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 . In the case of steady-state incompressible fluid flow, with F( x , y , z ) the fluid velocity at ( x , y , z ) on σ , Φ can be interpreted as _____ .
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 . In the case of steady-state incompressible fluid flow, with F( x , y , z ) the fluid velocity at ( x , y , z ) on σ , Φ can be interpreted as _____ .
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
.
In the case of steady-state incompressible fluid flow, with
F(
x
,
y
,
z
)
the fluid velocity at
(
x
,
y
,
z
)
on
σ
,
Φ
can be interpreted as _____
.
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.
Consider the vector field F(x, y, z) = (-6y, -6x, 2z). Show that F is a gradient vector field by determining a function V which satisfies F = VV. Your function V should satisfy V(0,0,0) = 0.
V(x, y, z) =
Find the counterclockwise circulation of the vector field
F(2, y) = (e" - )i + (2aye" + )3
Ji+ (2aye" +
)j around the boundary of the region bounded by the
lines y= 2x, y= 8x, and y = 4.
A. 42/36
O B. -41/36
C. 43/48
D. 32/44
E. 47/24
F. 45/40
G. 33/40
H. 40/32
The vector field F(x,y)= can be written as:
O1. F(x,y)=(xy+x2 )i - (8)j
O II. F(x,y)=(xy+x² )i + (2-10)j
O II. (x,y)=(xy-2)i + (x² -10)j
O IV. F(x,y)=x(y+x)i + (2 -10)j
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