Question

Given S is the surface of the paraboloid z= 4-x^2-y^2 and C is the curve of intersection of the paraboloid with the plane z=0. Verify stokes theorem for the field F=2zi+xj+y^2k.( you might verify it by checking both sides of the theorem)

Answer #1

In the following problems, the surface S is the part of the
paraboloid z= x^2 + y^2 which lies below the plane z= 4, and
includes the circular intersection with this plane. This single
surface S could also be described as being contained inside the
cylinder x^2+y^2= 4.
(a) Iterate, but do not evaluate, the integral ∫∫S(z+x) dS in
terms of two parameters. Write the integrand in simplest form.
(b) Use Stoke’s theorem to rewrite ∫S(delta X F) · ndS...

Use Stokes" Theorem to evaluate (F-dr where F(x, y, z)=(-y , x-z
, 0) and the surface S is the part of the paraboloid : z = 4- x^2 -
y^2 that lies above the xy-plane. Assume C is oriented
counterclockwise when viewed from above.

Consider the part of the paraboloid x = 4 − y ^2 − z ^2 that
lies in front of the plane x = 0.
(a) What is its mass if its density is ρ(x, y, z) = y^2 + z ^2
g/cm^2 ?
(b) What is its surface area?

Let S be the boundary of the solid bounded by the paraboloid
z=x^2+y^2 and the plane z=16
S is the union of two surfaces. Let S1 be a portion of the plane
and S2 be a portion of the paraboloid so that S=S1∪S2
Evaluate the surface integral over S1
∬S1 z(x^2+y^2) dS=
Evaluate the surface integral over S2
∬S2 z(x^2+y^2) dS=
Therefore the surface integral over S is
∬S z(x^2+y^2) dS=

Given the level surface S defined by f(x, y, z) = x −
y3 − 2z2 = 2 and the point P0(−4,
−2, 1).
Find the equation of the tangent plane to the surface S at the
point P0.
Find the derivative of f at P0in the direction of
r(t) =< 3, 6, −2 >
Find the direction and the value of the maximum rate of change
greatest increase of f at P0;
(d) Find the parametric equations of the...

For V = [2(x^2)y] i-( (z^3) + y) j + (3xyz) k, show that Stokes'
theorem
holds by calculating both sides of the equation for a square in the
x-y plane
with corners at (0; 0; 0), (3; 0; 0), (3; 3; 0), (0; 3; 0) .
Confirm that Stokes' theorem only depends on the boundary line by
integrating over the surface of a cube with an open bottom The
bounding line is the same as before.

Problem 10. Let F = <y, z − x, 0> and let S be the surface
z = 4 − x^2 − y^2 for z ≥ 0, oriented by outward-pointing normal
vectors.
a. Calculate curl(F).
b. Calculate Z Z S curl(F) · dS directly, i.e., evaluate it as a
surface integral.
c. Calculate Z Z S curl(F) · dS using Stokes’ Theorem, i.e.,
evaluate instead the line integral I ∂S F · ds.

Compute ∫∫S F·dS for the vector field F(x,y,z)
=〈0,0,x2+y2〉through the surface given by x^2+y^2+z^2= 4, z≥0 with
outward pointing normal.
Please explain and show work.
Thank you so much.

Use Stokes' Theorem to evaluate
S
curl F · dS.
F(x, y, z) = x2 sin(z)i + y2j + xyk,
S is the part of the paraboloid
z = 1 − x2 − y2
that lies above the xy-plane, oriented upward.

Find the area of the surface. The part of the paraboloid
z=1-x^2-y^2 that lies above the plane z=-2
(Please post hand writing one) thank you

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