Evaluate ∫ C ( y + sin x ) d x + ( z 2 + cos y ) d y + x 3 d z where C is the curve r ( t ) = 〈 sin t , cos t , sin 2 t 〉 , 0 ≤ t ≤ 2 π .[ Hint: Observe that C lies on the surface z = 2 x y ]
Evaluate ∫ C ( y + sin x ) d x + ( z 2 + cos y ) d y + x 3 d z where C is the curve r ( t ) = 〈 sin t , cos t , sin 2 t 〉 , 0 ≤ t ≤ 2 π .[ Hint: Observe that C lies on the surface z = 2 x y ]
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
Evaluate
∫
C
(
y
+
sin
x
)
d
x
+
(
z
2
+
cos
y
)
d
y
+
x
3
d
z
where C is the curve
r
(
t
)
=
〈
sin
t
,
cos
t
,
sin
2
t
〉
,
0
≤
t
≤
2
π
.[Hint:Observe that C lies on the surface
z
=
2
x
y
]
Evaluate √(2² + yz sin(xyz))dx+(y²+xz sin(xyz))dy+(x+xysin(xyz))dz where C
is the curve following the outline for the triangle from (1,0,0) to (0,1,0) to (0, 0, 1)
and back to (1,0,0).
Find the parametric equations of the tangent line LT to the curve
C:r(t) = 2 cos(t)i + sin(t)j + 2tk
at the point (-2, 0, 27)
b) Find the directional derivative (D.) of the function at P in the direction of PQ
f (r, y) = sin 2x cos y, P(T, 0), Q()
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