Math 237 Worksheet: line integrals and gradient vector fields ...

Math 237 Worksheet: line integrals and gradient vector fields
Problem 1. Referring to the figure below, arrange the following integrals in order from
smallest to largest:
∫
F ⃗ · d⃗r, F ⃗ · d⃗r, F ⃗ · d⃗r, F ⃗ · d⃗r, F ⃗ · d⃗r, F ⃗ · d⃗r.
∫C 1
∫C 2
∫C 3
∫C 4
∫C 5
Let F ⃗ (x, y) =y⃗i +2⃗j. Calculate ∫ ⃗ C
F · d⃗r, where C is the portion of the parabola y = x 2 that starts at (0, 0)
and ends at (3, 9).
Examples and Exercises
C 6
1. Given to the right is a vector field ⃗ F (x, y), together with
oriented curves C 1 ,C 2 ,C 3 ,C 4 ,C 5 , and C 6 .
(a) Is ∫ ⃗F · d⃗r positive, negative, or zero? Explain.
C6
C
5
C
2
C
6
C
1
C
3
C
4
∫
If C = C 1 + C 2 + C 3 + C 4 , is
(b) Let C be the closed curve C 1 + C 2 + C 3 + C 4 . Is ∫ C ⃗ F · d⃗r positive, negative, or zero? Explain.
C
F · d⃗r positive, negative, or zero?
∫
Problem 2. Let F ⃗ = −y⃗i + x⃗j. Compute the integrals F ⃗ · d⃗r, F ⃗ · d⃗r and
∫C 1
∫C 2
where C 1 , C 2 and C 3 are the paths
C 3
⃗ F · d⃗r
C 1 : t⃗i +⃗j − 1 ≤ t ≤ 1, C 2 : t⃗i + t⃗j − 1 ≤ t ≤ 1, C 3 : cos t⃗i + sin t⃗j 0 ≤ t ≤ 2π.
Problem 3. Decide whether each of the vector fields shown below is conservative in two
ways: 1) graphically, without doing any calculation and 2) by applying the curl test. If the
vector field is conservative, find a potential function.
Let ⃗ F and ⃗ G be the vector fields
whose formulas and graphs are
shown to the right.
2
⃗F (x, y) =−y⃗i + x⃗j
2
⃗G(x, y) =x⃗i + y⃗j
(a) Is ⃗ F conservative? Justify
your answer graphically, without
doing any calculations.
1
0
1
0
1
1
(c) Arrange the following quantities in order from smallest to largest:
2
∫
∫
∫
∫
∫
⃗F · d⃗r, ⃗F · d⃗r, ⃗F · d⃗r, ⃗F · d⃗r,
2 1 0 1 2
⃗F · d⃗r,
∫
2
2 1 0 1 2
⃗F · d⃗r
C1
C2
C3
4 Developed by Jerry Morris
C4
Problem 4. Use the fundamental theorem for line integrals to evaluate
∫
(2x⃗i + 2y⃗j) · d⃗r
C 1
and
∫
(2x⃗i + 2y⃗j) · d⃗r
C 2
C5
where C 1 is the line segment from (0, 4) to (3, 1) and C 2 is the parabola y = (x − 2) 2 from
(0, 4) to (3, 1). Could we apply this same theorem to the line integrals from Problem 2?
C6
ities to accompany Calculus, Hughes-Hallett et al, Wiley, 2013 17

Problem 5. Consider the vector field ⃗ F =
−y
√
x2 + y 2 ⃗i +
conservative? What does the curl test say about this vector field?
x
√
x2 + y 2 ⃗j. Is this vector field
2