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Short magenta segment represents \(\Delta z\). Long green segment represents \(F'(z)\).
Multiply these together (arcs top right) to get the blue segment. $$\Delta F \approx F'(z) \Delta z$$ As \(t\) changes, F(z) moves in the direction of the blue segment \(\Delta F\). $$F(z+\Delta z) \approx f(z)+F'(z) \Delta z$$ As \(z\) follows \(\Delta z\) you approximate the parametric curve \(z_t=\Phi(t)\). As \(F(z)\) follows \(\Delta F\), you approximate the parametric curve \(F(\Phi(t))\) [Turn on trace to see this curve form]. |
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The final z point at \(t=1\) is \(z_{1}=\Phi(1)\)
We can find this end point by adding up all the \(\Delta z\)s. $$z_1\approx z_0+\sum\Delta z \approx z_0+\sum \Phi'(t) \Delta t $$ Taking the limit... $$z_1 = z_0+\int_0^1 dz = z_0+\int_0^1 \Phi'(t)dt $$ We can find the final \(F(z_1)\) by adding up all the \(\Delta F\)s $$F(z_1)\approx F(z_0)+\sum\Delta F$$ Taking the limit... $$F(z_1) = F(z_0)+\int_\Phi dF = F(z_0)+\int_\Phi F'(z)dz = \int_0^1 F'(\Phi(t))\Phi'(t)dt $$ rewritten as $$ \int_\Phi F'(z)dz = F(z_1) - F(z_0) $$ In other words, the total change in \(F\) along the path \(F(\Phi)\) from \(F(z_0)\) to \(F(z_1)\) is the same as the change in \(F\) along the straight line segment from \(F(z_0)\) to \(F(z_1)\). Which, when you think about it, is really just the fundamental idea of tip to tail vector addition. |
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