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5DUQ3-Y_gX4 | So we'll call it a
vector space as well. |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: Yeah. |
5DUQ3-Y_gX4 | But it's incredibly
useful, though. |
5DUQ3-Y_gX4 | Because very often, especially
in physical sciences, |
5DUQ3-Y_gX4 | you have something where
conceptually you're |
5DUQ3-Y_gX4 | solving for a functions. |
5DUQ3-Y_gX4 | So you're solving for the
fluid flow or something |
5DUQ3-Y_gX4 | around an airplane wing. |
5DUQ3-Y_gX4 | And what you want is that
then take that fluid flow |
5DUQ3-Y_gX4 | and compute the drag
on the airplane wing. |
5DUQ3-Y_gX4 | And then in order
to optimize it, |
5DUQ3-Y_gX4 | you want the derivative
of the drag with respect |
5DUQ3-Y_gX4 | to that flow field, with
respect to the function, |
5DUQ3-Y_gX4 | or with respect to the
shape of the airplane, which |
5DUQ3-Y_gX4 | is a function. |
5DUQ3-Y_gX4 | So it's very, very nice to
be able to take derivatives |
5DUQ3-Y_gX4 | connected to functions and
work with vector functions |
5DUQ3-Y_gX4 | as vector spaces. |
5DUQ3-Y_gX4 | And very soon we're going
to be able to do that |
5DUQ3-Y_gX4 | with this notion
of a derivative. |
5DUQ3-Y_gX4 | Because we're going to be able
to define linear operators, |
5DUQ3-Y_gX4 | functions that act on functions. |
5DUQ3-Y_gX4 | And linear operators
are functions. |
5DUQ3-Y_gX4 | But that's getting a bit
too far ahead of ourselves. |
5DUQ3-Y_gX4 | OK. |
5DUQ3-Y_gX4 | So the point is that
the 18.01-- so far, we |
5DUQ3-Y_gX4 | haven't done any derivatives
more than 18.01, at least |
5DUQ3-Y_gX4 | in my half. |
5DUQ3-Y_gX4 | Alan went a bit further. |
5DUQ3-Y_gX4 | But already we can start
to see, hopefully, how |
5DUQ3-Y_gX4 | this is going to generalize. |
5DUQ3-Y_gX4 | So if you have a
function f of x and you |
5DUQ3-Y_gX4 | make a small change
in the input, delta x, |
5DUQ3-Y_gX4 | and you ask for the small
change in the output |
5DUQ3-Y_gX4 | to first order, which we can
denote with this d notation, |
5DUQ3-Y_gX4 | the derivative is
the linear operator |
5DUQ3-Y_gX4 | that gives us that,
the linearization |
5DUQ3-Y_gX4 | of that function for a
small change in the input. |
5DUQ3-Y_gX4 | And that is exactly equivalent
to what you learned in 18.01. |
5DUQ3-Y_gX4 | But it's going to be easier
now to generalize this |
5DUQ3-Y_gX4 | to other kinds of inputs
and other kinds of outputs, |
5DUQ3-Y_gX4 | where, in 18.01, we move this to
the side where we take df, dx. |
5DUQ3-Y_gX4 | We divide them. |
5DUQ3-Y_gX4 | For numbers, that's fine. |
5DUQ3-Y_gX4 | For other kinds of things, that
becomes a little bit weirder |
5DUQ3-Y_gX4 | to talk about. |
5DUQ3-Y_gX4 | Of course, you could
define it as notation. |
5DUQ3-Y_gX4 | But I think it's a
lot clearer if you |
5DUQ3-Y_gX4 | think of it in this sense
once you start generalizing |
5DUQ3-Y_gX4 | to other kinds of objects. |
5DUQ3-Y_gX4 | So with that said,
let's do that. |
5DUQ3-Y_gX4 | Now, let's revisit 18.02. |
5DUQ3-Y_gX4 | And let me do it in two parts. |
5DUQ3-Y_gX4 | So part one is going
to be functions-- |
5DUQ3-Y_gX4 | the first thing you
usually do in 18.02, |
5DUQ3-Y_gX4 | which is functions that take a
vector in or multiple variables |
5DUQ3-Y_gX4 | in. |
5DUQ3-Y_gX4 | But we'll think of it as a
vector in and a scalar out. |
5DUQ3-Y_gX4 | So we're going to have a scalar. |
5DUQ3-Y_gX4 | My output is blue, right? |
5DUQ3-Y_gX4 | Yes. |
5DUQ3-Y_gX4 | We'll keep the same
color scheme, good. |
5DUQ3-Y_gX4 | So we're going to have a scalar
function f of a vector input x. |
5DUQ3-Y_gX4 | And I'll put a little
vector sign above it. |
5DUQ3-Y_gX4 | I won't always do that, but
it's nice to be clear sometimes, |
5DUQ3-Y_gX4 | which is a vector,
which is a scalar. |
5DUQ3-Y_gX4 | So x is going to live in our m. |
5DUQ3-Y_gX4 | So this is going to be an
m component column vector. |
5DUQ3-Y_gX4 | OK. |
5DUQ3-Y_gX4 | And what we want
to do is imagine |
5DUQ3-Y_gX4 | what happens to the
output when you change |
5DUQ3-Y_gX4 | the input by a little bit. |
5DUQ3-Y_gX4 | So we're going to
take f of x plus dx. |
5DUQ3-Y_gX4 | Think of this as a
really small change. |
5DUQ3-Y_gX4 | It's infinitesimal. |
5DUQ3-Y_gX4 | We're going to
drop anything that |
5DUQ3-Y_gX4 | goes like dx squared or anything
like that, any higher terms. |
5DUQ3-Y_gX4 | Let me just move
it to [INAUDIBLE].. |
5DUQ3-Y_gX4 | It's black-- minus f of x. |
5DUQ3-Y_gX4 | And we're going to define
this as f prime of x dx. |
5DUQ3-Y_gX4 | So we wanted to note we have an
arbitrary change in the inputs. |
5DUQ3-Y_gX4 | dx is an arbitrary,
very, very small vector. |
5DUQ3-Y_gX4 | And we want to ask, what's
the change in the output |
5DUQ3-Y_gX4 | differential, df? |
5DUQ3-Y_gX4 | And the answer is
going to be that this |
5DUQ3-Y_gX4 | is going to be, for
a very small dx, |
5DUQ3-Y_gX4 | we can approximate this by
a linear operator on dx. |
5DUQ3-Y_gX4 | So this is going to be a
linear operator, always |
5DUQ3-Y_gX4 | going to be a linear operator. |
5DUQ3-Y_gX4 | And what is that
linear operator do? |
5DUQ3-Y_gX4 | This one takes a vector
in and gives you a scalar. |
5DUQ3-Y_gX4 | So this has to equal a-- |
5DUQ3-Y_gX4 | df is a scalar,
but dx is a vector. |
5DUQ3-Y_gX4 | So what this has to be is it
has to be kind of a row vector. |
5DUQ3-Y_gX4 | You can think of it more
as a one-row matrix. |
5DUQ3-Y_gX4 | Or there's fancier names
for this, like covector |
5DUQ3-Y_gX4 | or dual vector. |
5DUQ3-Y_gX4 | We won't really use that. |
5DUQ3-Y_gX4 | I just want to throw
them out there. |
5DUQ3-Y_gX4 | So if you want to take a vector
in and take a vector out, |
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