# Normalizing a plot by introducing a variable

I am trying to plot the graph between P and x. My teacher wants me to normalize this graph by introducing a constant $$k$$ to get $$P/k^2$$ on the y-axis and $$xk$$ on the x-axis. I did that by doing a short cut, by manually seeing the max value of P and plotted the graph by doing $$P/0.0035$$ which was the maximum value of $$P$$ from graph. But how can I do it efficiently by using mathematical commands? My code is below:

gamma = 1/Sqrt[1 - D[r (-1 + 1/v) + ArcTanh[r], r]^-2]


Then I plotted $$p-x$$ in the limits, $${x,-1,1}$$ as follows: Definition:

power = 1/(6 Pi) D[gamma, r]^2


Plot

pow = Plot[{power/0.0035 /. v -> 1/2, power/0.0035 /. v -> 1/3,
power/0.0035 /. v -> 1/4}, {r, -1, 1},
AxesLabel -> {Style["x\[Kappa]", Black, FontFamily -> "Times",
FontSize -> 15], Style["P/\[Kappa]^2", Black, FontFamily -> "Times", FontSize -> 15]},
LabelStyle -> {FontSize -> 12, FontFamily -> "Times", Black, Bold},
PlotStyle -> {{Red, Thickness[0.005]}, {Purple,
Thickness[0.005]}, {Yellow, Thickness[0.005]}},
PlotLegends ->
Placed[LineLegend[{Directive[Thickness[0.1], Red],
Directive[Thickness[0.1], Purple],
Directive[Thickness[0.1], Yellow]}, {Style["s=1/2", 11, Bold],
Style["s=1/3", 11, Bold], Style["s=1/4", 11, Bold]},
LegendMarkerSize -> {{21, 15}}], {0.6, 0.6}]]


If I introduce a $$k^2$$ quantity, my x-axis should also be scaled. How can I completely normalize this graph with both axes calling through the quantity $$k$$ using the mathematica commands? I have mathematica 9.0.

• k = Sqrt[MaxValue[{power, 1/4 <= v <= 1/2, -1 <= r <= 1}, {r, v}]] which evaluates to (2 Sqrt[2/π])/27 The approximate value of k^2 is 0.00349311 Nov 5, 2023 at 0:53

You may use "Maximize" to get the r value of the maximum. With this you get the maximal value for P: pmax:

gamma = 1/Sqrt[1 - D[r (-1 + 1/v) + ArcTanh[r], r]^-2];
power = 1/(6 Pi) D[gamma, r]^2;
pmax = power /. v -> 1/2 /.
Maximize[{power /. v -> 1/2, -1 <= r <= 1}, r][[2]];
pow = Plot[{power/pmax /. v -> 1/2, power/pmax /. v -> 1/3,
power/pmax /. v -> 1/4}, {r, -1, 1},
AxesLabel -> {Style["x\[Kappa]", Black, FontFamily -> "Times",
FontSize -> 15],
Style["P/\[Kappa]^2", Black, FontFamily -> "Times",
FontSize -> 15]},
LabelStyle -> {FontSize -> 12, FontFamily -> "Times", Black, Bold},
PlotStyle -> {{Red, Thickness[0.005]}, {Purple,
Thickness[0.005]}, {Yellow, Thickness[0.005]}},
PlotLegends ->
Placed[LineLegend[{Directive[Thickness[0.1], Red],
Directive[Thickness[0.1], Purple],
Directive[Thickness[0.1], Yellow]}, {Style["s=1/2", 11, Bold],
Style["s=1/3", 11, Bold], Style["s=1/4", 11, Bold]},
LegendMarkerSize -> {{21, 15}}], {0.6, 0.6}]]