Evaluating a Line Integral of a Vector Field In Exercises 11–24, find the value of the line integral ∫ C F ⋅ d r . ( Hint: If F is conservative, the integration may be easier on an alternative path.) F ( x , y ) = 2 x y i + x 2 j ( a ) r 1 ( t ) = t i − t 2 j , 0 ≤ t ≤ 1 ( b ) r 2 ( t ) = t i + t 3 j , 0 ≤ t ≤ 1
Solution Summary: The author compares the vector function F(x,y)=mathrmcos
Evaluating a Line Integral of a Vector Field In Exercises 11–24, find the value of the line integral
∫
C
F
⋅
d
r
.
(Hint: If F is conservative, the integration may be easier on an alternative path.)
F
(
x
,
y
)
=
2
x
y
i
+
x
2
j
(
a
)
r
1
(
t
)
=
t
i
−
t
2
j
,
0
≤
t
≤
1
(
b
)
r
2
(
t
)
=
t
i
+
t
3
j
,
0
≤
t
≤
1
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.
Using Green's Theorem on this vector field problem, compute a) the circulation on the boundary of R in terms of a and b, and b) the outward flux across the boundary of R in terms of a and b.
a. Show that the outward flux of the position vector field F = x i + y j + z k through a smooth closed surface S is three times the volume of the region enclosed by the surface.
b. Let n be the outward unit normal vector field on S. Show that it is not possible for F to be orthogonal to n at every point of S
Flux of a vector field?
Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.
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01 - What Is an Integral in Calculus? Learn Calculus Integration and how to Solve Integrals.; Author: Math and Science;https://www.youtube.com/watch?v=BHRWArTFgTs;License: Standard YouTube License, CC-BY