Let F(x, y, z) = z tan^−1(y^2)i + z^3 ln(x^2 + 7)j + zk. Find
the flux of F across S, the part of the paraboloid x2 + y2 + z = 29
that lies above the plane z = 4 and is oriented upward.
3. Let F : X → Y and G: Y → Z be functions.
i. If G ◦ F is injective, then F is injective.
ii. If G ◦ F is surjective, then G is surjective.
iii. If G ◦ F is constant, then F is constant or G is
constant.
iv. If F is constant or G is constant, then G ◦ F is
constant.
The curried version of let f (x,y,z) = (x,(y,z)) is
let f (x,(y,z)) = (x,(y,z))
Just f (because f is already curried)
let f x y z = (x,(y,z))
let f x y z = x (y z)
Let
F(x, y,
z) = z
tan−1(y2)i
+ z3
ln(x2 + 8)j +
zk.
Find the flux of F across S, the part
of the paraboloid
x2 +
y2 + z = 28
that lies above the plane
z = 3
and is oriented upward.
S
F · dS =
Let F(x,y,z) = < z tan-1(y2),
z3 ln(x2 + 1), z >. Find the flux of F
across S, the top part of the paraboloid x2 +
y2 + z = 2 that lies above the plane z = 1 and is
oriented upward. Note that S is not a closed surface.
5. Let X, Y and Z be sets. Let f : X ! Y and g : Y ! Z
functions.
(a) (3 Pts.) Show that if g f is an injective function, then f
is an injective function.
(b) (2 Pts.) Find examples of sets X, Y and Z and functions f
: X ! Y and g : Y ! Z such that g f is
injective but g is not injective.
(c) (3 Pts.) Show that if...
Let f(x,y) be a scalar function, and let F(x,y,z) be a vector
field. Only one of the following expressions is meaningful. Which
one?
a) grad f x div F
b) div(curl(grad f))
c) div(div F)
d) curl(div(grad f))
e) grad(curl F)
Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )
Let f(x) = ln x for 1 ≤ x ≤ 3. Write down the formula for
Simpson's Rule with n = 4 that approximates the integral of ln x on
[1, 3].
Without doing numerical calculations, tell whether in
approximating the integral of ln on [1, 3] by the Trapezoidal Rule
with n = 8, the value T the Trapezoidal Rule gives is larger than,
or is smaller than, the integral. Justify your answer.
Let f: X-->Y and g: Y-->Z be arbitrary maps of sets
(a) Show that if f and g are injective then so is the
composition g o f
(b) Show that if f and g are surjective then so is the
composition g o f
(c) Show that if f and g are bijective then so is the
composition g o f and (g o f)^-1 = g ^ -1 o f ^ -1
(d) Show that f: X-->Y is...