с
E Verify
stoke's thm for
F = Cy, ?, x) , if S is paraboloi
=fox, y) as t-Ox?ty2), 220
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Answer:
Stoke’s Theorem states:
∮
C
F
⋅
d
r
=
∬
S
(
∇
×
F
)
⋅
d
S
,
where
∇
×
F
=
∣
∣
∣
∣
∣
i
j
k
∂
∂
x
∂
∂
y
∂
∂
z
P
Q
R
∣
∣
∣
∣
∣
=
(
∂
R
∂
y
−
∂
Q
∂
z
)
i
+
(
∂
P
∂
z
−
∂
R
∂
x
)
j
+
(
∂
Q
∂
x
−
∂
P
∂
y
)
k
is the curl of the vector field
F
.
The symbol
∮
indicates that the line integral is taken over a closed curve.
We assume there is an orientation on both the surface and the curve that are related by the right hand rule. That is, if you were to walk around the curve in its preferred direction with your head pointing in the same direction as the normal vector
n
to the surface, then the surface would always
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