Question

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.

Answer #1

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.

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)

10.) (23 pts.) Verify the Divergence Theorem for P(x, y, z) =
(y) i+ (—x) j + (—xz) k , where the solid D is enclosed by the
paraboloid z = x^2 + y^2 and the plane z = 1.

Let F(x,y,z) = ztan-1(y^2) i + (z^3)ln(x^2 + 8) j + z k. Find
the flux of F across the part of the paraboloid x2 + y2 + z = 20
that lies above the plane z = 4 and is oriented upward.

Use Stokes' Theorem to evaluate the integral
∮CF⋅dr=∮C8z^2dx+8xdy+2y^3dz where C is the circle x^2+y^2=9 in the
plane z=0 .

URGENT
Can someone answer these two questions within the next hour?
Use Green’s Theorem to find the integral of the vector field F~
(x, y) = (5y + 4x)~i + (3y − 7x)~j counterclockwise around the
ellipse x 2 9 + y 2 = 1. Hint: The area of the ellipse with
equation x 2/ a 2 + y 2 /b 2 = 1 is πab.
Use Stokes’ Theorem to compute Z C F~ · d~s where F~ (x, y,...

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

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.

Evaluate the following.
f(x, y) = x + y
S: r(u, v) = 5
cos(u) i + 5 sin(u)
j + v k, 0 ≤ u
≤ π/2, 0 ≤ v ≤ 3

Compute the line integral of f(x, y, z) = x 2 + y 2 −
cos(z) over the following paths:
(a) the line segment from (0, 0, 0) to (3, 4, 5)
(b) the helical path → r (t) = cos(t) i + sin(t) j + t k from
(1, 0, 0) to (1, 0, 2π)

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