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What is Vector Area of an object? 
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#1
Jul2013, 04:26 AM

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I read about them in Topic of FLUX,what are vector areas,how they are different from surface areas,apart from the fact that they are perpendicular to the surface area,but why is that so? and what is the physical significance of them? why they are used so?



#2
Jul2013, 09:15 AM

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The "vector area" of a portion of a surface (not every "object" has "area" much less "vector area!) is a vector whose length is equal to the area of the surface and whose direction is perpendicular to the surface.
Strictly speaking, only a portion of a plane has a "vector area" since only a plane would have a unique normal direction. But given any curved surface we can talk about the "differential vector area" a "vector" whose length is the differential of area at a given point on the surface and whose direction is that of the normal vector at that point. For example, the sphere with radius R and center at the origin can be written in parametric equations as [itex]x= Rcos(\theta)sin(\phi)[/itex], [itex]y= Rsin(\theta)sin(\phi)[/itex] and [itex]z= Rcos(\phi)[/itex]. That is the same as saying that the "position vector" or any point on surface is [tex]\vec{v}= Rcos(\theta)sin(\phi)\vec{i}+ Rsin(\theta)sin(\phi)\vec{j}+ Rcos(\phi)\vec{k}[/tex] The derivatives with respect to [itex]\theta[/itex] and [itex]\phi[/itex], [tex]\vec{v}_\theta= Rsin(\theta)sin(\phi)\vec{i}+ Rcos(\phi)sin(\phi)\vec{j}[/tex] and [tex]\vec{v}_\phi= Rcos(\theta)cos(\phi)\vec{i}+ Rsin(\theta)cos(\phi)\vec{j} Rsin(\phi)\vec{k}[/tex] are vectors lying in the tangent plane to the surface at each point. Their cross product (I'll leave it to you to calculate that) is a vector perpendicular to both and so perpendicular to the tangent plane and perpendicular to the sphere at each point. Its length is the "differential of area" for the sphere and so the vector itself is the "vector differential of area". 


#3
Jul2013, 01:47 PM

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#4
Jul2413, 03:59 AM

P: 93

What is Vector Area of an object?



#5
Jul2413, 07:50 AM

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Just like you can add infinitesimal areas to get a total area, you can add infinitesimal vectors over a region to get a total vector as a result.
Its knowing what the infinitesimal is and what you are actually adding with regard to the integral that is key in understanding the above statement. 


#6
Jul2413, 07:52 AM

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Chet 


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