(a) Use Stokes’ Theorem to evaluate ∫ c F ⋅ d r , where F ( x , y , z ) = x 2 z i + x y 2 j + z 2 k and C is the curve of intersection of the plane x + y + z = 1 and the cylinder x 2 + y 2 = 9 , oriented counterclockwise as viewed from above. (b) Graph both the plane and the cylinder with domains chosen so that you can see the curve C and the surface that you used in part (a). (c) Find parametric equations for C and use them to graph C .
(a) Use Stokes’ Theorem to evaluate ∫ c F ⋅ d r , where F ( x , y , z ) = x 2 z i + x y 2 j + z 2 k and C is the curve of intersection of the plane x + y + z = 1 and the cylinder x 2 + y 2 = 9 , oriented counterclockwise as viewed from above. (b) Graph both the plane and the cylinder with domains chosen so that you can see the curve C and the surface that you used in part (a). (c) Find parametric equations for C and use them to graph C .
Solution Summary: The author explains Stokes' Theorem: Let S be an oriented piecewise-smooth surface that is bounded by a simple, closed, piece-wise, smooth boundary curve C with positive orientation.
(a) Use Stokes’ Theorem to evaluate
∫
c
F
⋅
d
r
,
where
F
(
x
,
y
,
z
)
=
x
2
z
i
+
x
y
2
j
+
z
2
k
and
C
is the curve of intersection of the plane
x
+
y
+
z
=
1
and the cylinder
x
2
+
y
2
=
9
, oriented counterclockwise as viewed from above.
(b) Graph both the plane and the cylinder with domains chosen so that you can see the curve
C
and the surface that you used in part (a).
(c) Find parametric equations for
C
and use them to graph
C
.
Find the parametric equations for the line through the point of intersection of the lines
L1:x = 2 + 3t, y = 5t, z = 2 +t and L2: x = 3+ 2t, y = 4+ t, z = 5 – 2t and
perpendicular to the plane 3x – 4y + 2z = 7
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