Boundary Value Problems/Vector Field and Flow

Return Boundary Value Problems

Vector Analysis: Basics

We are interested in operations on sets in a multidimensional space. The focus is on physical problems so think about three spatial dimensions, ${x,y,z}$, and the dimension of time $t$. For notation, I will use $x_{1}=x$ ,$x_{2}=y$ and $x_{3}=z$. This is easily scaled to $n$ - dimensions for larger spaces. So $R^{n}$ corresponds to $R^{3}$ for most of the time, or less.

Two types of fields

Scalar Field

A scalar x is a real number, this is denoted as $x\in \mathbf {R}$. A vector in 3-D $v\in \mathbf {R^{3}}$ is a "3-tuple" of real numbers $v=\{v_{1},v_{2},v_{3}\}$ where $v_{i}\in \mathbf {R}$ for $i=1,2,3$. Example: $v=\{1,1,-1\}$