Math 227
February 2016
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Mathematics 227
Midterm 1
February 10, 2016
There are four questions worth a total of 40 marks.
No calculators or notes are allowed.
10 40
1. The curve C is given by
r(t) = et (cos t − sin t) , et (cos t + sin t) , et
0≤t≤π
(a) Find the arc length of the curve.
(b) At t = 0, find the unit tangent vector T̂, the unit principal normal vector N̂, the unit
binormal vector B̂ and the curvature κ.
(c) Find parametric equations for the osculating circle (i.e. the best fitting circle) to C at
t = 0.
Math 227
10 40
February 2016
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~ = z ı̂ı + z ̂ − (3x + y)k̂ which passes through (1, 1, 2).
2. Find the streamline of F
Math 227
10 40
February 2016
Name:
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3. (a) For which value(s) of the constants a, b is the vector field
F~ = 2x sin(πy) − ez ı̂ + ax2 cos(πy) − 3ez ̂ − x + by ez k̂
conservative?
(b) Let F be a conservative field from part (a). Find all functions φ for which F = ∇φ.
Math 227
10 40
4. Evaluate
February 2016
R
C
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~ · d~r where
G
~ = (2x sin(πy) − ez ) ı̂ + πx2 cos(πy) − 3ez ̂ − xez k̂
G
and C is the intersection of y = x and z = ln(1 + x) from (0, 0, 0) to (1, 1, ln 2).
Math 227
February 2016
Name:
This page just provides extra work space.
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Math 227
February 2016
Name:
Page 6 of 6
Math 227 Cheatsheet
Curves
◦ a(t) =
d2 s
dt2 (t) T̂(t)
◦ κ(t) =
|v(t) × a(t)|
|v(t)|3
+ κ(t)
2
ds
dt (t)
N̂(t)
(t)
v(t) × a(t)] · da
dt
◦ τ (t) =
|v(t) × a(t)|2
◦ T̂(t) =
◦
v(t)
|v(t)|
dT̂
ds (s) =
dN̂
(s) =
ds
dB̂
ds (s) =
dT̂
(t)
N̂(t) = B̂(t) × T̂(t) = dt dT̂ (t)
κ(s) N̂(s)
τ (s) B̂(s) − κ(s) T̂(s)
−τ (s) N̂(s)
dt
B̂(t) = T̂(t) × N̂(t) =
v(t) × a(t)
|v(t) × a(t)|