PERTEMUAN 25-26
Teorema Stokes
1. Sketch the region of integration
and write an equivalent double
integral with the order of
integration reversed.
Ans
2.
2 Sketch the region of integration,
determine the order of integration,
and evaluate the integral.
Ans : ( e8 – 1 ) /4
3 Find the volume of the solid whose base is
the region in the xy-plane that is bounded
by the parabola y = 4 - x 2 and the line y =
3x, while the top of the solid is bounded by
the plane z = x + 4.
Ans : 625 / 12
Evaluate the improper integral .
Ans :
1
Sketch the region bounded by the parabola x = y - y 2 and the line y = -x. Then
find the region's area as an iterated double integral. Ans . 4/3
4
Write six different iterated triple integrals for the volume of the tetrahedron cut from
the first octant by the plane 4x + 2y + 6z = 12. Evaluate one of the integrals.
Ans.
5
6
2
2
Ans. 6
Find the volume of the wedge cut from the cylinder x + y = 1 by the planes z = - y
and z = 0
Ans. 2/3
Find the volume of the region in the first octant bounded by the coordinate planes
and the surface z = 4 - x 2 - y.
ans
( 128 / 15 )
STOKE'S THEOREM
Stoke's theorem states that, under conditions normally met in practice, the circulation
of a vector field around the boundary of an oriented surface in space in the directions
counterclockwise with respect to the surface's unit normal vector field n equals the
integral of the normal component of the curl of the field over the surface.
STOKE'S THEOREM
The circulation of F = M i + N j + P k around the boundary of C of an oriented
surface S in the direction counterclockwise with respect to the surface's unit normal
vector n equals the integral of ∇ × F • n over S.
NOTE:
If two different oriented surfaces S1 and S2 have the same
boundary D, then their curl integrals are equal:
NOTE:
If C is a curve in the xy-plane, oriented counterclockwise, and R is
the region in the xy-plane bounded by C, then d = dx dy an
and Stoke's theorem becomes
Notice that this is the circulation-curl form of Green's theorem.
EXAMPLE 1: Calculate the circulation of the field F = x 2 i + 2x j + z 2 k
around the curve C: the ellipse 4x 2 + y 2 = 4 in the xyplane, counterclockwise when viewed from above.
SOLUTION:
Since it is in the xy-plane, then n = k and (∇ × F) • n = 2.
We are working with the ellipse 4x 2 + y 2 = 4 or x 2 + y 2/4 = 1, so I will use
the transformation x = r cos θ and y = 2r sin θ to transform this ellipse into a
circle. I will also have to use the Jacobian to find the integrating factor for
this integral.
EXAMPLE 2:
Calculate the circulation of the field F = y i + xz j + x 2 k
around the curve C: the boundary of the triangle cut from
the plane x + y + z = 1 by the first octant,
counterclockwise when viewed from above.
SOLUTION: Using the shortcut formula
where M = y, N = xz, and P = x 2, I will find ∇ × F.
∇ × Φ = (0 - x) i + (0 - 2x) j + (z - 1) k = -x i -2x j + (z - 1) k
The triangle that we are looking at from above is in the plane x + y + z = 1,
and the vector perpendicular to the plane is p = i + j + k.
Let f = x + y + z - 1, and since the shadow is in the xy-plane, let p = k.
When z = 0, then x + y = 1 or y = 1 - x. When y = 0, then x = 1. Finally, we
have to get rid of the z in the integrand, so solve x + y + z = 1 for z. z = 1 - xy
EXAMPLE Use the surface integral in Stoke's theorem to calculate the flux
3:
of the curl of the field F = 2z i + 3x j + 5y k across the surface r
(r, θ ) = (r cos θ ) i + (r sin θ ) j + (4 - r 2) k, 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π
in the direction of the outward unit normal n.
SOLUTION: Before we start to solve this problem, we need a fact from
integration of parametric surfaces, and here is the fact.
FACT:
Now apply this to ∇ × F • n dσ .
STOKE'S THEOREM FOR SURFACES WITH HOLES
DEFINITION:
A region D is simply connected if every closed path in D can be
contracted to a point in D without leaving D. (See figure 1)
figure 1
THEOREM:
If ∇ × F = 0 at every point of a simply connected open region
D in space, then on any piecewise smooth closed path C in D,
Soal soal
Divergence
Use the Divergence theorem to evaluate
Bila F = ( x2z , – y , xyz )
dan S dibatasi oleh kubus : 0 < x < a , 0 < y < a , 0 < z < a
Stokes Theorem
Verify Stokes Theorem where
F = ( z – y , x – z, x- y ) dan S : z = 4 – x2 – y2 , 0 < z
Use the surface integral in Stoke's theorem to calculate the flux of the curl of
the field F = 2y i + (5 - 2x) j + (z 2 - 2) k across the surface r (ϕ , θ ) = (2sin ϕ
cos θ ) i + (2sin ϕ sin θ ) j + (2cos ϕ ) k, 0 ≤ ϕ ≤ π /2, 0 ≤ θ ≤ 2π in the
direction of the outward unit normal n.