Math 221
Quiz 9
Spring 2013
Several Variable Calculus
1. The final examination is on what date and at what time?
Solution. The final examination takes place on the third of May from 15:00
to 17:00, or, more memorably, 3–5 on 5/3.
2. Find curl 𝐹⃗ and div 𝐹⃗ [that is, ∇ × 𝐹⃗ and ∇ ⋅ 𝐹⃗ ] when
̂
𝐹⃗ (𝑥, 𝑦, 𝑧) = (sin 𝑥)̂𝚤 + (cos 𝑥)̂𝚥 + 𝑧2 𝑘.
Solution. This problem is a routine computation:
|
|
𝚥̂
𝑘̂ |
| 𝚤̂
| 𝜕
𝜕
𝜕 ||
|
∇ × 𝐹⃗ = |
|
| 𝜕𝑥
𝜕𝑦 𝜕𝑧 ||
|
|sin 𝑥 cos 𝑥 𝑧2 |
|
|
| 𝜕
| 𝜕
|
| 𝜕
𝜕 |
𝜕 ||
𝜕 ||
|
|
|
|
|
|
|
= 𝚤̂ | 𝜕𝑦 𝜕𝑧 | − 𝚥̂ || 𝜕𝑥 𝜕𝑧 || + 𝑘̂ | 𝜕𝑥
𝜕𝑦 |
|cos 𝑥 𝑧2 |
|sin 𝑥 cos 𝑥|
|sin 𝑥 𝑧2 |
|
|
|
|
|
|
|
|
|
|
̂
= −(sin 𝑥)𝑘,
and
𝜕
𝜕
𝜕 2
∇ ⋅ 𝐹⃗ =
(sin 𝑥) +
(cos 𝑥) +
(𝑧 )
𝜕𝑥
𝜕𝑦
𝜕𝑧
= cos 𝑥 + 2𝑧.
3. Set up an integral for the surface area of the parametric surface given by
̂
𝑟⃗ (𝑢, 𝑣) = 𝑣2 𝚤̂ − 𝑢𝑣̂𝚥 + 𝑢2 𝑘,
0 ≤ 𝑢 ≤ 1,
0 ≤ 𝑣 ≤ 2.
(Do not attempt to evaluate the integral!)
Solution. The element of surface area in parametric form is d𝑢 d𝑣 times
𝜕 𝑟⃗
𝜕 𝑟⃗
the area of the parallelogram determined by the tangent vectors
and :
𝜕𝑢
𝜕𝑣
𝜕 𝑟⃗ 𝜕 𝑟⃗
namely, the length of the cross product
× . Here is the computation.
𝜕𝑢 𝜕𝑣
𝜕 𝑟⃗
= 0̂𝚤 − 𝑣̂𝚥 + 2𝑢𝑘̂
𝜕𝑢
April 11, 2013
and
Page 1 of 2
𝜕 𝑟⃗
̂
= 2𝑣̂𝚤 − 𝑢̂𝚥 + 0𝑘,
𝜕𝑣
Dr. Boas
Math 221
Quiz 9
Spring 2013
Several Variable Calculus
so
| 𝚤̂
𝚥̂ 𝑘̂ ||
𝜕 𝑟⃗ 𝜕 𝑟⃗ ||
|
̂
×
= | 0 −𝑣 2𝑢| = 2𝑢2 𝚤̂ + 4𝑢𝑣̂𝚥 + 2𝑣2 𝑘.
|
|
𝜕𝑢 𝜕𝑣 |
|
|2𝑣 −𝑢 0 |
√
The length of this vector is 4𝑢4 + 16𝑢2 𝑣2 + 4𝑣4 . Therefore the surface area
is given by the integral
2
1
√
∫0 ∫0
or
2
2
∫0 ∫0
1
4𝑢4 + 16𝑢2 𝑣2 + 4𝑣4 d𝑢 d𝑣,
√
𝑢4 + 4𝑢2 𝑣2 + 𝑣4 d𝑢 d𝑣.
(This integral cannot be evaluated in terms of elementary functions.)
April 11, 2013
Page 2 of 2
Dr. Boas