## I am having trouble with this problem. If anyone could help that would be great. Let M be the capped cylindrical surface which is the

Question

I am having trouble with this problem. If anyone could help that would be great.
Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2+y^2=16, 0≤z≤1, and a hemispherical cap defined by x^2+y^2+(z−1)^2=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.

in progress 0
6 months 2021-07-27T16:06:49+00:00 1 Answers 12 views 0

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
.