Math 2E: Vector Calculus Midterm Answers
1.
(a) ds = x (t)2 + y (t)2 dt = (1 cos t)2 + sin2 t dt =
the slide is
ds =
2 2 cos t dt
C
(b)
0
2 2 cos t dt, hence the length of
0
2 2 cos t dt =
4 cos(t/2)
0
0
2 2(1 2 sin2 (t/2) dt =
0
4 sin2 (t/2) dt =
0
2 sin
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MATH 2E
FALL 2016
Practice Midterm 1
Name, ID, and Discussion Number:
Instructions: No notes, books, calculators, or supplies allowed except pencils, pens, and erasers. Show your work (only writing the final answer is not
enough to get full credit).
Sugge
Partial Differentiation To differentiate f ( x, y) with respect to x, treat y as a constant
Examples
1. Let f ( x, y) = 2x2 y3 + sin x. Then
f
= f y ( x, y) = 6x2 y2
y
f
= f x ( x, y) = 4xy3 + cos x,
x
2. Let f ( x, y, z) = x2 + yz1 . Then
f y ( x, y, z)
16.6
Parametric Surfaces and their Areas
Can dene surfaces similarly to spacecurves: need two parameters u, v instead of just t.
Denition. Let x, y, z be functions of two variables u, v, all with the same domain D. The parametric surface
dened by the co-o
16.7
Surface Integrals
Recall: if D is a 2D-region, then
D
f dA = Area( D ) f av
For surfaces: approximate S by parallelograms Sij located at points Pij . On each parallelogram,
f av f ( Pij ), and so it seems sensible to have
S
f dS
f ( Pij ) Area(Sij
16.5
Curl and Divergence
Many ways to differentiate a vector eld F( x, y, z) =
the total derivative or Jacobian matrix:
P P P
x
y
Q
y
R
y
DF = Q
x
R
x
P( x,y,z)
Q( x,y,z)
R( x,y,z)
. The nine scalar 1st-derivatives form
z
Q
z
R
z
Entries of DF can com
16.4
Greens Theorem
Unless a vector eld F is conservative, computing the line integral
C
F dr =
C
P dx + Q dy
is often difcult and time-consuming. For a given integral one must:
1. Split C into separate smooth subcurves C1 , C2 , C3 .
2. Parameterize each
16.3
The Fundamental Theorem of Line Integrals
Recall the Fundamental Theorem of Calculus for a single-variable function f :
b
a
f ( x ) dx = f (b) f ( a)
It says that we may evaluate the integral of a derivative simply by knowing the values of the functi
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MATH 2E
Fall 2016
Practice problems for the final
Note: The final exam is cumulative. About 40% of the final is from the topics
we covered up to midterm II and 60% is from the topics after. The following
are practice problems for the second portion of the
MATH 2E
FALL 2016
Practice Midterm 2
Name, ID, and Discussion Number:
Instructions: No notes, books, calculators, or supplies allowed except pencils, pens, and erasers. Show your work (only writing the final answer is not
enough to get full credit).
Sugge
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Math 2E: Vector Calculus Midterm v1
1 Find the values of the three line integrals
2 sin t
helix r(t) = 2 cos t from t = 0 to 2
t
R
C
z ds,
R
C
z dx,
R
C
z dz where C is the portion of the
(9)
2 Consider the change of co-ordinates x = 2uv, y = u2 v2
(a) Ca
Math 2E: Vector Calculus Midterm v2
1 Find the values of the three line integrals
3 cos t
helix r(t) = 3 sin t from t = 0 to 2
t
R
C
z ds,
R
C
z dx,
R
C
z dz where C is the portion of the
(9)
2 Consider the change of co-ordinates x = 2uv, y = v2 u2
(a) Ca
Math 2E Lec A
Quiz 1
Solution
(4 pts) 1) Find a parametric equation of the line through the points (1, 1, 2 ) and
( 1,0,1) .
The standard parameterization for the line connecting these two points is
l ( t ) = t 1,0,1 + (1 t ) 1, 1, 2
l ( 0 ) = 1, 1, 2 and
Math 2E Midterm 2
Solution
1. A semicircular wire is in the shape of the lower half of a circle of radius a centered at
the origin. The composition of the wire is such that the linear density of the wire is equal
to y 2 . Find the mass of this wire.
( x,
Math 2E Lec A
Quiz 2 (Redux)
Solution
1) (4 pts) Find the arc-length for the curve r ( t ) =
t
t
, ln ( cos ( t ) ) ,
2
2
between the
1
points ( 0, 0, 0 ) and
, ln
.
,
2 4 2
4 2
For your reference, here's a graph of this curve:
0.1
0.0
0.2 0.3
0.8
0.6
Math 2E Lec A
Quiz 3
Solution
1) (5 pts) Determine whether or not the vector field is conservative. If it is, find the
potential function.
F ( x, y ) = y cos x,sin x y
We could start by noticing that F ( x, y ) = y cos x,sin x y = M ( x, y ) , N ( x, y )
Math 2E Lec A
Quiz 6
Solution
1) (10 pts) Evaluate the integral
0 z 4.
X
z dS where the surface X is z 2 = x 2 + y 2 with
The surface is a cone, shown below:
To parameterize, we first look at the z -trace, which is a circle centered at the origin with
rad