Question

In what way is the measurement of the net flow of a vector
field

Field[x, y] = {m[x, y], n[x, y]}

along C related to the measurement of the net flow of its spin
field

spinField[x, y] = {-n[x, y], m[x, y]}

across C?

Answer #1

**Answer :**

Field[x, y] = {m[x, y], n[x, y]}

The integral of field is used to measure the area of enclosed region by this vector field.

So in this case , integral

If this net integral is positive then field is in counterclockwise direction.

and if negative then field is in clockwise direction.

spin Field[x, y] = {-n[x, y], m[x, y]}

Integral of spin field :

If this integral is positive then the field is inside to outside and

if this integral is negative than the field is outside to inside as

**Spin field is always perpendicular to the vector
field.**

So,

Trajectories of field vector and spin vector are at right angle.

1. Given the vector field v = 5ˆi, calculate the vector flow
through a 2m area with a normal vector
•n = (0,1)
•n = (1,1)
•n = (. 5,2)
•n = (1,0)
2. Given the vector field of the form v (x, y, z) = (2x, y, 0)
Calculate the flow through an area of area 1m placed at the
origin and parallel to the yz plane.
3. Given a vector field as follows v = (1, 2, 3)....

Consider the vector field F = <2 x
y^3 , 3 x^2
y^2+sin y>. Compute
the line integral of this vector field along the quarter-circle,
center at the origin, above the x axis, going from the point (1 ,
0) to the point (0 , 1). HINT: Is there a potential?

Calculate the outward flux of the vector field F(x,y) =
x[i] + y^2[j] across the square
bounded by x= 1, x= -1, y=
1 and y= -1.

8. Use the Divergence Theorem to compute the net outward flux of
the field F= <-x, 3y, z> across the surface S, where S is the
surface of the paraboloid z= 4-x^2-y^2, for z ≥ 0, plus its base in
the xy-plane.
The net outward flux across the surface is ___.
9. Use the Divergence Theorem to compute the net outward flux of
the vector field F=r|r| = <x,y,z> √x^2 + y^2 + z^2 across the
boundary of the region...

Sketch the vector field F⃗ (x,y)=−5i and calculate the line
integral of F⃗ along the line segment from (−5,3) to (0,4).

The function f(x, y) = x^−2 y^3 is a potential for a vector
field F. Use this to evaluate ∫ C F · dr where C is a curve from
(1, 1) to (2, 2).

Find the divergence of the following vector field: C(x, y) = (xi
+ yj) * log((x^2) + (y^2))

Consider the vector field.
F(x, y,
z) =
6ex
sin(y),
7ey
sin(z),
5ez
sin(x)
(a) Find the curl of the vector field.
curl F =
(b) Find the divergence of the vector field.

Consider the vector field. F(x, y, z) = 9ex sin(y), 9ey sin(z),
2ez sin(x) (a) Find the curl of the vector field. curl F = (b) Find
the divergence of the vector field.

Sketch the vector field vec F (x,y)=xi +yj and calculate the
line integral of along the line segment vec F from (5, 4) to (5,
8)

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