University of Essex Gauss Theorem Two Questions

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I need help with the following two questions, with solutions and workings please

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10. Gauss’ theorem states that
SfF.ds – SSS..
VFdV.
For the vector field given by
F = xzi + 5y?j – z?k,
(a) Calculate div F (equivalently V. F).
[3 marks]
(b) Verify Gauss’ theorem for the vector field F and the cuboid
05×51, 0 Sys 1, 05751, given that
9
F. ds – 2
2
SS
where S is the surface of the cuboid.
[3 marks]
(c) Calculate the contributions to
SS.F. ds
through the three surfaces of the cuboid with outward pointing normals
i, j and k respectively, hence deduce the contributions through the remaining
surfaces.
[4 marks]
9. (a) Calculate 7 x F for the vector field
F = 2z?i + yzj + xyzk.
[4 marks]
(b) Hence calculate
SS (vxF). d
for the vector field F = 2z2i + yzj + xyzk and the unit square
05xs1, y = 0 and 0 szs1 (i.e. ds = dxdzj).
[6 marks]

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Tags:
vector field

unit square

side surface

surface of the cuboid

outward pointing normals

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