Question

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. Find the flux of the vector field F = 2yi - z] + x?k across the surface of
the parabolic cylinder y = 8x in the first octant bounded by the planes
y = 4 and z = 6.
[Ans. : 132]​

Answer 1

Step-by-step explanation:

Parametrize your surface, say by:

r⃗ (u,v):=(u28,u,v),0≤u≤4,0≤v≤6⟹

r′u→=(u4,1,0),r′v→=(0,0,1)⟹r′y→×r′v→=∣∣∣∣∣i⃗ u40j⃗ 10k⃗ 01∣∣∣∣∣=(1,−u4,0)

Then, since n⃗ =r′y→×r′v→∥∥∥r′y→×r′v→∥∥∥ ( and observe that the norm of the vectorial product cancels in the following!) and also F(r⃗ (u,v))=(2u,−3,u464) , we get that:

∬sF⃗ ⋅n⃗ dS⃗ =∬DF⃗ (r⃗ (u,v))⋅(r′y→×r′v→)dA=∫40∫60(2u,−3,u464)⋅(1,−u4,0)dA=

=∫40∫60114udvdu=664∫40udu=334⋅16=132