Question

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.

Answer #1

% matlab code

% Flow Rate Inward is given by double integral;

f=@(x,y) -2.*y.*x.^3-2.*y.^3;

q = integral2(f,0,1,0,1);

disp('The Flow Rate Inwards in units^3/sec : ')

disp(q)

disp('The Flow Rate Outwards in units^3/sec : ')

disp(-q)

Find the flux of the vector field F(x, y, z) = x, y, z through
the portion of the parabaloid z = 16 - x^2-y^2 above the
plane ? = 7 with upward pointing normal.

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

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.

Let F(x, y, z) = z tan−1(y^2)i + z^3 ln(x^2 + 7)j + zk. Find the
flux of F across S, the part of the paraboloid x^2 + y^2 + z = 29
that lies above the plane z = 4 and is oriented upward.

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.

Flowing through all space is an electric field E(x, y, z) =
αyz(x hat) + αxz(y hat) + αxy(z hat). Show that the curl of the
electric field vanishes, ∇ × E = 0. Use the definition of electric
potential to find the potential difference between the origin and r
= x(x hat) + y(y hat) + z(z hat), V (r) − V (0) = −(integral from 0
to r of (E · dl)). As the line integral is independent...

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.

Let F(x, y,
z) = z
tan−1(y2)i
+ z3
ln(x2 + 9)j +
zk. Find the flux of
F across S, the part of the paraboloid
x2 + y2 +
z = 7 that lies above the plane
z = 3 and is oriented upward.

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