Use Stokes’s Theorem to evaluate ∮ C F . d r . F( x , y , z ) = ( z + sin x ) i + ( x + y 2 ) j+ ( y + e z )k; C is the intersection of the sphere x 2 + y 2 + z 2 = 1 and the cone z = x 2 + y 2 with counterclockwise orientation looking down the down the positive z - axis.
Use Stokes’s Theorem to evaluate ∮ C F . d r . F( x , y , z ) = ( z + sin x ) i + ( x + y 2 ) j+ ( y + e z )k; C is the intersection of the sphere x 2 + y 2 + z 2 = 1 and the cone z = x 2 + y 2 with counterclockwise orientation looking down the down the positive z - axis.
F(
x
,
y
,
z
)
=
(
z
+
sin
x
)
i
+
(
x
+
y
2
)
j+
(
y
+
e
z
)k;
C
is the intersection of the sphere
x
2
+
y
2
+
z
2
=
1
and the cone
z
=
x
2
+
y
2
with counterclockwise orientation looking down the down the positive z-axis.
Match each parametrization with the corresponding surface.
(i)
(u, cos (u), sin (v))
Z
(iv)
(u, v³, v)
(ii)
Answer Bank
(u, u + v, v)
(cos (u) sin (v), 3 cos (u) sin (u), cos (v))
(v)
(iii)
(u, u (2+ cos (v)), u (2+ sin (u)))
Find the parametric equations of the line tangent to the curve of intersection of the
surfaces x² + y² = z and x² + y² = 4 at the point (√2, √2,4).
Let C be the upper half of the circle whose equation is x? + (y - 2)2 = 1. Let S be the surface
that is generated by revolving curve C along the x-axis. Describe S using a vector function of 2
3 3V3
variables and find an equation of the plane tangent to the surface S at the point (0, ).
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