Concept explainers
Stokes’ Theorem for evaluating line
13. F = 〈x2 – z2, y, 2xz〉; C is the boundary of the plane z = 4 – x – y in the first octant.
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- Set-up the integral being asked in the problem. No need to evaluate. Show all solutions.arrow_forwardEvaluate the line integral PF • dr by evaluating the surface с integral in Stokes' Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation when viewed from above. F = (-3y, -z,x) C is the circle x² + y² = 26 in the plane z = 0.arrow_forwardCal 3arrow_forward
- Evaluate the line integral (ci+ 3xyj – (x + z)k) · dr where C is the parametric curve r(t) = (1 – t)i + (4 + t)j + (2 – t)k, 0arrow_forwardLet F =. Compute the flux of curl(F) through the surface z = 1- x² - y² for x² + y² ≤ 11 oriented with an upward-pointing normal. Flux = help (fractions) (Use symbolic notation and fractions where needed.) Hint: Stokes' Theorem shows a direct computation can be done in an alternative fashion.arrow_forwardEvaluate the line integral by stokes theorem where F(vector)=[ 0, xyz ,0] , C is the boundary of a triangle with vertices (1,0,0),(0,1,0),(0,0,1).arrow_forwardStokes’ Theorem on closed surfaces Prove that if F satisfies theconditions of Stokes’ Theorem, then ∫∫S (∇ x F) ⋅ n dS = 0,where S is a smooth surface that encloses a region.arrow_forward(c) Verify the line integral and surface integral by relating them to the Stokes' Theorem where C is the circle x² + y² 1 on xy-plane with a counterclockwise orientation looking down the positive z-axis. = √ x²ydx + xdyarrow_forwardChannel flow The flow in a long shallow channel is modeled by the velocity field F = ⟨0, 1 - x2⟩, where R = {(x, y): | x | ≤ 1 and | y | < 5}.a. Sketch R and several streamlines of F.b. Evaluate the curl of F on the lines x = 0, x = 1/4, x = 1/2, and x = 1.c. Compute the circulation on the boundary of the region R.d. How do you explain the fact that the curl of F is nonzero atpoints of R, but the circulation is zero?arrow_forwardUse Stokes' theorem to calculate f, F.ds, where F = (z,x, y) and C is the boundary of a triangle with vertices (0,0,1), (3,0, -2) and (0,1,2).arrow_forwardVerify the formula fF-T ds = [ (curl F) -n ds in Stokes' Theorem by evaluating the line integral and the surface integral. Assume that the surface has an upward orientation. F(x, y, z) = 4x i+ 4y j+9z k Va? - -. where o is the upper hemisphere z = fF.T ds = | (curl F) -n ds = |arrow_forwardDevarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning