Concept explainers
(a) Use Stokes’ Theorem to evaluate
(b) Graph both the plane and the cylinder with domains chosen so that you can see the curve
(c) Find parametric equations for
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Chapter 16 Solutions
Calculus (MindTap Course List)
- a. Find a parametrization for the hyperboloid of one sheet x2 + y2 - z2 = 1 in terms of the angle u associated with the circle x2 + y2 = r2 and the hyperbolic parameter u associated with the hyperbolic function r2 - z2 = 1. (Hint: cosh2 u - sinh2 u = 1.) b. Generalize the result in part (a) to the hyperboloid (x2/a2 ) + (y2/b2 ) - (z2/c2 ) = 1.arrow_forwardUse Stokes' Theorem to evaluate F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = xyi + 4zj + 6yk, C is the curve of intersection of the plane x + z = 7 and the cylinder x2 + y2 = 36.arrow_forwardconsider the curve formed by the intersection of the surface (x+z)^2+y^2 =9 and the plane x-2z = 0. (a) Find a parametrisation r of the curve of the form r(t) = x(t)i + y(t)j+z(t)k, expressed in terms of the circular functions. (b) Find all of the points on the curve where r'(t) and r"(t) are perpendicular.arrow_forward
- Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y2)i + (y + z2)j + (z + x2)k, C is the triangle with vertices (7, 0, 0), (0, 7, 0), and (0, 0, 7).arrow_forwardUse Stokes' Theorem to evaluate F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y2)i + (y + z2)j + (z + x2)k, C is the triangle with vertices (3, 0, 0), (0, 3, 0), and (0, 0, 3).arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage