Vector Fields
Line Integrals
Independence
of Path
Green's
Theorem
10
10
10
10
20
20
20
20
30
30
30
30
40
40
40
40
QUESTION:
Determine the type of input and output for
gradient, divergence, and curl.
ANSWER:
Gradient: Input – scalar, output – vector
Divergence: Input – vector, output – scalar
Curl: Input – vector, output – vector
QUESTION:
Find 𝑑𝑒𝑙 𝑓 : 𝑓 𝑥, 𝑦, 𝑧 = ln |𝑥𝑦𝑧|
ANSWER:
1 1 1
, ,
𝑥 𝑦 𝑧
QUESTION:
Find 𝑑𝑖𝑣 𝐹 and 𝑐𝑢𝑟𝑙 𝐹 :
2
2
𝐹 𝑥, 𝑦, 𝑧 = 𝑥 𝑖 − 2𝑥𝑦𝑗 + 𝑦𝑧 𝑘
ANSWER:
𝑑𝑖𝑣 𝐹 = 2𝑦𝑧
2
𝑐𝑢𝑟𝑙 𝐹 = 𝑧 , 0, −2𝑦
QUESTION:
Find 𝑔𝑟𝑎𝑑 𝑑𝑖𝑣 𝐹 :
𝑥
𝑥
𝐹 𝑥, 𝑦, 𝑧 = 𝑒 cos 𝑦 , 𝑒 sin 𝑦 , 𝑧
ANSWER:
𝑥
2𝑒 , − sin 𝑦 , 0
QUESTION:
What is 𝑑𝑠 equal to?
ANSWER:
𝑥′ 𝑡
2
+ 𝑦′ 𝑡
2 𝑑𝑡
QUESTION:
3
𝑥
+
𝑦
𝑑𝑠;
C
is
the
curve
𝑥
=
𝐶
3
3𝑡, 𝑦 = 𝑡 , 0 ≤ 𝑡 ≤ 1
ANSWER:
14(2 2 − 1) ≈ 25.598
QUESTION:
𝑥
+
2𝑦
𝑑𝑥
+
𝑥
−
2𝑦
𝑑𝑦;C
is
𝐶
the line segment from (1,1) to (3,-1)
ANSWER:
0
QUESTION:
Find the work done by F: 𝐹 𝑥, 𝑦 =
𝑥 + 𝑦, 𝑥 − 𝑦 ; C is the quarter
ellipse, 𝑥 = a cos 𝑡 , 𝑦 = 𝑏 sin 𝑡 , 0 ≤
𝑡 ≤ 2𝜋
ANSWER:
2
𝑎 +𝑏
−2
2
QUESTION:
What determines if 𝐶 𝐹 ∙ 𝑑𝑟 is
independent of path?
ANSWER:
𝐹 is conservative or a gradient
vector field. (𝑀𝑦 = 𝑁𝑦 )
QUESTION:
Is F conservative? 𝐹 𝑥, 𝑦 =
10𝑥 − 7𝑦 𝑖 − (7𝑥 − 2𝑦)𝑗
ANSWER:
Yes
QUESTION:
Is F conservative?𝐹 𝑥, 𝑦 =
−2𝑥
−2𝑧
2
2 𝑖−( 2
2 )𝑘
𝑥 +𝑧
ANSWER:
No
𝑥 +𝑧
QUESTION:
Find a function for which
𝐹 𝑥, 𝑦 = 10𝑥 − 7𝑦 𝑖 − (7𝑥 −
2𝑦)𝑗 is the gradient.
ANSWER:
2
2
𝑓 𝑥, 𝑦 = 5𝑥 − 7𝑥𝑦 + 𝑦 + 𝐶
QUESTION:
𝐶
2𝑥𝑦𝑑𝑥 + 𝑦 2 𝑑𝑦; C is the closed
curve formed by 𝑦 =
𝑥
,𝑦
2
ANSWER:
64
−
≈ 4.2667
15
= 𝑥
QUESTION:
𝑥𝑦𝑑𝑥
+
𝑥
+
𝑦
𝑑𝑦;
C
is
the
𝐶
triangle with vertices (0,0), (2,0),
(2,3)
ANSWER:
−1
QUESTION:
2
2
Find the flux of 𝐹 = 𝑥 + 𝑦 , 2𝑥𝑦
across the boundary of the square
with vertices at (0,0), (0,1), (1,1), (1,0)
ANSWER:
2
QUESTION:
Find the work done by 𝐹 =
2
2
𝑥 + 𝑦 𝑖 − 2𝑥𝑦𝑗 moving
clockwise around the square with
vertices (0,0), (0,1), (1,1), (1,0)
ANSWER:
2