Concept explainers
Motion of a Liquid In Exercises 85 and 86, the motion of a liquid in a cylindrical container of radius 4 is described by the velocity
. Find
where S is the upper surface of the cylindrical container.
Want to see the full answer?
Check out a sample textbook solutionChapter 15 Solutions
Calculus (MindTap Course List)
- Using Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)arrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (6, 0), and (0, 6) a) 216 b) 72 c) 0 d) 36arrow_forwardUsing Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. IF = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (8, 0), and (0, 7) a) 112 b) 392 c) 0 d) 56arrow_forward
- Use Stokes’ Theorem to evaluate ∫ F*dr where C is oriented counter-clockwise as viewed from above. F(x,y,z) = yi-zj+x2k C is the triangle with vertices (1,0,0), (0,1,0), and (0,0,1) Note: The triangle is a portion of the plane x+y+z=1arrow_forwardUsing Stokes' theorem, solve the line integral of G(x, y, z) - (1, x + yz, xy-√z) around the boundary of surface S, which is given by the piece of the plane 3x + 2y + z = 1 where x, y, and z all ≥ 0.arrow_forwardA. State the Fundamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = x^2 + 2y^2 + 3z^2 and F = grad f. Find the line integral of F along the line C with parametric equations x = 1 + t, y = 1 + 2t, z = 1 + 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forward
- y = ex, y = 0, x = 0, x = 3 Find the exact coordinates of the centroid. (x, y) =arrow_forwardSurface integral of a vector field? Let T be the upper surface of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 4. Calculate the integral of the image below, where S is the face of T that is in the xy plane.arrow_forwardFlux of a vector field? Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.arrow_forward
- A. State the F undamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = xy + 2yz + 3zx and F = grad f. Find the line integral of F along the line C with parametric equations x = t, y = t, z = 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forwardUse Stokes's Theorem to evaluate C F · dr. C is oriented counterclockwise as viewed from above. F(x, y, z) = (cos(y) + y cos(x))i + (sin(x) − x sin(y))j + xyzk S: portion of z = y2 over the square in the xy-plane with vertices (0, 0), (a, 0), (a, a), and (0, a)arrow_forwardUsing cylindrical coordinates evaluate ʃ ʃ ʃE sqrt ((x2 + y2)) dV where E is the solid bounded by the circular paraboloid z = 1 – 16 (x2 + y2) and the xy-plane.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning