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# Parametrization of a reverse path

Understanding how to parametrize a reverse path for the same curve. Created by Sal Khan.

## Want to join the conversation?

• I'm very lost at , where does he get x(a+b-t) from? •  choosing the quantity (a+b-t) is a nice trick that satisfies
* when t=a, (a+b-t)=(a+b-a)=b
* when t=b, (a+b-t)=(a+b-b)=a
That is, (t) and (a+b-t) reverse values at a and at b
• During . Shouldn't it be b<= t <= a? not a<= t <= b? I mean I understand it is a <= t <= b if you think that the line starts at a, but when I think of it graphically I mean t increases as we move on the positive x-axis direction I think its b<= t <= a? I'm confused which way to think about it, but for now I'll think of it that t strats at the top being t=a and we move down to t=b. • I think the key thing to realize is that T is what does not change between the two integrals. it's starting value and ending value are the same in both cases.
Therefore, in order to make the integral move in the reverse direction you have to change how the integral interprets the end points (if that makes sense).
Now, if you went and messed with the value of T directly, you couldn't make a direct comparison between the integrals without substituting in and getting back the A+B-T.
• Guys, how do you link all those knowledge together? Is there scheme in khan academy for mathematics? I think I start losing the connection because of my lacking in the English language. I have a weak relationship with it. • Everything is presented in order.
At the top of the page on the left, beside the Khan Academy logo is a drop down list of subjects. Click on that, then click on math, if you are not already there. The suggested program of study is there.

Now it is possible it could be a language barrier thing, but consider this too:
Math at this level isn't just about doing the calculations, it is more thinking about how these things work the way they do, and that introspective analysis takes time. In a typical university setting, the three courses, pre, differential and integral calculus will take you a year and a half to complete (a year if your university has the trimester system) before getting into multivariable calculus. In that year and a half you are practicing, thinking doing research, usually with a 3 month break to continue reflecting on what you have learned, all over an 18 month period. On average, the typical student cannot internalize all this in less than a year, so it may be that you are rushing a bit. Only you can know the answer.

If you haven't already, please download a calculus textbook and start doing problems, and not just the "differentiate the following" types of problems, but the more advanced ones that force you to think about what you know and apply it - these are the problems that get your mind's internalization machinery working, these are where the connections are forged.

Perhaps take a break from multivariable calculus and go back to single variable calculus, but this time from an analytical point of view, where we talk about the why and how of mathematics. Real Analysis may be what you are looking for. Here are some textbooks made available free by the author. Try the elementary book:

Silly question:
Have you completely finished the pre-calc, differential calc and integral calc sections?

Keep up the great work. I have been following you for a while. You are awesome, RD!
• I don't get what Sal does at .
He says that the curve on the left and the curve on the right are essentially the same, with the difference being only in the case of directions, which is true.
It is true also that they should both yield the same value for x = t and y = t
This means that x = x(t) = x(a + b - t) and y = y(t) = y(a + b - t)
This means that t = a + b - t which implies that t = (a + b)/2 which can only be possible when 'a', 't' and 'b' are in AP, which they surely aren't.
It may work for the points where t = 'a' and t = 'b', but that is obvious, since at 't' = 'a' or 'b', t = (a + b)/2 will be satisfied.
This is just as logical as saying 2 terms are in an AP. And even if they are (which is not possible), there is no progression taking place, since d = 0.
This is the first time I've found it difficult to believe Sal.
Can someone clarify? • It is true that both curves are generated with the functions x=x(u) and y=y(u). It is not true that x(t) = x(a + b - t). For both curves, c and -c t does go from a to b, but in the first curve, c, the argument goes from a to b with t, in the second curve, -c, the argument goes from b to a. Its true they cover all the same points, but in the opposite order.

Another way of looking at how Sal derived the second parametrization for the reverse path is this:

To follow the same path but in reverse you know you want your argument to go from b to a, but were still assuming that t goes from a to b.

So, to start at b you can plug in (b - (t - a)). That way when t starts at a the t-a term is zero and the argument is b:
(b - (a - a)) = (b - (0)) = b

As t begins increasing the argument starts going down from b, until t reaches b and then our argument evaluates to a:

(b - (b - a)) = (b - b + a) = (0 + a) = a

See? If t goes from a to b, and your argument to your function is (b - (t - a)), then your argument starts at b and goes down to a. And if you distribute out the minus sign there you get:

(b - (t - a)) = (b - t + a) = (a + b - t)

And the last expression is the argument Sal uses.
• What's a scalar field? Maybe I missed the video or part of a video he went over it. • It seems to me that the second graph uses a,b in two different contexts. The first is to represent the start and end points on the curve while the second is the actual coordinates of a and b namely (x(a),y(a)) and x(b),y(b)). This is very confusing as it implies that when t=a, it's coordinates are actually x(b),y(b)). Is all of this done simply to prevent a negative t progression? That is, if t progresses positively from a to b then it would have to progress negatively from b to a. Is this the point of the confusion? • Where do you go over parameterizing curves? • The (a + b - t) trick is weakly explained. It seems very important mathematically and the crux of the whole proof here. Can anyone help us all out on this point? I can see many others have stumbled on this part.   