Question

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**.

Answer #1

Answer :

Consider the given vector field F(x,y) = x**[i]** +
y^2**[j]**

Let M = x and N = y^{2}

Then ∂M/∂x = 1 and ∂N/∂y = 2y

The outward flux of the vector field F is

The outward flux is 4

Use Green's Theorem to find the counterclockwise circulation
and outward flux for the field
F=(3x−y)i+(y−x)j and curve C: the square bounded by x=0,
x=4,y=0, y=4.
find flux and circulation

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...

Use Green's theorem to find the counterclockwise circulation and
outward flux for the field F=(4x-9y)i +(8y-9x)j and the curve C:
the triangle bounded by x=0,x=8,y=0 ,y=8 The flux is ??? The
circulation is ???

. a. [2] Compute the divergence of vector field F = x 3y 2 i +
yj − 3zx2y 2k
b. [7] Use divergence theorem to compute the outward flux of the
vector field F through the surface of the solid bounded by the
surfaces z = x 2 + y 2 and z = 2y

Evaluate the outward flux ∫∫S(F·n)dS of the vector
fieldF=yz(x^2+y^2)i−xz(x^2+y^2)j+z^2(x^2+y^2)k, where S is the
surface of the region bounded by the hyperboloid x^2+y^2−z^2= 1,
and the planes z=−1 and z= 2.

Find the flux of the vector field F (x, y, z) =< y, x, e^xz
> outward from the z−axis and across the surface S, where S is
the portion of x^2 + y^2 = 9 with x ≥ 0, y ≥ 0 and −3 ≤ z ≤ 3.

Use the divergence theorem to calculate the flux of the vector
field F = (y +xz) i+ (y + yz) j - (2x + z^2) k upward through the
first octant part of the sphere x^2 + y^2 + z^2 = a^2.

Find the flux of the vector field F =
x i +
e6x j +
z k through the surface S given
by that portion of the plane 6x + y +
3z = 9 in the first octant, oriented upward.
PLEASE EXPLAIN STEPS. Thank you.

Use Stokes’ Theorem to calculate the flux of the curl of the
vector field F = <y − z, z − x, x + z> across the surface S
in the direction of the outward unit normal where S : r(u, v)
=<u cos v, u sin v, 9 − u^2 >, 0 ≤ u ≤ 3, 0 ≤ v ≤ 2π. Draw a
picture of S.

Evaluate the surface integral
S
F · dS
for the given vector field F and the oriented
surface S. In other words, find the flux of
F across S. For closed surfaces, use the
positive (outward) orientation.
F(x, y, z) = xy i + yz j + zx k
S is the part of the paraboloid
z = 2 − x2 − y2 that lies above the square
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
and...

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