Question

Stokes' Theorem is the generalization of the circulation form of Green's Theorem in the x y-plane. Use Stokes' Theorem to write the circulation form of Green's Theorem in the y z-plane.

Answer #1

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Stokes' Theorem is the generalization of the circulation form of
Green's Theorem in the x y-plane. Use Stokes' Theorem to write the
circulation form of Green's Theorem in the y z-plane.
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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

Suppose F⃗ (x,y)=〈x^2+5y,7x−3y^2〉. Use Green's Theorem to
calculate the circulation of F⃗ around the perimeter of the
triangle C oriented counter-clockwise with vertices (10,0), (0,5),
and (−10,0).

Derive curl form of Green’s Theorem in xy-plane from Stokes’
Theorem

Use the surface integral in Stokes' Theorem to calculate the
circulation of the field
F=x2i+5xj+z2k
around the curve C: the ellipse
25 x squared plus 4 y squared equals
25x2+4y2=2 in the xy-plane, counterclockwise
when viewed from above.

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.

Use Green's theorem to find the counterclockwise circulation and
outward flux for the field F=(4x-9y)i +(8y-9x)j and the curve C:
the triangle bounded by x=0,x=8,y=0 ,y=8 The flux is ??? The
circulation is ???

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.

Use Stokes' Theorem to evaluate ∫ C F · dr where F = (x +
5z) i + (3x + y) j + (4y − z) k and C is the curve of
intersection of the plane x + 2y + z = 16 with the coordinate
planes

Use Stokes' Theorem to compute the flux of
curl(F) through the
portion of the plane x37+y33+z=1 where
x, y, z≥0, oriented with an
upward-pointing normal, for F = <yz,
0, x>.
(Use symbolic notation
and fractions where needed.)
Flux =

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