Teorema Stokes Pertemuan 25 - 26 Matakuliah : Kalkulus II Tahun : 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. 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

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 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

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. 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

and Stoke's theorem becomes 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

Since it is in the xy-plane, then n = k and ( × F) • n = 2. 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

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

SOLUTION: Using the shortcut formula 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

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

When z = 0, then x + y = 1 or y = 1 - x. When y = 0, then x = 1 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

SOLUTION: Before we start to solve this problem, we need a fact from 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

Now apply this to • • • • Bina Nusantara University

• • Bina Nusantara University

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

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

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

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