Line Integrals Let F ( x , y ) = 2 x i + x y 2 j and consider the curve y = x 2 from ( 0 , 0 ) to ( 2 , 4 ) in the xy -plane. Set up and evaluate line integrals of the forms ∫ C F · d r and ∫ C M d x + N d y . Compare your results. Which method do you prefer? Explain.
Line Integrals Let F ( x , y ) = 2 x i + x y 2 j and consider the curve y = x 2 from ( 0 , 0 ) to ( 2 , 4 ) in the xy -plane. Set up and evaluate line integrals of the forms ∫ C F · d r and ∫ C M d x + N d y . Compare your results. Which method do you prefer? Explain.
Let
F
(
x
,
y
)
=
2
x
i
+
x
y
2
j
and consider the curve
y
=
x
2
from
(
0
,
0
)
to
(
2
,
4
)
in the xy-plane. Set up and evaluate line integrals of the forms
∫
C
F
·
d
r
and
∫
C
M
d
x
+
N
d
y
. Compare your results. Which method do you prefer? Explain.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Find the slope of the tangent in the positive x-direction to the surface z =
3x3 – 6xy at the point (2, 1, 12).
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Calculate the line integral of the vector field F = (y, x,x² + y² ) around the boundary curve, the curl of the vector field, and the
surface integral of the curl of the vector field.
The surface S is the upper hemisphere
x² + y + z? = 25, z 2 0
oriented with an upward-pointing normal.
(Use symbolic notation and fractions where needed.)
F. dr =
curl(F) =
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