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Gauss' theorem states that [[F. ds = [[[ V. FdV. (a) Calculate div F (equivalently V-F) for the vector field F = 4xyi − y²j + zzk. (b) Verify Gauss' theorem for the vector field F in part (a) and the cuboid given by 0≤x≤2, 0≤ y ≤3, 0≤ ≤4, where S is the surface of the cuboid and F.ds = 96. (c) Given that the contributions to FdS from the surfaces with outward pointing normals -i, -j,-k are equal to zero, calculate the individual contributions from the other three surfaces.

Gauss' theorem states that [[F. ds = [[[ V. FdV. (a) Calculate div F (equivalently V-F) for the vector field F = 4xyi − y²j + zzk. (b) Verify Gauss' theorem for the vector field F in part (a) and the cuboid given by 0≤x≤2, 0≤ y ≤3, 0≤ ≤4, where S is the surface of the cuboid and F.ds = 96. (c) Given that the contributions to FdS from the surfaces with outward pointing normals -i, -j,-k are equal to zero, calculate the individual contributions from the other three surfaces.

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Vectors In Two And Three Dimensions
Section9.FOM: Focus On Modeling: Vectors Fields
Problem 11P
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Gauss' theorem states that JF.ds= V.FdV. (a) Calculate div F (equivalently V-F) for the vector field F = 4xyi − y²j + zzk. (b) Verify Gauss' theorem for the vector field F in part (a) and the cuboid given by 0≤x≤2, 0≤ y ≤3, 0≤x≤4, where S is the surface of the cuboid and F.ds = 96. (c) Given that the contributions to FdS from the surfaces with outward pointing normals -i, -j, -k are equal to zero, calculate the individual contributions from the other three surfaces.
Transcribed Image Text:Gauss' theorem states that JF.ds= V.FdV. (a) Calculate div F (equivalently V-F) for the vector field F = 4xyi − y²j + zzk. (b) Verify Gauss' theorem for the vector field F in part (a) and the cuboid given by 0≤x≤2, 0≤ y ≤3, 0≤x≤4, where S is the surface of the cuboid and F.ds = 96. (c) Given that the contributions to FdS from the surfaces with outward pointing normals -i, -j, -k are equal to zero, calculate the individual contributions from the other three surfaces.
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