1. Verify the divergence theorem for: (a) = 4ri - 2y2j+z2k in the cylindrical region r² + y² = 4 bounded below by the ry plane and above by the plane z 3. = (b) = zi− y) + zk in the spherical region r² + y² + 2² = 1. (c) F = zi - 2y² + rk over the region bounded by the paraboloid z = 4-2-y² and the zy plane. ==
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- Show that R(u, v) = (5u + 2, u − v, 9u + v) parametrizes the plane 2x − y − z = 4. Then: (a) Calculate eu, ev, and Ñ(u, v). (b) Find the area of S = R(D), where D = (u, v): 0 ≤u ≤ 7,0 ≤ v ≤ 5. (c) Express f(x, y, z) = yz in terms of u and v and evaluate [[ f(x, y, z) ds. (a) eu = , ev = (b) Area(S) = (c) ff f(x, y, z) ds = = ‚ Ñ(u, v) =Use Stokes' Theorem to evaluate of intersection of the plane x + 3y +z = 12 with the coordinate planes. (Assume that C is oriented counterclockwise as viewed from above.) F. dr where F = (x + 6z)i + (8x + y)j + (10y −z) k_and C is the curveCalculate F · dr cylinder x2 + where F = xzi + 2zj – xyk and C is the intersection of the plane y = z+ 2 with y? = 4
- Consider the surface S consisting of the portion of the graph of the function z = x2 + y? above the annulus 64The intersection of the ellipsoid x2 +4y2 + z2 = 4 with the plane z= 3x gives an ellipse C. A) Write a parametrization of this ellipse r(t) for 0 sts2r. r(t)= ( cos t,sint,cos t). B) In order to do the line integral of the function 4 f(x,y,z) = 16 + 4 over this ellipse one must find the integral of (enter the appropriate number in each term which appears in the integrand) S. J, ds = JOsin 2t + cos ?) dtLet F = -9zi+ (xe#z– 2xe**)}+ 12 k. Find f, F·dĀ, and let S be the portion of the plane 2x + 3z = 6 that lies in the first octant such that 0 < y< 4 (see figure to the right), oriented upward. Z Explain why the formula F · A cannot be used to find the flux of F through the surface S. Please be specific and use a complete sentence.The intersection of the ellipsoid x² + 4y? + z? = 36 with the plane Z = 3x gives an ellipse C. A) Write a parametrization of this ellipse r(t) for 0st<2n. r(t)= (cost,sint, cos t). B) In order to do the line integral of the function 4 f(x.у,2) 3D 16 + 36 36 over this ellipse one must find the integral of (enter the appropriate number in each term which appears in the integrand) 2n So |sin2t + cos 2t) dt f ds =(a) Find a vector parametric equation for the ellipse that lies on the plane 2y - 5x + z = 2 and inside the cylinder x² + y² = 16. F(u, v) = for 0 ≤ u≤ 4 and 0 ≤ v ≤ 2. (b) dA = Tux Tv = = ||A|| = ||₁|| = (d) Set up and evaluate a double integral for the surface area of the ellipse. (c) dA= Surface area =Find both parametric and rectangular representations for the plane tangent to r(u,v)=u2i+ucos(v)j+usin(v)kr(u,v)=u2i+ucos(v)j+usin(v)k at the point P(4,−2,0)P(4,−2,0).One possible parametric representation has the form⟨4−4u⟨4−4u , , 4v⟩4v⟩(Note that parametric representations are not unique. If your first and third components look different than the ones presented here, you will need to adjust your parameters so that they do match, and then the other components should match the ones expected here as well.)The equation for this plane in rectangular coordinates has the form x+x+ y+y+ z+z+ =0Suppose f(x, y) satisfies the basic existence and uniqueness theorem in some rectangular region Rof the- xy- plane . Explain why two distinct solutions of the DE y' = f (x, y) cannot intersect or be tangent to each other at a point (x,, yo) eR.2. Verify Stokes's Theorem for F(x,y, z) = y i+ zj+xk and the plane x + y + z = 2 in the first octant, oriented upward. a) First, find the sum of the normal component component of the curl F over the surface. b) Then, find the sum of the tangential component of F along the bounding curve of the surface.Consider the surface S consisting of the portion of the graph of the function z = x? + y above the annulus 4SEE MORE QUESTIONSRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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