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#### Concept explainers

**Verifying Stokes’ Theorem** *Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S. and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation.*

**9.** **F** = 〈*y* – *z*, *z* – *x*, *x – y*〉; *S* is the cap of the sphere *x*^{2} + *y ^{2}* +

*z*= 16 above the plane

^{2}*C*is the boundary of

*S.*

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# Chapter 14 Solutions

Calculus: Early Transcendentals (2nd Edition)

# Additional Math Textbook Solutions

University Calculus: Early Transcendentals (3rd Edition)

Glencoe Math Accelerated, Student Edition

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)

Thomas' Calculus: Early Transcendentals (14th Edition)

Calculus and Its Applications (11th Edition)

- Evaluate Curlvñ , where v= 2xyi +(x² – 2y)j+ xzk and ñ is a unit vector normal to the surface shown in the figure: (i) (ii) surface z=i surface y=1
*arrow_forward*Use Stokes' Theorem to evaluate F. dr where F = (x + 7z)i + (10x + y)j + (8y − z) k_and C is the curve of intersection of the plane x + 3y + z = 18 with the coordinate planes. (Assume that C is oriented counterclockwise as viewed from above.)*arrow_forward*Check that the point (1, 1, 1) lies on the given surface. Then, viewing the surface as a level surface for a function f(x, y, z), find a vector normal to the surface and an equation for the tangent plane to the surface at (1, 1, 1). 2x2 – 2y + 4z? = 4 vector normal = tangent plane: = Z*arrow_forward* - Calculate 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_forward*The tangent plane at a point Po (f(u0.Vo) 9(40.vo).h(4o.vo)) u(uo.vo) xry (uo.vo), the cross product of the tangent vectors ru (uo.vo) and r, (uo.vo) at Po. Find an equation for the plane tangent to the surface at Po. Then find a Cartesian on a parametrized surface r(u,v) = f(u,v) i+ g(u,v) j+ h(u,v) k is the plane through P, normal to the vector equation for the surface and sketch the surface and tangent plane together. The circular cylinder r(0,z) = (4 sin (20)) i+ (8 sin 0) j+z k at the point Po (2/3,2,2) corresponding to (0,z) =*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_forward*4. Consider the vector function r(z, y) (r, y, r2 +2y"). (a) Re-write this vector function as surface function in the form f(1,y). (b) Describe and draw the shape of the surface function using contour lines and algebraic analysis as needed. Explain the contour shapes in all three orthogonal directions and explain and label all intercepts as needed. (c) Consider the contour of the surface function on the plane z= for this contour in vector form. 0. Write the general equation*arrow_forward*Vector F is mathematically defined as F = M x N, where M = p 2p² cos + 2p2 sind while N is a vector normal to the surface S. Determine F as well as the area of the plane perpendicular to F if surface S = 2xy + 3z.*arrow_forward* - Verify Stokes's Theorem for F = z²î+ x²j + y²k and S is the surface z2 = x2 + y2, y 2 0, and 0
*arrow_forward*Asap*arrow_forward*sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. Surface: x2+y2=4; z=x2 Parameter: x=2sin(t)*arrow_forward*

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- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning