Let S1 be the part of the paraboloid z = 1 − x ^2 − y ^2 that lies above the plane z = 0. Let S2 be the part of the cone z = √ x ^2 + y ^2 + 2(sqrt till y^2) that lies inside of the cylinder x ^2 + y^ 2 = 1. Let S3 be the part of the cylinder x ^2 + y ^2 = 1 that lies between these surfaces. If S...
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rough the surface 4. o pm) What is the value of the flux of the vector field F(x,y)j+z ioriented with upward- pointing normal vector? (A) 0 (B) 2n/3 (C) π (D) 4T/3 (E) 2π Use Stokes, Theorem to evaluateⅡcurl F.dS, where F(x, y, z)-(x2 sin Theorem to evaluate Jceun F'.asS , where Fl.e)(', ») and 5. (5pts.) F,y, sin z, y', xy) and s is the part of the paraboloid : -...
3. Sketch S and compute where S is the part of the cone z-Vx+y* between z-1 andz -3, oriented by the unit normal with negative z-component. S is the oriented surface given by the parametrization ф(II,'')-(11+1, 112-r ,uv) and (11, v) E [0.1] x [0.1] S is upper unit hemisphere, oriented by the unit normal pointing away from the origin.
3. Sketch S and compute where S is the part of the cone z-Vx+y* between z-1 andz -3, oriented by...
1. Let F(x,y,z) =< 32, 5x, – 2y >. Use Stokes's Theorem to evaluate the integral Scurl F.ds, where S is the part of the paraboloid z = x² + y2 that lies below the plane z = 4 with upward- pointing normal vector.
Verify Stokes, Theorem for the surface S that is the paraboloid given by z = 6-x2-y2 that lies above the plane z 2 (oriented upward) and the vector field F(x, y, z)2yzi+yj+3xk.
Verify Stokes, Theorem for the surface S that is the paraboloid given by z = 6-x2-y2 that lies above the plane z 2 (oriented upward) and the vector field F(x, y, z)2yzi+yj+3xk.
5. Calculate the surface area of the portion of the sphere x2+y2+12-4 between the planes z-1 and z ะไ 6. Evaluate (xyz) dS, where S is the portion of the plane 2x+2y+z-2 that lies in the first octant. 7. Evaluate F. ds. a) F = yli + xzj-k through the cone z = VF+ア0s z 4 with normal pointing away from the z-axis. b) F-yi+xj+ek where S is the portion of the cylinder+y9, 0szs3, 0s r and O s y...
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your own personal paraboloid to investigate, let T be the three-dimensional solid region bounded y2 and above by the plane z 5y + 6 below by the paraboloid zx2+ Find the volume V of the solid oblique paraboloid T. Sketch a picture of T. Can you see that T is symmetric with respect to the yz-plane? Describe the region R in the yg plane that is the vertical projection of T. This plane region will...
(1 point) Let S be the part of the plane z 4 y which lies in the first octant, oriented upward. Evaluate the flux integral of the vector field F 2i + j + 3k across the surface S (with N being the unit upward vector normal to the plane). B.I 48 C. I 72 E. 1 24
(1 point) Let S be the part of the plane z 4 y which lies in the first octant, oriented upward. Evaluate...
2. Evaluate 1,(1,0, 2) . ds, where s is the cone z = VE4y2 with 0 < z < 2, Upward 1,0,2) ds, where S is the pointing normal. 3. Use a surface integral to find the area of the region of the plane z2y +3 with
2. Evaluate 1,(1,0, 2) . ds, where s is the cone z = VE4y2 with 0
Let S be part of the paraboloid \(z=x^{2}+y^{2}\) that lies under the plane \(z=4\). Evaluate the surface integral \(\iint_{S} z d S\).