Answer:
Ok... I hope this is correct
Step-by-step explanation:
Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^(2)+y^(2)=16
Center: (
0
,
0
)
Vertices: (
4
,
0
)
,
(
−
4
,
0
)
Foci: (
4
√
2
,
0
)
,
(
−
4
√
2
,
0
)
Eccentricity: √
2
Focal Parameter: 2
√
2
Asymptotes: y
=
x
, y
=
−
x
Then 0≤z≤1, and a hemispherical cap defined by x^2+y^2+(z−1)^2=16, z≥1.
Simplified
0
≤
z
≤
1
,
x
^2
+
y
^2
+
z
^2
−
2
^z
+
1
=
16
,
z
≥
1
For the vector field F=(zx+z^2y+4y, z^3yx+3x, z^4x^2), compute ∬M(∇×F)⋅dS in any way you like.
Vector:
csc
(
x
) , x
=
π
cot
(
3
x
) , x
=
2
π
3
cos
(
x
2
) , x
=
2
π
Since
(
z
x
+
z
^2
y
+
4
y
,
z
^3
y
x
+
3
x
,
z
^4
x
^2
) is constant with respect to F
, the derivative of (
z
x
+
z
^2
y
+
4
y
,
z
^3
y
x
+
3
x
,
z
^4
x
2
) with respect to F is 0
.
