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.

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: $$v = \{1,1,-1\} $$