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) Φ is the value of the surface integral _ _ _ _ _ . (b) If σ is the unit sphere and n is the outward unit normal, then 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) Φ is the value of the surface integral _ _ _ _ _ . (b) If σ is the unit sphere and n is the outward unit normal, then 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)
Φ
is the value of the surface integral
_
_
_
_
_
.
(b) If
σ
is the unit sphere and n is the outward unit normal, then 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.
U (x, y)
4c(y + 1)i + xyj, and V(x, y)
=
Consider two vector fields in the xy plane, given in the Cartesian coordinates as:
cyi - xj, where c is a constant. Find
where in the xy plane the vectors of these two fields are parallel to one another, and
where they are mutually orthogonal.
=
4. Consider the vector function r(z, y) (r, y, r2 +2y").
(a) Re-write this vector function as surface function in the form f(1,y).
(b) Describe and draw the shape of the surface function using contour lines and algebraic analysis
as needed. Explain the contour shapes in all three orthogonal directions and explain and label
all intercepts as needed.
(c) Consider the contour of the surface function on the plane z=
for this contour in vector form.
0. Write the general equation
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