Question

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).

• Calculate the field flow through a side 1 cube (oriented as you
wish).

• Assume that the previous cube is missing one of its faces (the
one you want), in that case, how much will be the total flow
through it.

Answer #1

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 parametrize for the vector field that goes through a
point P at t=0
1) F(x,y) = i + xj, P = (-2,2)
2) F(x,y) = -yi + xj, P= (1,0)

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 = 4 − x2 − y2 that lies above the square
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
and has...

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 = 6 − x2 − y2 that lies above the square
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
and has...

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 = 6 − x2 − y2 that lies above the square
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
and...

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...

Differential Geometry
3. In each case, express the given vector field V in
the standard form (b) V(p) = ( p1, p3 - p1, 0)p for all p. (c) V =
2(xU1 + yU2) - x(U1 - y2 U3). (d) At each point p, V(p)
is the vector from the point ( p1, p2, p3) to the point (1 + p1,
p2p3, p2). (e) At each point p, V(p) is the vector from p to the
origin.

compute the flux of the vector field F through the parameterized
surface S. F= zk and S is oriented upward and given, for 0 ≤ s ≤ 1,
0 ≤ t ≤ 1, by x = s + t, y = s – t, z = s2 +
t2.
the answer should be 4/3.

1.)The velocity field of a flow is given by
V=a(x2-y2)i-2axyj
(a) Determine the convective acceleration in the y
direction.
(b) Check if continuity equation is satisfied.
(c)determine the vorticity.
(d) Find the stream function.

1.) The velocity field of a flow is given by
V=2(-x+y)i+(3x2+2y)j
(a) find the stream function.
(b) Determine the vorticity.
(c)Find the velocity potential if possible.

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