# What is Vector Area of an object?

by kashan123999
Tags: object, vector
 P: 93 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?
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,503 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".
HW Helper
Thanks
PF Gold
P: 5,187
 Quote by 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?
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.

P: 93
What is Vector Area of an object?

 Quote by Chestermiller 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.
 P: 4,573 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.