Hi, over the next few minutes we are going to discuss this topic called direction
fields. They also sometimes referred to as slope fields. Okay. So the idea here
with fields is because we're going to have a sense of direction. So we want to think
of the slopes. So you could have a positive slope this is a sketch of a
line with a positive slope. This here would be sketching a line that has
slope equal to zero. It's flat and horizontal. Then you can have a line that has
slope that is negative. Right? So this is positive slope, zero, ok? So, bigger than 0,
equal to zero, less than zero. In terms of slope M is a common notation
for slope. So you can have positive, equal to zero, or negative. Okay so that's the
idea with this sense of direction that is going to define some sketch and we're
going to use the information we are familiar with. Specifically information
that we can derive from a differential equation. So we want to think of the
right hand side of the differential equation as a function of two variables.
So again here we're assuming we're writing dy/dx that's the same as writing,
excuse me, writing y prime is the same as writing in Leibniz notation dy/dx. And
so, what we want to think that this here on the right hand side is an expression
that involves both x and y, right? So this would be an expression of x and y but we
are always making the assumption when we write this that y is a function of x. X being
the independent variable and Y is an expression
of X. But, we can have a right hand side that involves both of them. What would be
good examples? Well how about something that looks like 2x times y, or you know, that's
this is one type of example. So the idea also here is that we want to take
information, after all we are working on the two-dimensional plane, so we're going
to think of these coordinates really as if they were coordinates in a sense of X
comma Y, but where Y is a function of X. So if we know x we know y. And so now,
we want to think in terms of x zero y zero. A pair on the two-dimensional plane. Maybe
x one y one and maybe x two y two and x three y three and what do we do with all these
pairs? Well, we'd like to evaluate. Okay? We want to evaluate the right hand
side into each one of those. So we evaluate f at x0 y0, we evaluate the
right hand side at x1 y1, and we want to connect that to the meaning of this.
Which is dy/dx and so when we're evaluating we are
obtaining information. Okay, so let me write here dy/dx equals f of that. So what we're
saying there is that at x equals x 1 and y equals y 1 we are obtaining
information about the slope of the tangent. And in some sense slope of the
tangent at at this point this point x1 y1.
So here we have a complete example. An expression the right hand side that is y
squared minus 2x plus 5. So what happens if we evaluate dy/dx at x equals 1, y
equals 0. So that means that we are going to evaluate this expression. Okay, y is 0
x is 1, and this gives us minus 2 plus 5 equals 3. Now this three is going to
have the meaning of the value of a slope, when we think of slope of a
tangent to a curve. Okay that curve would be defined in the two-dimensional plane and
that slope here being positive we want to think of this as we're going to draw
an arrow that illustrates a line with a positive slope. So what happens if we now
evaluate dy/dx at, so we worked the first one, what happens if we evaluate now this at x
equals 2 and y equals 1. So we square, multiply by 2, and add five right so y in this case y
is 1 x is 2. Okay, so this gives us 1 minus 4 plus 5. Okay, and this value is
minus 3 plus 5 is 2. So, this is still giving us a value of the slope that is
positive. So that takes care of the second one and now we do another
evaluation using x equals 3 and y equals 1. So when we evaluate 3, 1 in the right
hand side then we obtain a value of zero and this would
be representing a slope equal zero of some tangent line. So with these
evaluations obtained a positive value of positive slope, positive value positive slope,
zero which is a flat line. The slope of a flat line, a horizontal line. And so, the
idea is that we could continue this evaluation in many many points in the
two-dimensional plane, and so that set of points in the two-dimensional plane, if we
were going to do this consistently, would look like this. And so, this is the type
of job that you could do by hand, but is actually a very good job for a computer.
Which is exactly how this figure was generated here and the point that I'm
trying to make is that we want to be able to interpret these
lines here are somehow, well if i put the error is probably too much right now, but
each of these lines, okay the position of each of these lines, is representing
values of the slope of a tangent line to a curve. I have not said to you what is
the curve, but the point is that when you see these type of graphic this is what
we are going to call a slope field. Okay? And let's perhaps identify from those
points we are working with. The last point was (3,1) so it had x had a value of
3 and y had a value 1. So if we follow this that point three one is somewhere
around here. Right? And (3,1). So what we found out was that (3,1) here is 3 in the x
axis horizontal axis and that's the height is about one and so at that point
(3,1) the value we obtained was 0 and so we would draw
and we draw a line that's kind of like a flat line. Okay, but if we identify some
of the other points, the second point was (2,1) so the x-coordinate 2 is right here.
So I just move this line here, and you can see that that is position on the
x coordinate equals two, and at the point (2,1) the value we obtained was a positive
value. And you can see how that's exactly around this point the values of slope
are quite positive. Right so we're going to if we were going to draw the slope
exactly at that point then we would sketch something that looks with that
type of inclination that type of angle or slope. Right and so, that's the idea
with these type of plots. And more importantly is that we could sketch the
slopes relative to the right hand side of the differential equation
and then we can do even more. We could decide, you know what I'm going to
choose a point on this this plane. For example, we'll maybe I should talk about
this point here, this point here would be zero approximately (0,1) and that point in
the two-dimensional plane with an x coordinate and a y coordinate could then
could then serve as information that we use in the initial conditions. For example,
I'm going to specify that I want an initial condition with x equals 0 and y
equals 1, and then I'm going to take a look at if i start with such values on
the initial condition, this curve appearing here in red is telling us
information about the so-called trajectory and this trajectory
is the trajectory of the solution of the differential equation. So I could choose
to start at different at the different values a minus 2x equals minus 2, and y
equals 3 for example then that would give me another initial condition. So
some, the convenience of using some of these tools, these computational tools us
that you could generate a lot of these solutions by basically only point and
click, right. So this offers each of these curves is a so-called trajectory which
is associated the differential equation and now when you see the slopes and you
see the curves then those slopes can convey the message. Basically by reading
the slope field you can have a sense of the so-called behavior of the solutions
of the differential equation. So this here is the direction field, or the slope
field for a different differential equations. A differential equation
which the right-hand side dy/dx equals sine and the argument of the sine
function, the trigonometric function, the argument is the difference between y and
x. Y minus x. Okay, so right now this is for initial conditions at (0,0) we would
see that that solution appears. And so this is a quite nice because of the
definition of the sine function this gives some curves that quite smooth. Right and so the idea of the slopes and in this field is to kind of paint a
picture of what the trajectories what the trajectories would look like