Drawing more Complicated Graphs in Mathematica

[hufflepup]

I have a function, S[q] which is obtained by integrating another function, A[r,q] over r.

My problem is that the lower limit of integration,
rmin depends on q in a complicated manner. Mathematica can however solve this equation to give several routes, only one of which is real which is the one I want.

I have a second function too, L[q] which is also obtained by integrating B[r,q] over r. Again the lower limit of integration depends on q in the same manner.

What I want to do is for a given value of q, solve
the equation for rmin, and use this to calculate
the corresponding L[q]. I then wish to calculate
S[q] in the same manner.

I want Mathematica to do this for many values of
q over a range and use the results to plot
S[q] against L[q]. Does anyone know an efficient
way to do this?

Thank you for your help.

 

[Dale]

I am not sure I follow your plan very well, but you certainly can make a table where each entry is the solution for a given value of q.

 

[hufflepup]

Would it be possible to get Mathematica to plot a curved graph from a table like that if you do not know what it is supposed to look like to start with?

 

[Dale]

You can either use ParametricPlot or ListParametricPlot depending on if you want to explicitly generate the table of data points, or if you want Mathematica to handle that internally.

 

[sauberss]

@hufflepup:dude i too have the same problem..
r u able to get nw?
let me know if u cn..

 

[hufflepup]

Hi sauberss,

you can solve a problem like this as follows:

If F[q] = integral^Infinity_rmin A[r,q]dq

G[q] = integral^Infinity_rmin B[r,q] dq

where ^Infinity and _rmin represent the upper and lower limits of integration respectively, and
rmin= a(q)

Mathematica code:

F[q_]:= NIntegrate[A[r,q], {r, rmin, Infinity}]

G[q_]:=NIntegrate[B[r,q], {r, rmin, Infinity}]

The_]:= notation defines what is called a pattern, and basicly means that these expression won't be evaluated until they are called later, when a specific value of q will be provided.

As the lower limit of integration depends on rmin you now have 2 options, which is easiest will depend on your specific setup:

1) Get Mathematica to solve for rmin in terms of q, rmin=z(q), then replace rmin in the above equations with z(q).

2) Solve for q in terms of rmin, q=w(rmin) and replace the q in the integrals with w(rmin) in which case your equations would look like:

F[rmin_]:= NIntegrate[A[r,rmin], {r, rmin, Infinity}]

G[rmin_]:=NIntegrate[B[r,rmin], {r, rmin, Infinity}]

Note that these are of the form F[rmin_] not F[w(rmin)_] becuse the expression in the []
just telss Mathematica which variables will be replaced with numbers later.


To plot your graphs you then do:

graphname = ParametricPlot[{F[q], G[q]}, {q, qmin, qmax}]

OR

graphname = ParametricPlot[{F[rmin], G[rmin]}, {rmin, rminmin, rminmax}]

depending whether you went down path 1) or 2)

where qmin/rminmin and qmax/rminmax define the range of q/rmin over which you want to plot the graph.

I hope this helps you.