Temperature is given by T=x^3+y^3-3xy, what is the hottest point on the square plate bounded by x=0, x=2, and y=0,y=3?

Question

anonymous · Answer

Are you familiar with the gradient?

anonymous · Answer

yes i am

anonymous · Answer

Okay, well the definition of the gradient is that it gives you a vector which points in the direction of the maximum value.

So I would take the gradient of the function you were given. Then apply the boundary conditions and you will have the maximum temperature value for this plate.

anonymous · Answer

so the gradient would be x^3+y^3-3xy, 3x^2-3y,3y^2-3x, but how should I apply the boundaries just plug them in? @Pompeii00

anonymous · Answer

Oops, I'm sorry I misread the prompt you gave, I apologize.

you want the point of maximum temperature. So, the gradient points as a vector in the direction of max increase. If you are at the highest point, you can't point towards the point you're on. So the gradient is 0.

you need to take your gradient here and set it equal to zero. Then simply take the solutions you find that lie within the region of the plate. 

I apologize for the confusion.

anonymous · Answer

so then the critical points (0,0) and (1,1)? @Pompeii00

anonymous · Answer

Yep!

anonymous · Answer

They are both the hottest points? Also how would I find coldest points @Pompeii00

anonymous · Answer

The solution is here on 6b http://math.sci.ccny.cuny.edu/document/show/1702 if you want to check it but I can't really make much sense off that.

anonymous · Answer

Wow I haven't seen this in forever it seems haha. Good memories. I forgot about that directional derivative stuff.

So yeah, the basic idea is solving for all the critical points. This requires you to solve like we did for (0, 0) and (1, 1). You can use the directional derivative equation to determine what is at these points. As per solutions, you should find a saddle point and minimum.

Then you have to go around the boundaries and on each side of the rectangle, solve again for critical points (this would be 4 steps, since there are 4 sides to do).

Then you simply plug in all of the critical point coordinates you find into the T(x,y). This will allow you to see which point is the hottest (hint: it has the largest T(x,y) value).

anonymous · Answer

gotcha thank you :)