This

**preview**shows page 1. to view the full**5 pages of the document.**1.(a) [5 marks] Define by . Suppose

32

:RR →f),,(),( 2vuvevuvu u++=f22

:RR →g

is of class C,

1)1,0()1,1(

=

g, and . Compute

−

=12

11

)1,1(gD)1,1)(( gf

ο

D.

)1,1()1,0()1,1)(( gfgf DDD =ο

−

+

=12

11

1

0

12

)1,0(

uv

e

u

u

−

=12

11

11

01

10

−=

03

11

12

(b) [5 marks] Suppose stysxyxfu

=

=

=,2),,( . Assuming f is of class C, find

2

ts

u

∂∂

∂2

in

terms of x , y and the partial derivatives of f over x and y .

t

y

y

f

t

x

x

f

t

u

∂

∂

∂

∂

+

∂

∂

∂

∂

=

∂

∂s

y

f

x

f

∂

∂

+

∂

∂

=0y

f

s∂

∂

=

=

∂

∂

∂

∂)( t

u

s

=

∂

∂

∂

∂)( y

f

s

s

=

∂

∂

∂

∂

+

∂

∂

∂∂

∂

+

∂

∂)( 2

22

s

y

y

f

s

x

yx

f

s

y

f=

∂

∂

+

∂∂

∂

+

∂

∂

2

22

2y

f

st

yx

f

s

y

f

2

22

y

f

y

yx

f

x

y

f

∂

∂

+

∂∂

∂

+

∂

∂

=

2

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2. Assume that the equation 0).,(

=

zyxF , where F is of class , defines z implicitly as a

1

C

function of x and y , that is ),( yxgz

=

. Suppose 0)2,2,1(

=

−

F and )2,3,1()2,2,1(

=

−F∇.

(a) [4 marks] Evaluate ∂ in the direction of

)2,1( −g

u)1,1(

2

1−=

u.

)2,3,1()2,2,1( =−∇F⇒1)2,2,1( =−

∂

∂

x

F, 3)2,2,1( =−

∂

∂

y

F , 2)2,2,1( =−

∂

∂

z

F

Hence 2

1

)2,1( −=−

∂

∂

x

g , 2

3

)2,1( −=−

∂

∂

y

g and

2

1

)1,1(

2

1

)

2

3

,

2

1

()2,1( −=−⋅−−=−∂ g

u

(b) [6 marks] Suppose the surface is parametrized by . S),,(),( 2vuvuevu vu +−= +

f

Is the surface ),(:

1yxgzS

=

tangent to at the point ? Justify. S)2,2,1( −

Remark: Two surfaces and are tangent at the point aS1

S1

SS

∩

∈

if the tangent planes

at a to both surfaces coincide.

By the assumption ( and

1

)2,2,1 S∈− S

∈

−

)2,2,1( since it corresponds to )1,1(),(

−

=

vu .

The normal to the surface at the point

1

S)2,2,1(

−

is n =

1)2,3,1()2,2,1( =

−

∇

F. The

normal to is S

)1,1(

11

21)1,1)((

−

+

+

−

=−

∂

∂

×

∂

∂

=

vu

vu

e

ue

vu

kji

ff

n

111

211

−

−=

kji

=)2,3,1( −

−

−

and it is parallel to . Hence the tangent planes coincide and the surfaces are tangent to

1

n

each other at the point . )2,2,1( −

3

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