Question

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)

Answer #1

A fluid is flowing through space following the vector field F(x,
y, z) = yi − xj + zk. A filter is in the shape of the portion of
the paraboloid z = x^2 + y^2 having 0 <= x <= 3 and 0 <= y
<= 3, oriented inwards (and upwards). Find the rate at which the
fluid is moving through the filter.
PLEASE SOLVE ON MATLAB, when I did it by hand I got 18.

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) = yi − xj + 4zk, S is the hemisphere x^2 +
y2^ + z^2 = 4, z ≥ 0, oriented downward

Find vector and parametric equations for:
a) the line that passes through the point P(9,-9,6) parallel to
the vector u = <3,4,-2>
b) the line passing through the point P(6,-2,6) parallel to the
line x=2t, y = 2 - 3t, z = 3 +6t
c) the line passing through the point P(5, -2,1) parallel to the
line x = 3 - t, y = -2 +4t, z = 4 + 8t

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)....

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) = yi − xj + 2zk,
S is the hemisphere
x2 + y2 + z2 = 4,
z ≥ 0,
oriented downward

Let C be a closed curve parametrized by r(t) = sin ti+cos tj
with 0 ≤ t ≤ 2π. Let F = yi − xj be a vector field.
(a) Evaluate the line integral xyds. C
(b) Find the circulation of F over C. (c) Find the flux of F
over C.

(1 point) Find the simplest vector parametric expression r⃗ (t)
for the line that passes through the points P=(5,3,−2) at time t =
4 and Q=(8,−1,−3) at time t = 10

For each vector field F~ (x, y) = hP(x, y), Q(x, y)i, find a
function f(x, y) such that F~ (x, y) = ∇f(x, y) = h ∂f ∂x , ∂f ∂y i
by integrating P and Q with respect to the appropriate variables
and combining answers. Then use that potential function to directly
calculate the given line integral (via the Fundamental Theorem of
Line Integrals):
a) F~ 1(x, y) = hx 2 , y2 i Z C F~ 1...

The Flux of the Vector field F(1, -2, 0) through any closed
surface is zero? Why?
How do you find a vector field that the flux through any closed
surface is equal to the volume enclosed?

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.

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