:PROPERTIES: :ID: 68ccac6b-4af9-4512-a776-57c44086227c :mtime: 20231021010311 20231019011446 :ctime: 20231019011444 :END: #+title: vector area #+filetags: :public:project: * Definition of Vector Area *vector area* (also called the *oriented area*) refers to the idea of tracking both the area and the orientation of an oriented surface. The reason we call it "vector" area is because we can use [[id:5864974d-0edf-4757-9b1f-31b159c9aa7a][Vectors]] to keep track of the oriented area. We use $\vec{S} = A\hat{n}$ to track the vector area. The magnitude of $\vec{S}$ is equal to the area of the surface, and the direciton of $\vec{S}$ is pointing in the direction of the normal vector to the oriented surface. * Outward Normal convention When talking about the vector area of a closed surface, it is a convention to use the outward-facing normal vector. * Closed body simplification For any closed surface, regardless of the shape, the sum of all the vector areas must be $0$. ** Closing shape trick If you are asked to find the vector area of an open surface, you can use a trick, where you add an imaginary surface to close the open surface, and then subtract. * Components of vector area \[\vec{S} = (A\hat{n}_{x}, A\hat{n}_{y}, A\hat{n_{y}})\] The components of the vector area are its projections onto the planes $x= 0 , y=0, z=0$, respectively. The area is independent of the orientation, BUT the components of the vector area are dependent on orientation. * Equations about the vector area \[\vec{S} = A\hat{n}\] \[\vec{S} = (A\hat{n}_{x}, A\hat{n}_{y}, A\hat{n_{y}})\] \[\vec{S} = A(\cos\theta_{x},\cos\theta_{y},\cos\theta_{z})\]