In these exercises, F( x , y , z ) denotes a vector field defined on a surface σ oriented by a unit normal vector field n ( x , y , z ) , and Φ denotes the flux of F across σ . (a) Assume that σ is the graph of a function z = g ( x , y ) over a region R in the x y -Plane and that n has a positive k-component for every point on σ . Then a double integral over R whose value is Φ is _____ . (b) Suppose that σ is the triangular region with vertices ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , and ( 0 , 0 , 1 ) with upward orientation. T hen the flux of F ( x , y , z ) = x i + y j + z k across σ is Φ = _ _ _ _ _ .
In these exercises, F( x , y , z ) denotes a vector field defined on a surface σ oriented by a unit normal vector field n ( x , y , z ) , and Φ denotes the flux of F across σ . (a) Assume that σ is the graph of a function z = g ( x , y ) over a region R in the x y -Plane and that n has a positive k-component for every point on σ . Then a double integral over R whose value is Φ is _____ . (b) Suppose that σ is the triangular region with vertices ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , and ( 0 , 0 , 1 ) with upward orientation. T hen the flux of F ( x , y , z ) = x i + y j + z k across σ is Φ = _ _ _ _ _ .
In these exercises,
F(
x
,
y
,
z
)
denotes a vector field defined on a surface
σ
oriented by a unit normal vector field
n
(
x
,
y
,
z
)
,
and
Φ
denotes the flux of
F
across
σ
.
(a) Assume that
σ
is the graph of a function
z
=
g
(
x
,
y
)
over a region R in the
x
y
-Plane
and that n has a positive k-component for every point on
σ
. Then a double integral over R whose value is
Φ
is
_____
.
(b) Suppose that
σ
is the triangular region with vertices
(
1
,
0
,
0
)
,
(
0
,
1
,
0
)
,
and
(
0
,
0
,
1
)
with upward orientation. T hen the flux of
F
(
x
,
y
,
z
)
=
x
i
+
y
j
+
z
k
across
σ
is
Φ
=
_
_
_
_
_
.
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.
Compute the flux of the vector field F(x,y, z) = yi– 2xy k over the triangle T with
vertices (1,0,0), (0, 1,0), and (0, 0, 2).
Consider the vector field F(x,y,z) = (3x²y)ỉ + (x³ + 1)j + (9z²) and the curve C
made up of the quarter circle cut from the sphere x² + y² + z² = 25 by the plane
x = 3, from (3,4,0) to (3,0,4) traveling counterclockwise when looking from the
positive x -axis.
a.
b.
Use a line integral (as in Section 15.2) to compute the work done by the
vector field in moving a particle along C.
Show that the vector field is conservative, then compute the work done
by the vector field in moving a particle along C using a potential function
(as in Section 15.3) for the vector field.
Find the flux of the vector field F = (y, z, x) across the
-
part of the plane z = 4+3x+2y above the rectangle
[0,2] × [0, 5] with upwards orientation.
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