PHY2061
R. D. Field
Solutions
Vector Analysis
Page 1 of 5
Vector Analysis Solutions
Problem 1:
Calculate the
gradient
of the scalar function
3
2
yz
x
f
=
.
Answer:
z
yz
x
y
z
x
x
xyz
f
ˆ
3
ˆ
ˆ
2
2
2
3
2
3
+
+
=
∇
!
Solution:
The gradient is given by
z
yz
x
y
z
x
x
xyz
z
z
f
y
y
f
x
x
f
f
ˆ
3
ˆ
ˆ
2
ˆ
ˆ
ˆ
2
2
3
2
3
+
+
=
∂
∂
+
∂
∂
+
∂
∂
=
∇
!
.
Problem 2:
Let
y
x
x
y
F
ˆ
ˆ
+
=
!
.
(a) Calculate
r
d
F
!
!
⋅
∫
along
Path
1
and
Path 2
from point
P
1
=(0,0)
to
P
2
=(1,1)
as
shown in the Figure.
(b) Calculate the
divergence
,
F
!
!
⋅
∇
, and the
curl
,
F
!
!
×
∇
.
Answers:
∫
=
⋅
1
r
d
F
!
!
0
0
=
×
∇
=
⋅
∇
F
F
!
!
!
!
Solution:
(a) Along
Path 1
we have
y = x
and
dy = dx
and hence,
∫
∫
∫
∫
=
=
+
=
+
=
⋅
1
1
0
1
1
1
2
Path
Path
y
x
Path
xdx
xdy
ydx
dy
F
dx
F
r
d
F
!
!
.
Path 2 consists of two parts Path 2a and Path 2b.
Along
Path 2a
we have
y
= 0 dy = 0
and along
Path 2b
we have
x = 1 dx = 0
.
Thus,
1
0
1
0
2
2
2
=
+
=
+
=
+
=
⋅
∫
∫
∫
∫
dy
xdy
ydx
xdy
ydx
r
d
F
a
Path
b
Path
Path
!
!
.
(b) The
divergence
is given by
0
0
0
0
=
+
+
=
∂
∂
+
∂
∂
+
∂
∂
=
⋅
∇
z
F
y
F
x
F
F
z
y
x
!
!
,
and the
curl
is given by
y
x
Path 1
Path 2
(1,1)
(0,0)

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