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 ΠR 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.
Search WWH ::




Custom Search