Double integrals (articles)
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Everything we've done up to now has dealt with Cartesian coordinates, even though you might have not realized that they were Cartesian coordinates, because I never called it that before just now. So what is a Cartesian coordinate? Let me draw the axis that you're hopefully familiar with by now. If you're not, review those videos. Let me just draw the y-axis and the x-axis. And so, Cartesian coordinates-- and actually, they can apply to more than just 2 dimensions. But 2 dimensions is what we tend to deal with. Then if I want to specify any point in 2 dimensional space, I just tell you how far in the x direction I have to go, and how far in the y direction. And I can specify that point. Like, let's see, if I said, well, let me give the Cartesian coordinates for that point right there. And I might say, well, to get to that point right there, I have to go to the right 3 steps. So this is 3, right here. And then I have to go up 4 steps. 1, 2, 3, 4. And you end up there. And by convention, we call this the x-coordinate, and we call this the y-coordinate, and we call this 3 comma 4. And this is just a convention where we say the first coordinate is how far in the x direction we go, and the second coordinate is how far in the up and down, or the y direction, we go. So this is 3 comma 4. And we could have gone 3 to the right and 4 up, or we could have gone 4 up and 3 to the right. And we'd have gotten to the exact same point. This is just one way to really specify a point in 2 dimensions. Another option, and you might have done this in everyday life, is to say, hey, let me just point you in a direction, and then you go a certain distance in that direction. So I could have said, you know what? Let me just point you in this direction. Let me just point you in that direction, and you need to go this far. You need to go this far in that direction. That would also get you to the same point. So how do I specify the direction? Well, how about I call this 0 degrees? How about I call this the x-axis? How about I call that 0 degrees? And we'll deal with degrees. But hopefully, you know how to convert between degrees and radians. So you could just as easily convert to radians. But you could specify the direction if you call this 0, saying, ok, this is theta degrees, and I want you-- I'm going to point you in a degree in-- you know, you could always imagine someone who's blindfolded. You point them in this direction and you say, walk r units. And then they'll end up at that same point. And so you could also specify this point-- instead of specifying, this is the Cartesian coordinates, this is x comma y-- you could also specify it-- maybe, if we can figure out a way to do it. And I'll do it in magenta. You can also specify it as r comma theta. Which essentially says, you know, walk r units in the theta direction. Well, let's figure out what that is. Because I think it's kind of abstract now, and might be a little difficult to understand. So we're going to break out a little trigonometry and actually just a little Pythagorean theorem. Can we figure out r and theta? Well, r, hopefully, is the easier one here. Because if you think about it, this is a right triangle. This is a right triangle. That has distance 3. This is distance 4. This is a right triangle. So what is r here? Pythagorean Theorem. 3 squared plus 4 squared is equal to the hypotenuse squared, or r squared. So we could say, 3 squared plus 4 squared is equal to r squared. That's 9 plus 16 is equal to r squared. 25 is equal to r squared. We don't want to deal with negative distances, so 5 is equal to r, r is equal to 5. Fair enough. So we already figured out that r is equal to 5. How do we figure out theta? So what do we know here? We know-- well, we're trying to figure out theta. And we know it's opposite side, right? This is just going back to SOHCAHTOA. Let me write down SOHCAHTOA. If this is completely unfamiliar to you, you might want to watch the basic trigonometry video. So we know the opposite side of theta, right? That's 4. And we also know the adjacent side. That's 3. So which of the trig functions uses the opposite and the adjacent? Well, the TOA. Opposite and adjacent. So tangent. So tangent of theta is equal to the opposite-- which is really the y-coordinate, which is equal to 4-- over the adjacent. Which is the x-coordinate, which is 3. So tangent of theta is equal to 4/3. And to solve this, you essentially just take the inverse tangent of both sides of this. So this is the same thing. And depending on your calculator, or the convention you use, you might write this. When you take the inverse tangent of both sides, you get-- well, I'll just write it out. You could write it as the arctan of the tangent of theta is equal to the arctan of 4/3. And this, of course-- the arctan of the tangent. This is the same thing as the inverse tan of the tangent of theta. This is just equal to theta. That just simplifies to theta. Is equal to arctan of 4/3. And then another way to write arctan is often-- they'll often write it-- I'll write it. So the same exact statement could be written as tan inverse to the negative 1 power. They sometimes say it. Although it's a little misleading sometimes, when they write it like this. Because you don't know. Oh, am I taking the whole tangent of theta to the negative power or something? But sometimes I'll write arctan as 10 to the negative 1 power. But either way. So we can figure out theta by taking the arctangent of 4/3. And most people do not have the arctangent of 4/3 memorized. And I don't know an easy way to figure it out, without getting my TI-85 out. So let's get it out. All right. So how do we know the arctan-- so we want to do this inverse-- let me get the calculator up a little higher, so you can see. We want to get this inverse tan right here. So you press second inverse tangent of 4 divided by 3. And I already set my calculator to degree mode. It gives you 53.13 degrees. So theta is equal to 53.13 degrees. So there you have it. We already knew that we could specify this point in the 2 dimensional plane by the point x is equal to 3, y is equal to 4. We can also specify it by r is equal to 5, and theta is equal to 53 degrees. So I'll write that. And polar coordinates, it can be specified as r is equal to 5, and theta is 53.13 degrees. So all that says is, OK, orient yourself 53.13 degrees counterclockwise from the x-axis, and then walk 5 units. And you'll get to the exact same point. And that's all polar coordinates are telling you. Let's do another one. Actually, before that, let's see if we can come up with something general here. Because once you have the general, then you can always solve for the particular. So let me just write an arbitrary point. Last time I gave a specific point. The point-- was it 3 comma 4? I think it was. So let's say I have the point x comma y. It's right there. So its x-coordinate is there, its y-coordinate is there. So how do we convert this to r, to polar coordinates? r, theta. Well, let's do the same thing we did in the last video. If this length is r, and this angle is theta. So just going into what we did in the last video. We use Pythagorean Theorem. We say that x squared plus-- because this distance here is also y, right? This distance here is x. Pythagorean Theorem, you get x squared plus y squared is equal to r squared. That's just the Pythagorean Theorem. And then you have the tangent of theta. The tangent of this angle. SOHCAHTOA. TOA. Opposite over adjacent. Tangent of theta is equal to the opposite-- which is y-- over the adjacent-- which is x. And this is really all you need. And if you wanted to go one step further, and this is actually something you should write at the side of your paper. How can we express y as a function if we wanted to go the other way? If we were given r and theta, how could we get y? Well, let's think about it. y-- if we want to know-- let's see, if we are given r and y and we want to figure out theta-- what deals with the r and the y, with respect to theta? r is the hypotenuse and y is the opposite. Let me write SOHCAHTOA down. So what deals with the opposite and the hypotenuse? Sine. So the sine of theta is equal to the opposite-- which is the y side, which is equal to y-- over the hypotenuse, which is r. Equal to r. And so if you multiply both sides by r, you get r sine theta is equal to y. And then let's do it the other way. What if we wanted to have an equation for theta, r, and x? x is adjacent to the angle. r is still the hypotenuse. So what deals with adjacent and hypotenuse? Well, that's CAH. Cosine. So we have the cosine of theta is equal to the adjacent, is equal to x, over the hypotenuse, which is r. Multiply both sides by r and you get x is equal to r cosine theta. And really, when you're equipped with this formula-- which really just comes from our trig identities-- this formula, which you know, is similar, just comes from SOHCAHTOA. This, which comes from SOHCAHTOA-- and really is, you know-- if you're given these first. If you're given these two, you can easily derive that one. And this one, which is just the Pythagorean Theorem, you're actually ready. You're fully equipped to convert between polar and rectangular coordinates. And we'll do that in the next video, because I just realized I'm out of time.