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Verifying Stokes’ Theorem Verify that the line
5. F = 〈y, – x, 10〉; S is the upper half of the sphere x2 + y2 + z2 = 1 and C is the circle x2 + y2 = 1 in the xy-plane.
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Chapter 17 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
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Glencoe Math Accelerated, Student Edition
- Identify the surface by eliminating the parameters from the vector-valued function r(u,v) = 3 cosv cosui + 3 cosv sinuj + Śsinvk a. plane b. sphere c. paraboloid d. cylinder e. ellipsoid d b a e (Darrow_forwardEvaluate Curlv•ñ, where v = 2xyi + (x² – 2y)j + xzk and i is a unit vector normal to the surface shown in the figure: (i) (ii) Z 1 surface z=1 surface y=1 1 1 1arrow_forwardLet S be the portion of the plane 2x + 3y + z = 2 lying between the points (−1, 1, 1), (2, 1, −5), (2, 3, −11), and (-1, 3, -5). Find parameterizations for both the surface S and its boundary S. Be sure that their respective orientations are compatible with Stokes' theorem. from (-1, 1, 1) to (2, 1,-5) from (2, 1, 5) to (2, 3, -11) from (2, 3, -11) to (-1, 3, -5) from (-1, 3, 5) to (-1, 1, 1) boundary S₁ (t) = S₂(t) = S3(t) = S4(t) = Φ(u, v) = te [0, 1) te [1, 2) te [2, 3) te [3, 4) UE [-1, 2], VE [1, 3]arrow_forward
- 6. Use Stokes theorem to evaluate §. F·dr, where F = (-3y² , 4z, 6x) and C is the triangle in the plane z = ½ y with vertices (2,0, 0), (0, 2, 1) and (0, 0, 0) with a counterclockwise orientation looking down the positive z-axis.arrow_forwardCalculate the curl(F) and then apply Stokes' Theorem to compute the flux of curl(F) through the surface of part of the cone √x² + y2 that lies between the two planes z = 1 and z = 8 with an upward-pointing unit normal, vector using a line integral. F = (yz, -xz, z³) (Use symbolic notation and fractions where needed.) curl(F) = flux of curl(F) = [arrow_forwarda. Sketch the surface x2 - y2 + z2 = 4. b. Find a vector normal to the surface at (2, -3, 3). Add the vector to your sketch. c. Find equations for the tangent plane and normal line at (2, -3, 3).arrow_forward
- iv) Given F(x, y, z) = -yzį + 2xj + xyzk be a vector function that passes through a surface S of z+x² + y² = 6 that lies above z = 2, which is positively oriented. Sketch the surface S based on the given information, then evaluate [ſ. curl F · n dS by using Stokes' Theorem. %3Darrow_forwardUse 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 curvearrow_forwardLet F = Use Stokes' Theorem to evaluate F. dr, where C is the triangle with vertices (6,0,0), (0,6,0), and (0,0,6), oriented counterclockwise as viewed from above.arrow_forward
- (6) A vector field oriented outward is given by Suppose that the surface of the plane xt y+ 2 =1 in the first octant contained within triangle C with vertices (1,0,0) , Co,1,0) and (0,0,1) transverse in anticlockwise direction. (i) Sketch the surface of the plane. Cii) Evaluate f P. dr bu C using Stoke's theorem. buisharrow_forwardc) Use Stokes' theorem to evaluate ſF dr where F = 22i+ yJ + xk and C is the triangle with vertices (2,0, 0), (0, 2,0) and (0,0, 2) with anticlockwise direction when viewed from above.arrow_forwardPath of steepest descent Consider each of the following surfaces and the point P on the surface. a. Find the gradient of ƒ. b. Let C’ be the path of steepest descent on the surface beginning at P, and let C be the projection of C’ on the xy-plane. Find an equation of C in the xy-plane. c. Find parametric equations for the path C’ on the surface. ƒ(x, y) = 4 - x2 - 2y2 (a paraboloid); P(1, 1, 1)arrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
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