Matakuliah
Tahun
: Kalkulus II
: 2008 / 2009
Teorema Stokes
Pertemuan 25 - 26
2 Sketch the region of integration, determine the order of integration,
and evaluate the integral.
Bina Nusantara University
3
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
Bina Nusantara University
4
Evaluate the improper integral
Ans : 1
Sketch the region bounded by the parabola x = y - y2 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.
Ans. 6
Bina Nusantara University
5
5 Find the volume of the wedge cut from the cylinder x 2 + y 2 = 1 by the
planes z = - y and z = 0
Ans. 2/3
6 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 )
Bina Nusantara University
6
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.
•
Bina Nusantara University
•
7
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.
Bina Nusantara University
8
EXAMPLE 1: Calculate the circulation of the field F = x2 i + 2x j + z2 k
around the curve C: the ellipse 4x2 + y2 = 4 in the xyplane,
counterclockwise when viewed from above.
SOLUTION:
Since it is in the xy-plane, then n = k and ( × F) • n = 2.
•
Bina Nusantara University
9
We are working with the ellipse 4x2 + y2 = 4 or x2 + y2/4 = 1, so I will use
the transformation x = r cos 0 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.
Bina Nusantara University
10
EXAMPLE 2: Calculate the circulation of the field F = y i + xz j + x2 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 = x2, I will find × F.
Bina Nusantara University
11
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.
•
Bina Nusantara University
12
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
•
Bina Nusantara University
13
EXAMPLE 3: Use the surface integral in Stoke's theorem to calculate
the flux 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 - r2) 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:
Bina Nusantara University
14
Now apply this to
•
Bina Nusantara University
•
•
•
15
•
•
Bina Nusantara University
16
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)
Bina Nusantara University
figure 1
17
THEOREM: If
at every point of a simply connected open
region D in space, then on any piecewise smooth closed path C in D,
Bina Nusantara University
18
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
Bina Nusantara University
19
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 + (z2 - 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.
Bina Nusantara University
20