Exact Equations Intuition 2 (proofy) More intuitive building blocks for exact equations.
Exact Equations Intuition 2 (proofy)
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- In the last video I introduced you to the idea of the chain
- rule with partial derivatives.
- And we said, well, if I have a function, psi, Greek letter,
- psi, it's a function of x and y.
- And if I wanted to take the partial of this, with respect
- to-- no, I want to take the derivative, not the partial--
- the derivative of this, with respect to x, this is equal to
- the partial of psi, with respect to x, plus the partial
- of psi, with respect to y, times dy, dx.
- And in the last video I didn't prove it to you, but I
- hopefully gave you a little bit of intuition that you can
- believe me.
- But maybe one day I'll prove it a little bit more
- rigorously, but you can find proofs on the web if you are
- interested, for the chain rule with partial derivatives.
- So let's put that aside and let's explore another property
- of partial derivatives, and then we're ready to get the
- intuition behind exact equations.
- Because you're going to find, it's fairly straightforward to
- solve exact equations, but the intuition is a little bit
- more-- well, I don't want to say it's difficult, because if
- you have the intuition, you have it.
- So what if I had, say, this function, psi, and I were to
- take the partial derivative of psi, with respect to x, first.
- I'll just write psi.
- I don't have to write x and y every time.
- And then I were to take the partial derivative with
- respect to y.
- So just as a notation, this you could write as, you could
- kind of view it as you're multiplying the operators, so
- it could be written like this.
- The partial del squared times psi, or del squared psi, over
- del y del, or curly d x.
- And that can also be written as-- and this is my preferred
- notation, because it doesn't have all this extra junk
- You could just say, well, the partial, we took the partial,
- with respect to x, first. So this just means the partial of
- psi, with respect to x.
- And then we took the partial, with respect to y.
- So that's one situation to consider.
- What happens when we take the partial, with respect to x,
- and then y?
- So with respect to x, you hold y constant to get just the
- partial, with respect to x.
- Ignore the y there.
- And then you hold the x constant, and you take the
- partial, with respect to y.
- So what's the difference between that and if we were to
- switch the order?
- So what happens if we were to-- I'll do it in a different
- color-- if we had psi, and we were to take the partial, with
- respect to y, first, and then we were to take the partial,
- with respect to x?
- So just the notation, just so you're comfortable with it,
- that would be-- so partial x, partial y.
- And this is the operator.
- And it might be a little confusing that here, between
- these two notations, even though they're the same thing,
- the order is mixed.
- That's just because it's just a different way of
- thinking about it.
- This says, OK, partial first, with respect to x, then y.
- This views it more as the operator, so we took the
- partial of x first, and then we took y, like you're
- multiplying the operators.
- But anyway, so this can also be written as the partial of
- y, with respect to x-- sorry, the partial of y, and then we
- took the partial of that with respect to x.
- Now, I'm going to tell you right now, that if each of the
- first partials are continuous-- and most of the
- functions we've dealt with in a normal domain, as long as
- there aren't any discontinuities, or holes, or
- something strange in the function definition, they
- usually are continuous.
- And especially in a first-year calculus or differential
- course, we're probably going to be dealing with continuous
- functions in soon. our domain.
- If both of these functions are continuous, if both of the
- first partials are continuous, then these two are going to be
- equal to each other.
- So psi of xy is going to be equal to psi of yx.
- Now, we can use this knowledge, which is the chain
- rule using partial derivatives, and this
- knowledge to now solve a certain class of differential
- equations, first order differential equations, called
- exact equations.
- And what does an exact equation look like?
- An exact equation looks like this.
- The color picking's the hard part.
- So let's say this is my differential equation.
- I have some function of x and y.
- So I don't know, it could be x squared times
- cosine of y or something.
- I don't know, it could be any function of x and y.
- Plus some function of x and y, we'll call that n, times dy,
- dx is equal to 0.
- This is-- well, I don't know if it's an exact equation yet,
- but if you saw something of this form, your first impulse
- should be, oh-- well, actually, your very first
- impulse is, is this separable?
- And you should try to play around with the algebra a
- little bit to see if it's separable, because that's
- always the most straightforward way.
- If it's not separable, but you can still put it in this form,
- you say, hey, is it an exact equation?
- And what's an exact equation?
- Well, look immediately.
- This pattern right here looks an awful
- lot like this pattern.
- What if M was the partial of psi, with respect to x?
- What if psi, with respect to x, is equal to M?
- What if this was psi, with respect to x?
- And what if this was psi, with respect to y?
- So psi, with respect to y, is equal to N.
- What if?
- I'm just saying, we don't know for sure, right?
- If you just see this someplace randomly, you won't know for
- sure that this is the partial of, with respect to x of some
- function, and this is the partial, with respect to y of
- some function.
- But we're just saying, what if?
- If this were true, then we could rewrite this as the
- partial of psi, with respect to x, plus the partial of psi,
- with respect to y, times dy, dx, equal to 0.
- And this right here, the left side right there, that's the
- same thing as this, right?
- This is just the derivative of psi, with respect to x, using
- the partial derivative chain rule.
- So you could rewrite it.
- You could rewrite, this is just the derivative of psi,
- with respect to x, inside the function of x,
- y, is equal to 0.
- So if you see a differential equation, and it has this
- form, and you say, boy, I can't separate it, but maybe
- it's an exact equation.
- And frankly, if that was what was recently covered before
- the current exam, it probably is an exact equation.
- But if you see this form, you say, boy, maybe
- it's an exact equation.
- If it is an exact equation-- and I'll show you how to test
- it in a second using this information-- then this can be
- written as the derivative of some function, psi, where this
- is the partial of psi, with respect to x.
- This is the partial of psi, with respect to y.
- And then if you could write it like this, and you take the
- derivative of both sides-- sorry, you take the
- antiderivative of both sides-- and you would get psi of x, y
- is equal to c as a solution.
- So there are two things that we should be caring you about.
- Then you might be saying, OK, Sal, you've walked through
- psi's, and partials, and all this.
- One, how do I know that it's an exact equation?
- And then, if it is an exact equation, which tells us that
- there is some psi, then how do I solve for the psi?
- So the way to figure out is it an exact equation, is to use
- this information right here.
- We know that if psi and its derivatives are continuous
- over some domain, that when you take the partial, with
- respect to x and then y, that's the same thing as doing
- it in the other order.
- So we said, this is the partial, with
- respect to x, right?
- And this is the partial, with respect to y.
- So if this is an exact equation, if this is the exact
- equation, if we were take the partial of this, with respect
- to y, right?
- If we were to take the partial of M, with respect to y-- so
- the partial of psi, with respect to x, is equal to M.
- If we were to take the partial of those, with respect to y--
- so we could just rewrite that as that-- then that should be
- equal to the partial of N, with respect to x, right?
- The partial of psi, with respect to y, is equal to N.
- So if we take the partial, with respect to x, of both of
- these, we know from this that these should be equal, if psi
- and its partials are continuous over that domain.
- So then this will also be equal.
- So that is actually the test to test if
- this is an exact equation.
- So let me rewrite all of that again and summarize it a
- little bit.
- So if you see something of the form, M of x, y plus N of x,
- y, times dy, dx is equal to 0.
- And then you take the partial derivative of M, with respect
- to y, and then you take the partial derivative of N, with
- respect to x, and they are equal to each other, then--
- and it's actually if and only if, so it goes both ways--
- this is an exact equation, an exact differential equation.
- This is an exact equation.
- And if it's an exact equation, that tells us that there
- exists a psi, such that the derivative of psi of x, y is
- equal to 0, or psi of x, y is equal to c, is a solution of
- this equation.
- And the partial derivative of psi, with respect to x, is
- equal to M.
- And the partial derivative of psi, with respect to y, is
- equal to N.
- And I'll show you in the next video how to actually use this
- information to solve for psi.
- So here are some things I want to point out.
- This is going to be the partial derivative of psi,
- with respect to x, but when we take the kind of exact test,
- we take it with respect to y, because we want to get that
- mixed derivative.
- Similarly, this is going to be the partial derivative of psi,
- with respect to y, but when we do the test, we take the
- partial of it with respect to x so we get that mixed
- This is with respect to y, and then with respect to
- x, so you get this.
- Anyway, I know that might be a little bit involved, but if
- you understood everything I did, I think you'll have the
- intuition behind why the methodology of
- exact equations works.
- I will see you in the next video, where we will actually
- solve some exact equations See
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