TerraNotes

Edo Nyland
Department of Physics
University of Alberta
Edmonton T6G 2A2
Canada
edo@phys.ualberta.ca

\(g\&⇀[
        r\&⇀; \(r\_1\)\&⇀] = \(F\&⇀\/m = \(\(-\ G\)\ \(m\_1\) \(\(r\_1\)\&⇀ - r\&⇀\)\/\(\(\ \)\(\(|\)\(\(r\_1\)\&⇀ - r\&⇀\)\( | \^3\)\)\) = \ ∇\([V[r\&⇀, \(r\_1\)\&⇀] = 
                G\ m\_1\ 1\/\(\(|\)\(\(r\_1\)\&⇀ - r\&⇀\)\(|\)\)]\)\)\)\)

(∂_z) (∫_(z + a)^(z + b))g[
            w] \[DifferentialD]w

test = ∫∫(1/(Sqrt[(u^2) + (v^2) + (w^2)])) \[DifferentialD]u \[DifferentialD]v //. {ArcTan[b_/a_] :> 
              ArcTan[a, b], 
            (1/2) I w Log[a_] - 
                (1/2) I w Log[
                    b_] :>  I w Log[(Sqrt[FullSimplify])[a/b]]};
f = Compile[{u, v, w}, Evaluate[test]];

\(g\_z = \(G\ ρ\ \(∫\_S\(∫\_\(-∞\)\%∞\( 
                    w\/\((u\^2 + v\^2 + w\^2)\)\^\(3/2\)\) du dv dw\)\) = 
        2\ G\ ρ \(∫\_S\( 
                  w\/\((v\^2 + w\^2)\)\^\(3/2\)\) ds\)\)\)
     =(∫_C) ArcTan[
              (v/w)] dw = (∫_C) Arg[
              v + i w] dw

\(2\ G\ ρ \(∑\+\(k = 1\)\%n\( b\_k\/\(1 + a\_k\^2\)\) \(1\/2\) 
            Log[\(x\_\(k + 1\)\^2 + z\_\(k + 1\)\^2\)\/\(x\_k\^2 + z\_k\^2\)]\) + \(\(\(a\_k\) b\_k\)\/\(1 + a\_k\^2\)\) \((ArcTan[
              x\_\(k + 1\)\/z\_\(k + 1\)] - 
            ArcTan[x\_k\/z\_k])\)\)
\(a\_k = \(x\_\(k + 1\) - x\_k\)\/\(z\_\(k + 1\) - z\_k\), \ 
    b\_k = \(x\_k\ z\_\(k + 1\) - \(x\_\(k + 1\)\) z\_k\)\/\(z\_\(k + 1\) - z\_k\)\)

\(g\&⇀[r\&⇀] = 
        G\ ρ \(∫\_V\(\((r\&⇀ - 
                      s\&⇀)\)\/\(\(|\)\(r\&⇀ - 
                      s\&⇀\)\( | \^3\)\)\) dv\_s\)\)=G ρ (Subscript[∇, r])(∫_V) 
                  (1/(|(Overscript[r, ⇀]) - 
                      (Overscript[s, ⇀])|)) d(Subscript[v, s])

\(TraditionalForm\`H\&⇀[
          r\&⇀] = \(∇\_r\(∫\_V M\&⇀·∇\_r\( 
                        1\/\(\(|\)\(r\&⇀ - 
                            s\&⇀\)\(|\)\)\) dv\_s\) = \(\(\(|\)\(M\&⇀\)\(|\)\(∂\_M\ \(∇\_r\(∫\_V\( 1\/\(\(|\)\(r\&⇀ - 
                              s\&⇀\)\(|\)\)\) dv\_s\)\)\)\) = \(\(\(|\)\(M\&⇀\)\(|\)\)\/\(G\ ρ\)\) ∂\_M\ g\&⇀[r\&⇀]\)\)\)

\(∂\_M\ □\ \ is\ the\ directional\ derivative\ along\ the\ direction\ of\ M\&⇀\)
\(∇\_r\ 
        is\ a\ gradient\ with\ respect\ to\ the\ field\ \((r\&⇀)\)\ \(\(coordinates\)\(.\)\)\)

One problem with this map is that the station labels are illegible but that can be cured by use of the zoom and navigate capability in the ImageRegion Palette