Suppose that the surface σ of the unit cube in the accompanying figure has an outward orientation. In each part, determine whether the flux of the vector field F( x , y , z ) = z j across the specified face is positive, negative, or zero. (a) The face x = 1 (b) The face x = 0 (c) The face y = 1 (d) The face y = 0 (e) The face z = 1 (f) The face z = 0
Suppose that the surface σ of the unit cube in the accompanying figure has an outward orientation. In each part, determine whether the flux of the vector field F( x , y , z ) = z j across the specified face is positive, negative, or zero. (a) The face x = 1 (b) The face x = 0 (c) The face y = 1 (d) The face y = 0 (e) The face z = 1 (f) The face z = 0
Suppose that the surface
σ
of the unit cube in the accompanying figure has an outward orientation. In each part, determine whether the flux of the vector field
F(
x
,
y
,
z
)
=
z
j
across the specified face is positive, negative, or zero.
(a)
The face
x
=
1
(b)
The face
x
=
0
(c)
The face y
=
1
(d)
The face y
=
0
(e)
The face z
=
1
(f)
The face z
=
0
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.
(b) Show that the vector field, F3
yz î + zx ŷ + xy 2 can be written both as the gradient of
a scalar and curl of a vector. Find the scalar and vector potential for this function.
Fill in the table and then sketch the vector field.
y F(x, y) =
-2
-2
-2
2
-2
-2
-2
3-
-2-
2.
/-
2.
Calculate the curl of the vector g
=
(x−y) ax+(y+x) ay, and visualize it. Explain
what does "curl of vector field" mean, in your own words. When does it equal
zero?
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