Boundary Value Problems/Vector Field and Flow

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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 $x\in 𝐑$. A vector in 3-D $v\in {𝐑}^{\mathrm{𝟑}}$ is a "3-tuple" of real numbers $v=\{v_1,v_2,v_3\}$ where $v_i\in 𝐑$ for $i=1,2,3$. Example: $v=\{1,1,-1\}$