01. Notation

Vector

A vector is an ordered tuple of cells which represents a orientated distance.

We write vectors as column vectors by default, that is:

${\bf v}:=\begin{pmatrix} x \\ y \\ z \end{pmatrix}$

A vector uniquely represents the line from the origin of the coordinate system to a point.

A field is a physical value which changes from place to place.

A scalar field does not discriminate the place or direction, it acts the same on any point in space.

For each point in space, a different vector value is possible.

$÷{d}{dx}F=\lim_{h\rightarrow 0} ÷{F(x+h)-F(x)}{h}$

$(∇⃗F)_i:=÷{d}{dx_i}F=\lim_{h\rightarrow 0} ÷{F(x_i+h)-F(x_i)}{h}$

$δ_{i,j}:={1,i=j; 0,i≠j}$

TODO path integral

TODO area integral

$∫_V ∇⃗∙D⃗⋅dV=∮_{δV} D⃗∙n⃗⋅dS$

$∫_{D} ((÷{∂}{∂x}g)-(÷{∂}{∂y}f))⋅dV=∮_{∂D} (f⋅dx+g⋅dy)$

$∫_Q (∇⃗⨯F⃗)∙n⃗_2⋅dS=∮_{∂Q} F⃗∙n⃗⋅ds$

(where n is the unit normal).

Author: Danny (remove the ".nospam" to send)

Last modification on: Thu, 09 May 2013 .