# 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, ${\displaystyle {x,y,z}}$, and the dimension of time ${\displaystyle t}$. For notation, I will use ${\displaystyle x_{1}=x}$ ,${\displaystyle x_{2}=y}$ and ${\displaystyle x_{3}=z}$. This is easily scaled to ${\displaystyle n}$ - dimensions for larger spaces. So ${\displaystyle R^{n}}$ corresponds to ${\displaystyle R^{3}}$ for most of the time, or less.

### Two types of fields

#### Scalar Field

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