Mark Minimum Value in DensityPlot

Question

Let's take this example:

DensityPlot[Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3}]

How do I retrieve the minimum value and position of the minimum of a plot from the plot? How do I tell Mathematica to mark the position of the minimum/minima within the plot?

And for ContourPlots

ContourPlot[Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3}]

Is there a way to mark the exact minimum position within the plot even though it's a contour plot?

Yes, Jens and Szabolcs, that would be the best way if i could manage to write the code for that. I'm still a beginner with mathematica and haven't managed so far to cope with that problem because....well i think i have to post a piece of my code to show it. I have deleted most of the variables and replaced them by constants for simplicity reasons:

eislexpl[epsilon_, n2_, x_] :=  28 (epsilon)^2 - 1 + 
((-2 Log[E^(1/2) x])/x^2 + (2.7(0.3 - epsilon))/x + 
(10 epsilon^2 n2^(3/2))/x^(3/2));

eislminx2[epsilon_?NumericQ, n2_?NumericQ] := 
(minx = FindArgMin[{eislexpl[epsilon, n2, x], 0 < x}, x];
eislexpl[epsilon, n2, minx]);

ueislexpl[epsilon_, h_, n1_, n2_] := 
n1 (40  epsilon^2 - 1) + (1 -1.27 ) (1 + (n1 - 1) HeavisideTheta[1 - n1] - 
(-1 + Exp[-(n1 - 1)]) HeavisideTheta[n1 - 1]) + n2  eislminx2[epsilon, n2] + 
(h - n1 - n2) (28  epsilon^2 - 1);

ueislexpl[epsilon_, h_, n1_, n2_] is the function i need to find the minimum for. But the problem for me is, that it calls this function eislminx[epsilon_?NumericQ,n2_?NumericQ]. So my problem is that in order to have a function that mathematica is able to find a minimum for, i have to plug n2 in eislminx[epsilon_?NumericQ,n2_?NumericQ], so that this part of ueislexpl[epsilon_, h_, n1_, n2_] determined, but at the same time i want to find the minimum for ueislexpl[epsilon_, h_, n1_, n2_] with respect to n2. So if anybody know how to find the minimum of ueislexpl[epsilon_, h_, n1_, n2_] with respect to n2 and n1 i would be happy to learn how.

Thanks for the help.

Well, my actual question has been answered by kguler, thanks again ;) And the question of how to find the minimum of that function above i will source out to here: Finding the minimum of a function

Answer 1

A quick-and-dirty way: get the contours for the .0000001st and .9999999th quantiles as approximations for min and max

dp = DensityPlot[Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3}];
cp1 = ContourPlot[Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3}, 
            ContourShading -> None, 
            ContourStyle -> {Directive[Red, Thick], Directive[Orange, Thick]}, 
            Contours -> Function[{min, max}, 
                          Rescale[{.00000001, .99999999999}, {0, 1}, {min, max}]]];
cp2 = ContourPlot[Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3},  ContourShading -> None];
Legended[Show[dp, cp1, cp2], 
             PointLegend[{Red, Orange}, {"min", "max"}, LabelStyle -> 16, 
                         BaseStyle -> PointSize[Large]]]

Answer 2

min = Minimize[
   {Sin[x] Sin[y], -4 < x < 4, -3 < y < 3},
   {x, y}][[1]]

-1

minPts = Cases[
  Reduce[{Sin[x] Sin[y] == min, -4 < x < 4, -3 < y < 3}, {x, 
    y}], (x == x1_ && y == y1_) :> {x1, y1}, Infinity]

{{-(Pi/2), Pi/2}, {Pi/2, -(Pi/2)}}

DensityPlot[
 Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3},
 Epilog ->
  {Red, AbsolutePointSize[6], Point[minPts]}]

ContourPlot[
 Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3},
 Epilog ->
  {Red, AbsolutePointSize[6], Point[minPts]}]