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Tutorial 1 (ELE 401)
1. Determine the gradient of the following fields and compute its value at the specified
point.
a)
)
.
,
.
,
.
(
z
e
V
y
x
4
0
2
0
1
0
,
5
cos
)
3
2
(
−
=
+
b)
)
,
,
(
e
T
z
0
3
/
2
,
sin
5
2
π
φ
ρ
−
=
c)
)
,
,
(
r
Q
2
/
6
/
1
,
sin
sin
2
θ
=
2. Find the divergence and curl of the following vectors:
a)
z
y
x
a
a
a
A
cos
sin
2
xz
xy
e
xy
+
+
=
b)
z
ρ
a
a
B
sin
os
2
2
z
c
z
+
=
c)
a
a
a
C
θ
r
sin
2
sin
1
os
2
r
r
c
r
+
−
=
3. Evaluate
and
if:
A
×
∇
A
×
∇
⋅
∇
a)
z
y
x
a
a
a
A
2
z
y
2
2
xz
y
x
−
+
=
b)
z
ρ
a
a
a
A
3
2
3
2
z
z
+
+
=
c)
a
a
A
r
cos
sin
2
2
r
r
−
=
4. Let
. Evaluate
z
ρ
a
a
D
cos
2
2
2
+
=
z
a)
∫
⋅
S
d
S
D
b)
∫
⋅
∇
v
dv
D
over the region defined by
2
0
,
1
1
5
0
<
<
≤
≤
−
≤
≤
z
,
.
5. Verify the divergence theorem:
∫
∫
⋅
∇
=
⋅
v
S
dv
d
A
S
A
for each of the following cases:
a)
and
S
is the surface of the cuboid defined by
.
z
y
x
a
a
a
A
z
y
y
2
3
2
+
+
=
y
x
1
0
1
0
1
0
<
<
<
<
<
<
z
,
y
,
x
b)
and
S
is the surface of the wedge
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Unformatted text preview: . z ρ a a a A cos 4 sin 3 2 − + = z z 5 45 2 < < < < < < z , , c) and S is the surface of a quarter of a sphere defined by θ r a a A cos sin 2 r r + = 2 / 2 / 3 < < < < < < , , r . 6. Find the flux of the curl of field a a a T θ r cos cos sin cos 1 2 + + = r r through the hemisphere 4 ≤ = , z r ....
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This note was uploaded on 04/17/2008 for the course ELE 401 taught by Professor Forgot during the Spring '08 term at Ryerson.
 Spring '08
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