Graphics Reference
In-Depth Information
4.2.7.
Prove that (0,1] is open in [-1,1].
4.2.8.
Prove that any set is open or closed in itself.
4.2.9.
Prove that the function f : (0,1] Æ
R
, f(x) = 1/x, is continuous but not uniformly
continuous.
Let
v
0
,
v
1
,
v
2
Œ
R
2
. Define a homeomorphism between the simplex
v
0
v
1
v
2
and the unit
disk
D
2
.
4.2.10.
4.2.11.
Show that the support of a function is the intersection of all closed sets
A
, where f
vanishes outside of
A
.
Section 4.3
4.3.1.
Prove Proposition 4.3.4.
4.3.2.
Prove Corollary 4.3.7(2).
Let f(t) = (t
2
,t), g(x,y) = y
2
- 4x, and G(t) = g(f(t)). Compute Df, Dg, and DG. Determine
DG in two ways: from its formula and by using the chain rule.
4.3.3.
Let f(x,y) = (x
3
- 2x + 1, xy + y
2
, x - 3y + 7). Compute the Jacobian matrix f¢(-1,5).
What is its rank?
4.3.4.
Let A, B : [a,b] Æ
R
3
be differentiable functions and define f : [a,b] Æ
R
3
4.3.5.
by f(t) = A(t)
¥ B(t). Prove that
¢
()
=
¢
()
¥
()
+
()
¥
¢
()
ft
At
Bt At
Bt.
(We are treating the 1 ¥ 3 Jacobian matrices as vectors here.) In short hand, the
differentiation rule for the cross-product is (A ¥ B)¢=A¢¥B + A ¥ B¢.
4.3.6.
Show that the function z = f(xy) satisfies the equation
∂
∂
z
x
∂
∂
z
y
x
-
y
= 0.
Show that the substitution x = e
s
and y = e
t
converts the equation
4.3.7.
2
2
∂
∂
u
x
∂
∂
u
y
∂
∂
u
x
∂
∂
u
y
2
2
x
+
y
+
x
+
y
=
0
2
2
into
2
2
∂
∂
u
s
∂
∂
u
+
=
0
.
2
2
t
4.3.8.
If f(x,y,z) = x sin z and
v
= (2,-1,3), find D
v
f(0,1,-1).
Show that the directional derivative of f(x,y) = y
2
/x at any point of the ellipse 2x
2
+ y
2
4.3.9.
= 1 in the direction of the normal to the ellipse at that point is zero.