Evaluate the surface integral SFdS because that the givcarolannpeacock.com vector field F and the oricarolannpeacock.comted surface ar S. In other words, discover the flux that F across S. For closed surfaces, usage the hopeful outward oricarolannpeacock.comtation. F(x,y,z)=zexyi3zexyj+xyk, S is the parallel of practice 5 through upward oricarolannpeacock.comtation.

You are watching: F(x, y, z) = zexyi − 3zexyj + xyk,

come determineTo evaluate:

The surface integral ∬S□F·dS for the provided vector field F and the oricarolannpeacock.comted surface S or find the flux the F across S by using optimistic (outward) oricarolannpeacock.comtation for closed surfaces.




1) Concept:

If F is a continuous vector field characterized on one oricarolannpeacock.comted surface ar S which is a vector duty ru, v=xu, vi+yu, vj+zu, vk climate the surface ar integral of F over S that is flux the F throughout S is


where D is the parameter domain

If k componcarolannpeacock.comt of ru×rv is positive thcarolannpeacock.com it gives upward oricarolannpeacock.comtation.

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2) Givcarolannpeacock.com:

Fx, y, z=zexyi-3zexyj+xy k, S is the parallelogram with parametric equations x=u+v, y=u-v, z=1+2u+v, 0≤u≤2, 0≤v≤1 through upward oricarolannpeacock.comtation

3) Calculations:

Givcarolannpeacock.com vector field F is Fx, y, z=zexyi-3zexyj+xy k

The oricarolannpeacock.comted surface ar S is the parallelogram through parametric equations x=u+v, y=u-v, z=1+2u+v, 0≤u≤2, 0≤v≤1 with upward oricarolannpeacock.comtation

That is ru, v=u+vi+u-vj+1+2u+vk






Fru, v=1+2u+veu+vu-vi-31+2u+veu+vu-vj+u+vu-vk


Use hopeful (outward oricarolannpeacock.comtation) because that closed surface.

Here, k- ingredicarolannpeacock.comt of ru×rv is negative, therefore for upward oricarolannpeacock.comtation use -(ru×rv)





By using concept;

The surface ar integral that F carolannpeacock.comd S that is flux the F throughout S is


D:{(u, v)|0≤u≤2, 0≤v≤1}

=∫01∫022u2-v2du dv

Integrate through respect come u, it gives


Using boundaries of integration,




Integrate v respect come v, the gives


Using limits of integration,





Thus, the surface ar integral ∬S□F·dS=4


The surface integral ∬S□F·dS because that the givcarolannpeacock.com vector ar F and the oricarolannpeacock.comted surface S is 4