Concept explainers
Work In Exercises 25-28, use Green’s Theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C.
Want to see the full answer?
Check out a sample textbook solutionChapter 15 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
- Using 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_forwarda) Find the work done by the force field F on a particle that moves along the curve C. F(x, y) = (x2 + xy)i + (y – x2 y)j C : x = t, y = 1/t (1 ≤ t ≤ 3)arrow_forwardGive an example of a rigid motion T in C n, T(0) = 0, which is not a linear transformation.arrow_forward
- Prove that the vector field F(x, y, z) = (x2 + yz)i − 2y(x + z)j + (xy + z2)k isincompressible, and find its vector potential function.arrow_forwardUse 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_forwardCalculate the flux of the vector function F(x, y, z) through the entire surface of the depicted cube. Tip: use Gauss's Theoremarrow_forward
- Use Green's theorem to calculate the work done by force F on a particle that is moving counterclockwise around closed path C. F(x, y)=(x^3/2 - 8y)i + (5x + 8 sqrt(y))j, C: boundary of a triangle with vertices (0,0), (5,0), and (0,5)arrow_forwardDetermine the total work, by Green's theorem, done by the force field F(x, y) =< 2x - y2, x² y>, on a particle following the path described in the graph below.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
- Using Green's Theorem on this vector field problem, compute a) the circulation on the boundary of R in terms of a and b, and b) the outward flux across the boundary of R in terms of a and b.arrow_forwardConsider the vector field ?(?,?,?)=(?+?)?+(2?+?)?+(2?+?)? F ( x , y , z ) = ( z + y ) i + ( 2 z + x ) j + ( 2 y + x ) k . a) Find a function ? f such that ?=∇? F = ∇ f and ?(0,0,0)=0 f ( 0 , 0 , 0 ) = 0 . ?(?,?,?)= f ( x , y , z ) = b) Suppose C is any curve from (0,0,0) ( 0 , 0 , 0 ) to (1,1,1). ( 1 , 1 , 1 ) . Use part a) to compute the line integral ∫??⋅?? ∫ C F ⋅ d r .arrow_forwardCompute the curl of the vector fields F(x, y) = cos(x + y)i + sin(x − y)jG(x, y,z) = (2y + 3z)i + 2xy 2j + (z x − y)karrow_forward
- 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