Is area a scalar or vector quantity – quora victaulic t gasket

A physical area can definitely be treated a vector because it can be oriented in different ways. If you don’t care about the direction, (like you assume you always know the orientation of a rug — flat on the floor) you can treat it as a scalar. But if, for example, you have a small loop that is a flow meter — it measures the amount of water passing through it per second — that area is clearly most conveniently treated as a vector. If you orient it so that the loop is perpendicular to the direction of flow, you get a big number — the flow. If you orient it so that the loop is parallel to the direction of flow, you get 0 since nothing passes through the loop. The amount you get at intermediate directions can easily be shown to be proportional to the sine of the angle between the normal to the loop and the direction of flow. This result is easily represented in terms of the math of vectors (a dot product).

The point to remember is that a physical area isn’t actually anything mathematical. It is what it is. We model it as a mathematical quantity when it has properties that look like that mathematical quantity. That allows us to think about a lot more characteristics of areas than just its magnitude.

That depends mostly on whether you are inside or outside of an integral. If you are inside an integral (in particular, a surface integral), then what you will be focusing on is an infinitesimal piece of your surface. In three dimensions, your local surface piece (which we will from here on assume to belong to a nice, i.e. smooth, surface) will have some sort of orientation which can be described by a normal vector. You can then construct a differential area vector whose length is equal to your infinitesimal surface’s area and whose direction is parallel to the unit normal vector. This vector then encodes both the magnitude and the direction of the surface.

If you are outside an integral, then what you are interested in is probably the total area of the surface. Because surfaces come in all different shapes, total area only really makes sense as a scalar. The reason the vector definition makes sense infinitesimally, then, is because all surfaces (at least the nice ones that we are considering) become planes if we zoom in enough and hence are all the same shape.

So when is surface orientation important? Consider a flat surface in a constant vector field (perhaps representing the flow of some fluid or of an electric field). What is the flux through the surface (i.e. how much fluid/electric field passes through the surface per unit time)? Well, that depends on how the surface is oriented. If the surface is parallel to the vector field, nothing is going to flow through it while at any other angle there will be some flux. If the surface is perpendicular to the vector field, then the flux will be maximized. If you treat your surface’s area as a vector as described previously, then the flux will be equal to the dot product between the vector field (which, because it is constant, we can treat as a single vector) and the area (which is also constant and hence can be treated as a single vector). In this example, as just mentioned, we can treat total area as a vector because the flux is constant across the surface but in general we will need to integrate over the whole surface in order to find the total flux.