Video transcript:

Tim – So you’ve got your rectangle and you’re considering the diagonal OB, so as you walk along there, the temperature is going to vary, and you’re going to draw a graph of that temperature variation.  So essentially, I mean x is really along this axis, but as you walk along here your x is changing and, since you’re fixed to a line, you know how y is changing as well in fact.  At every x value, your y is a half of the x value, because this is two and this is one.  So, that’s information that’s useful, and x is varying between 0 and 2, and, as you walk along here y varies between zero and one.  The function q(x) is the temperature along this line, so, as x varies, q(x) is going to be the temperature, and it’s actually going to be the intersection of the … , if you go, … where’s your mouse?  If you rotate that back to umm, so it’s the right way up.  Right, so you’ve got a plane intersecting this graph, and, in fact, a plane intersecting the two-dimensional graph that will give you the one-dimensional graph you want; in fact it’s going to start at zero and end at zero, and do something like that …

Student 1 – So you’re going to end up with a 2D kind of contour plot, or will we end up with...?

Tim – No you won’t end up with a contour plot, you’ll end up with a graph of this function q(x).  So you’re going to have x along here and q(x) along here, and its going to rise and then go back down again.  Imagine walking over this hill; you’re going to start at zero, you’re going to go up, and then come back down …  You can also link it with this, because, as you walk along here, at some point you will go through the centre of the plate, and you know what the temperatures are from the previous ones.  So your graph value at this point should correspond to your value k here; that’s just a little check, because, at some point you will go through the centre of the plate, so you should record the temperature there.  So what is q(x), how do you get q(x) from Φ(x,y)?  You’ve been working with Φ(x,y) all the way through, so what’s the link between q(x) and Φ(x,y)?

Student 2 – Is it with only either the x or the y variable as opposed to …, so it would be Φ(x) as supposed to Φ(x,y)?

Student 1 – Would it be that instead of plotting x against y, you’d have x against q(x)?

Tim – No what I’m asking you now is, you’ve been working with Φ(x,y) which is a function of two variables x and y varying in the rectangle.  Now you’re only interested in this line, which is just the temperature along here, which is a function of x.  So, how do you get from Φ(x,y), which is what you’ve been playing around with, to this thing that you want q(x); how do you get from Φ(x,y) to q(x)?

Student 2 – Well if you go back to Φ(x,y), where it’s got a y variable, and the equation change will replace it with ½ x.

Tim - Correct, exactly.

Student 2 – And then you just do a 2D plot?

Tim – No you won’t be doing a 2D plot because q(x) is just an ordinary function of one variable, so it’s not …, it’s just an ordinary …, like you did with the original ½ x, or whatever …

Student 2 – Ah right so just plot.

Tim – Well there’s various steps:  first of all you have to construct q(x), which, as you say, everywhere you see y …, instead of Φ(x,y), it’s going to be Φ(x), and you’re going to replace that y with ½ x, so that will be a subs command in Maple; subs y=½ x, and that will give you q(x).  In fact, what you’ll have is q(x) with an extra N, as you’ve always had, because you’re going to go to very big N according to accuracy.  So, in fact, what you’ll have is q[N](x):=subs(y=x/2,Phi[N](x,y)), which will give you q(x) with this extra N as usual, and then you’re going to plot that, and you’re going to have to subs for a value of N and change that value to get better plots.

Student 1 – And would 50 be enough for a … ?

Tim – Well, why don’t you just … , you’re going to play around with it.  Whatever value of N you have used before, I would use that value of N to plot the ... ; you want to be consistent.