2k)! x2k = 1 −

12

x2 + 124

x4 −· · ·

6. ln(x + 1) =∞

k =1

(−1)k +1

k xk = x −

12

x2 + 13

x3 −· · · (−1 < x ≤1)

7. arctan x =∞

k =1

(−1)k +1

(2k −1) x 2k − 1 = x −

13

x3 + 15

x5 −· · · (−1 ≤x ≤1)

Taylor’s formula for a function f : R 2 →R

f (a + h, b + k) = f (a, b ) + h f x (a, b ) + k f y (a, b ) + 12

h2 f xx (a, b ) + 2 hkf xy (a, b ) + k2 f yy (a, b ) + · · · =∞

n =0

1n !

h ∂ ∂x

+ k ∂ ∂y

nf (x, y )(a,b )

Tangent plane

Function z = f (x, y )

Equation of tangent plane through the point ( a,b,f (a, b ))

z = f (a, b ) + f x

(a, b )(x

−a ) + f

y(a, b )(y

−b)

Level surface F (x,y,z ) = C

Equation of tangent plane through the point ( a,b,c )

F x (a,b,c )(x −a ) + F y (a,b,c )(y −b) + F z (a,b,c )(z −c) = 0

Directional derivative

The directional derivative of a function f : R 3 →R at the point ( a,b,c ) and direction u (|u | = 1)

D u f (a,b,c ) = f u (a,b,c ) = u ·∇f (a,b,c ) = u ·(f x (a,b,c ), f y (a,b,c ), f z (a,b,c )).

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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 D uv in the uv -plane

x = x(u, v )

y = y(u, v ) ⇔

u = u(x, y )

v = v(x, y )

Then D

f (x, y ) dxdy = D uv

f (x (u, v ), y(u, v ))∂ (x, y )∂ (u, v )

dudv ,

with ∂ (x, y )

∂ (u, v ) =

x u xv

yu yv

= 0.

Polar coordinates

x = r cos θ

y = r sin θ

∂ (x, y )∂ (r, θ )

= r , D

f (x, y ) dxdy = D rθ

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 .

∆

f (x,y,z ) dxdydz = ∆ 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 ) = 0

Spherical coordinates

x = ρ sin φ cos θ

y = ρ sin φ sin θ

z = ρ cos φ

∂ (x,y,z )∂ (ρ,φ,θ )

= ρ2 sin φ

∆

f (x,y,z ) dxdydz = ∆ ρφθ

f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ) ρ2 sin φ dρ dφ dθ

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Line Integrals

Tangent line integral

Given a parametrized curve C : r(t) = ( x (t), y(t ), z (t )) , a ≤ t ≤b and a vector eld F = ( P,Q,R ).

C

F

·dr

=

b

a(P,Q,R ) ·(x (t), y (t ), z (t )) dt =

b

aP x (t) + Q y (t ) + R z (t ) dt

Line integral with respect to arc length

Assume a curve C as above and a function f : R 3 →R .

C

f (x,y,z ) ds =b

a

f (x (t), y(t), z (t )) (x (t ))2 + ( y (t ))2 + ( z (t ))2 dt, (ds = d|r |)

Green’s theorem

Given a plane, closed, positively oriented curve C that encloses a region D and a eld F = ( P, Q ).

C

F ·dr = D

∂Q∂x −

∂P ∂y

dxdy

Surface Integrals

General parametrized surface

S : r = r (u, v ) = ( x (u, v ), y(u, v ), z (u, v )) , (u, v ) ∈ D .

S

f (x,y,z ) dS = D

f (r (u, v ))∂ r

∂u × ∂ r

∂vdudv

Parametrization of (parts of) the surface x2 + y2 + z2 = R 2 (R > 0).

S : r = ( x,y,z ) = ( R sin φ cos θ, R sin φ sin θ, R cos φ), (φ, θ )∈D .

S

f (x,y,z ) dS = D

f (r (φ, θ ))∂ r

∂φ × ∂ r

∂θdφ dθ =

D

f (r (φ, θ )) R 2 sin φ dφ dθ .

Function graph z = h (x, y ).

S : r = ( x,y,z ) = ( x,y,h (x, y )) , (x, y ) ∈ S xy .

S

f (x,y,z ) dS = S xy

f (x,y,h (x, y )) 1 + h2x + h2

y dxdy

Gauss’ theorem

Given a space region K with (closed) surface boundary S and a eld F .n is the outer unit normal vector of the surface.

S

F ·n dS = K ∇ ·F dx dy dz .