Download MATH 2040 F16 Vector calculus problems

MATH 2040
F16
Vector calculus problems
On this problem set, assume that all functions are C ∞ on their domains.
Vector-valued functions
1. Do the exercises in the book for Section 1.5. (For the cross-product, just do one
component).
2. Let α(t) = x0 + t2 v where x0 and v are vectors which do not depend on t. Compute
α0 (t).
3. Let α(t) = (et , 3t2 sin t3 , t2 + 4t). Find α0 (t).
Rt
4. In the previous exercise, find 0 α(s)ds.
2
5. Find α0 (t) if α(t) = a(t) 3t, sin t, et if
(a) a(t) is an unknown function (your final answer will have the unknown functions
a(t) and a0 (t) left in it)
(b) a(t) is constant?
6. Find α0 (t) if α(t) = (sin t)v(t) in the following cases:
(a) v(t) is an unknown vector-valued function (your final answer will have the unknown
v(t) and v0 (t) in it)
(b) v(t) is a constant vector v0
7. Let f be a vector-valued function. Assume that f (t) is a vector function with constant
length; that is, hf (t), f (t)i = c for some constant c. Show that f 0 (t) is perpendicular to
f (t) for all t.
8. Let f (t) be a vector function and assume that f 00 (t) is parallel to f (t) for all t. Show
that
d
f (t) × f 0 (t) = 0.
dt
Partial derivatives
1. Let f (x, y) = ex sin y + y 2 ex . Compute
(a)
∂f
∂x
(b)
∂f
∂y
(c)
∂2f
∂x2
1
(d)
∂2f
∂x∂y
and
∂2f
∂y∂x
2. Let f (x, y, z) = cos xy 2 sin z + z 2 x3 y. Compute
(a)
∂f
∂x
(b)
∂f
∂y
(c)
∂f
∂z
(d)
∂2f
∂x∂z
(e)
∂3f
.
∂x∂y∂z
3. Let f (x, y) = x3 − 3xy 2 + ex cos y. Show that
∂2f
∂2f
+
= 0.
∂x2
∂y 2
Chain rule
p
1. In this question, let x = (x1 , x2 , x3 ) = (u1 , u2 , 1 − (u1 )2 − (u2 )2 ). Assume that u1
and u2 are functions of v 1 and v 2 . This relates to an example in class.
(a) Find an expression for
∂x1 ∂x2
∂x3
,
,
and
.
∂v 2 ∂v 2
∂v 2
Your final answer will contain the unknown derivatives of u1 and u2 , as in class.
(b) Now assuming that
√
3 2
1 1
v
u = v −
2√
2
3 1 1 2
u2 =
v + v
2
2
1
compute
∂x1 ∂x2
∂x3
,
,
and
∂v 2 ∂v 2
∂v 2
in terms of v 1 and v 2 (no other unknowns should appear in your final answer).
2. Let f (u1 , u2 ) be a function from R2 into R.
2
(a) If u1 and u2 depend on v 1 , v 2 and v 3 , write the general formulas for
∂f
∂v 2
and
∂f
.
∂v 3
2
(b) Now assume that u1 = v 1 sin (v 2 ) + 5v 3 and u2 = (v 1 )(v 3 )2 + e(3v ) . Find formulas
for
∂f
∂f
and
.
2
∂v
∂v 3
in terms of v 1 , v 2 , v 3 , and the derivatives of f .
3. Let w(x, y, z) be a function of x, y, and z. Let x, y, and z in turn be functions of u and
v. What are the formulas for
∂w
∂w
and
?
∂u
∂v
4. Let f (x, y, z) represent a function from R3 into R. Let x(t) = (x(t), y(t), z(t)) be a
vector-valued function of t. Show that
df
∂f ∂f ∂f
0
=
,
,
, x (t)
dt
∂x ∂y ∂z
3