What is Vector Area of an object?

[kashan123999]

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?

 

[HallsofIvy]

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
$x= Rcos(\theta)sin(\phi)$, $y= Rsin(\theta)sin(\phi)$ and $z= Rcos(\phi)$.
That is the same as saying that the "position vector" or any point on surface is
$$\vec{v}= Rcos(\theta)sin(\phi)\vec{i}+ Rsin(\theta)sin(\phi)\vec{j}+ Rcos(\phi)\vec{k}$$

The derivatives with respect to $\theta$ and $\phi$,
$$\vec{v}_\theta= -Rsin(\theta)sin(\phi)\vec{i}+ Rcos(\phi)sin(\phi)\vec{j}$$
and
$$\vec{v}_\phi= Rcos(\theta)cos(\phi)\vec{i}+ Rsin(\theta)cos(\phi)\vec{j}- Rsin(\phi)\vec{k}$$
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".

 

[Chestermiller]

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?

If you have a 2D curved surface S, you can focus on a differential element of area within the surface dA. The differential vector area associated with this differential element of area is defined as dA=ndA , where n is a unit normal to the surface. Suppose you have a fluid with velocity v flowing at the surface. If the fluid is not flowing normal to the surface, then the component of velocity tangent to the surface does not result in any flow through the surface. Only the component of velocity perpendicular to the surface results in fluid flow through the surface. The volumetric flow rate of fluid through the differential area element dA is equal to the velocity vector v dotted with the normal to the surface n times the differential area dA. But this is the same as the velocity vector v dotted with the differential vector area dA.

 

[kashan123999]

If you have a 2D curved surface S, you can focus on a differential element of area within the surface dA. The differential vector area associated with this differential element of area is defined as dA=ndA , where n is a unit normal to the surface. Suppose you have a fluid with velocity v flowing at the surface. If the fluid is not flowing normal to the surface, then the component of velocity tangent to the surface does not result in any flow through the surface. Only the component of velocity perpendicular to the surface results in fluid flow through the surface. The volumetric flow rate of fluid through the differential area element dA is equal to the velocity vector v dotted with the normal to the surface n times the differential area dA. But this is the same as the velocity vector v dotted with the differential vector area dA.

please can you explain it in lay-man's terms please :(

 

[chiro]

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.

 

[Chestermiller]

please can you explain it in lay-man's terms please :(

Imagine that it is raining out, and that there is no wind, so the rain is falling straight down. You have left three identical windows open, one on a vertical wall, one on a flat roof, and one on a slanted roof. The normal to the window on the vertical wall is also perpendicular to the direction the rain is falling, and no rain comes in that window. The normal to the window on the roof is parallel to the direction the rain is falling, and a maximum amount of rain comes in that window. The normal to the window on the slanted roof is at an angle to the direction that the rain is falling (neither parallel nor perpendicular), and an in-between amount of rain come in that window.

Chet