Question
In: Advanced Math
Use the Stoke’s theorem to evaluate Z Z S (∇×F)·nˆ·dS where F(x, y, z) = (x^2...
Use the Stoke’s theorem to evaluate Z Z S (∇×F)·nˆ·dS where F(x, y, z) = (x^2 z^2,y^2 z^2, xyz) and surface S is part of the paraboloid z = x^2 + y^2 that lies inside the cylinder x^2 + y^2 = 4, oriented upwards. Sketch the surface S and label everything.
Solutions
Colby Messinger answered 2 years agoRelated Solutions
Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi...
Use the Divergence Theorem to evaluate
S
F · dS,
where
F(x, y,
z) =
z2xi
+
y3
3
+ cos z
j +
(x2z +
y2)k
and S is the top half of the sphere
x2 +
y2 + z2 =
4.
(Hint: Note that S is not a closed surface.
First compute integrals over S1 and
S2, where S1 is the
disk
x2 +
y2 ≤ 4,
oriented downward, and S2 =
S1 ∪ S.)
Use the Divergence Theorem ∬SF⋅dS=∭D∇⋅FdV to find ∬SF⋅dS where F(x,y,z)=x^2i+y^2j+z^2k and S S is the surface...
Use the Divergence Theorem ∬SF⋅dS=∭D∇⋅FdV to find ∬SF⋅dS where
F(x,y,z)=x^2i+y^2j+z^2k and S S is the surface of the solid bounded
by x^2+y^2=9 , z = 0 , and z=6
Verify stokes theorem when S=(x,y,z): 9x^2+y^2=z^2 and 0 ≤z ≤2 and F(x,y,z)=0i+((9x^2)/2)j+((y^(3)*z)/3)k
Verify stokes theorem when S=(x,y,z): 9x^2+y^2=z^2 and 0 ≤z ≤2 and
F(x,y,z)=0i+((9x^2)/2)j+((y^(3)*z)/3)k
Verify stokes theorem when S=(x,y,z): 9x^2+y^2=z^2 and 0 ≤z ≤2 and F(x,y,z)=0i+((9x^2)/2)j+((y^(3)*z)/3)k
Verify stokes theorem when S=(x,y,z): 9x^2+y^2=z^2 and 0 ≤z ≤2 and
F(x,y,z)=0i+((9x^2)/2)j+((y^(3)*z)/3)k
Use spherical coordinates to evaluate the triple integral ∭e^−(x^2+y^2+z^2)/(x^2 + y^2 + z^2) dV , Where...
Use spherical coordinates to evaluate the triple integral
∭e^−(x^2+y^2+z^2)/(x^2 + y^2 + z^2) dV , Where E is the region
bounded by the spheres x^2 + y^2 + z^2 = 4 and x^2 + y^2 + z^2 =
9
Use the Divergence Theorem to evaluate ∫ ∫ S F ⋅ d S where F=〈2x^3,2y^3,4z^3〉 and...
Use the Divergence Theorem to evaluate ∫ ∫ S F ⋅ d S where
F=〈2x^3,2y^3,4z^3〉 and S is the sphere x2+y2+z2=16 oriented by
the outward normal.
1A) Use surface integral to evaluate the flux of F(x,y,z) =<x^3,y^3,z^3> across the cylinder x^2+y^2=1, 0<=z<=2...
1A) Use surface integral to evaluate the flux
of
F(x,y,z) =<x^3,y^3,z^3>
across the cylinder x^2+y^2=1, 0<=z<=2
1B) Use the Divergence Theorem to evaluate the
flux of F(x,y,z) =<x^3,y^3,z^3>
across the cylinder x^2+y^2=1, 0<=z<=2
Verify Stokes theorem for F =(y^2 + x^2 - x^2)i + (z^2 + x^2 - y^2)j...
Verify Stokes theorem for F =(y^2 + x^2 - x^2)i + (z^2 + x^2 -
y^2)j + (x^2 + y^2 - z^2)k over the portion of the surface x^2 +
y^2 -2ax + az = 0
Verify the Divergence Theorem for the vector field F(x, y, z) = < y, x ,...
Verify the Divergence Theorem for the vector field F(x, y, z) =
< y, x , z^2 > on the region E bounded by the planes y + z =
2, z = 0 and the cylinder x^2 + y^2 = 1.
By Surface Integral:
By Triple Integral:
Given function f(x,y,z)=x^(2)+2*y^(2)+z^(2), subject to two constraints x+y+z=6 and x-2*y+z=0. find the extreme value of f(x,y,z)...
Given function f(x,y,z)=x^(2)+2*y^(2)+z^(2), subject to two
constraints x+y+z=6 and x-2*y+z=0. find the extreme value of
f(x,y,z) and determine whether it is maximum of minimum.
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