Problem Set 4
MH2100
Solutions
E
Let
:
R
3
→
R
3
be a map given by
=
(
)
=
(
)
=
(
)
where
,
, and
are polynomials in the variables
.
(a) Prove that
is continuous on
R
3
.
(b) Is it true that
is also differentiable on
R
3
? Justify your answer.
S
(a) Since all component functions are polynomials, they are elementary func
tions defined on the whole
R
3
and are therefor continuous everywhere.
(b) All partial derivatives of all component functions are also polynomials.
They are
elementary functions defined on the whole
R
3
and continuous everywhere and
hence the map
is differentiable at any point.
E
What is the Jacobian matrix of a linear map
(
x
) =
A
x
,
:
R
→
R
. Here,
A
is an
by
matrix and
x
∈
R
is a vectorcolumn.
S
It’s
A
itself at any point.
Indeed, the definition of differentiability applied to
(
x
) =
A
x
with (supposedly)
(
x
) =
A
says
lim
Δ
x
→
0
A
(
x
+ Δ
x
)

A
x

A
Δ
x
Δ
x
= 0
which is true.
:)
1
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Questions on calculation
E
Write down the equation of the tangent plane to each of the following sur
faces at given points
(a)
=
2
+
2
at the point
(1 2 5)
.
(b)
2
+
2
+
2
= 169
at the point
(3 4 12)
.
(c)
=
+ ln
at the point
(1 1 1)
.
S
(a) Here,
= 2
,
= 2
, so
(1 2) = 2
and
(1 2) = 4
. Thus the tangent
plane is
= 5 + 2(

1) + 4(

2)
, which is
2
+ 4


5 = 0
.
(b) Here,
=
±
169

2

2
. Since at the given point we have
= 12
>
0
, we use
=
169

2

2
. Further,
=

169

2

2
=

=

169

2

2
=

Hence
(3 4 12) =

1
4
and
(3 4 12) =

1
3
. Thus, the answer is
= 12


3
4


4
3
⇔
3
+ 4
+ 12
= 169
(c) Here, it’s difficult to express
as a function of
, so we express
in
instead.
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 Spring '13
 Calculus, Derivative, Limit, lim

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