Welcome to Introduction to Vector Fields.
The goals of this video are to define a vector field
and then also to graph a vector field by hand
and with technology.
A vector is a quantity with magnitude and direction.
Here we see two black vectors that represent the speed
and the direction of the yellow and blue car.
So the arrow points in the direction of motion
and then because the yellow car has a longer vector,
its magnitude would be larger which means
the yellow car is going faster.
For example this car maybe going 75 miles per hour
and the blue car may only be going 35 miles per hour.
Again the arrow points in the direction of motion
and the length or the magnitude
of the vector represents how fast the car is going.
So functions that assign a vector to a point in a plane
or a point in space are called vector fields.
And they are useful in representing the various types
of force fields and velocity fields.
So let's take a look at a formal definition
of a vector field.
A vector field over a region R in a plane
or in two dimensions is given as we see here
where F of x, y is given as the vector
where the x component is a function of x and y
and the y component of the vector
would be given as a function of x, y
And f and g are continuous over the region R.
And for three dimensions, the vector field
will have three components given
by f, g, and h as functions of x, y, and z.
Again f, g, and h are continuous
over the three dimensional region or space V.
So one thing to notice is that the gradient
that we've already talked about is an example
of a vector field.
Before we graph our own vector fields, let's take a look
at some two-dimensional vector fields created
in Mathematica.
Here we see the force field given
by f of x, y equals the vector field x comma negative y.
Again, notice each vector has an arrow giving
it direction as well as a length which would be given
by the magnitude which represents the amount
of force at a given point.
What we see here in red is if we had a point in this field,
based upon the force given by the field,
the red path shows a direction that point would take.
So if we move this point somewhere else,
you can see the force would be in a different path.
And here's another two-dimensional vector field.
Again if we move the position of the point,
its path will change based upon the force of the field.
Let's go and take a look at one more.
Here's an interesting one.
So again if we move this point to some other location
in the field, its path will change based upon the forces
in that location.
Here's one more that I wanted to show.
Here's the equation of the vector field and here's the graph
of the vector field and you can see here
that it creates several circular motions.
Some of them are clockwise and some
of them are counterclockwise.
Now let's take a look at a few vector fields
in three dimensions.
Here is a graph of the vector field
F of x, y, z equals x comma y comma z.
And again each vector in the field gives us
a direction as well as a magnitude
where the length of the vector represents the amount
of force at that point.
So you might be able to think of this vector field
as an explosion from the center of this box.
Here's the three dimensional vector field given
by F of x, y, z equals negative y comma negative z comma x.
You can see here that the motion
of this vector field is more of a circular motion
where in the first example, the direction was outward.
Let's take a look at how we can graph
a two-dimensional vector field by hand.
To graph a vector field by hand,
we need to plot several vectors in the vector field
by selecting x and y values in the coordinate plane
and then plotting the corresponding vector.
And sometimes it is helpful to plot several vectors
with the same magnitude.
However, graphing software is very helpful
when plotting vector fields.
Let's take a look at a couple of examples.
Here we want to graph F of x, y
equals negative one comma one.
So notice that this vector field is not affected
by the values of x and y.
Every single vector in this vector field
would have an x component of negative one
and a y component of positive one.
So we'll start the origin and move left one unit
and then up one unit to represent our first vector
in the vector field.
Now what we'll do is go ahead and space
these one unit apart all along the x axis.
And now we'll go ahead and do the same along
the line y equals two.
So again the x component will be negative one
and the y component will be positive one.
So it looks just like this.
All along y equals two.
And let's go ahead and do the same at y equals negative two.
Now notice how I skipped y equals one
and y equals negative one and I
could have sketched vectors there
but notice how they would have run
into the vectors along the x axis.
So when creating a vector field,
you do have to decide how far you want
your vectors spaced apart because there are several ways
to represent an accurate vector field.
Here's the same vector field equated with Maple.
You can see here it's much more thorough
and obviously a lot faster.
Let's take a look at another example.
Here we have F of x, y equals zero comma negative x.
So again we'll pick points on the coordinate plane
and then sketch the corresponding vectors.
Notice that each vector is only affected by the value of x.
So what we'll do on this vector field is select a value
of x.
Let's say x equals one.
So all along this vertical line,
x equals one each vector would have an x component
of zero and a y component of negative one.
So if we start at positive four, we'd plot the vector zero,
negative one which would be here.
Let's go ahead and jump down to this point here
and plot the same vector.
Skip down to y equals negative one.
Plot the same vector.
Negative three, plot the same vector.
Now if x equals two, that would be along
this vertical line here.
And every vector along that vertical line
would have an x component of zero
and now a y component of negative two.
So if we started here at four,
we'd have a vector zero, negative two.
Now go and jump down to y equals one.
Plot the same vector.
Jump down to y equals negative two.
Plot the same vector.
Let's go ahead and try x equals three.
You probably can see the pattern now.
Every vector will have an x component of zero
and a y component of negative three.
So if we started up here at y equals four,
we'd go down to y equals negative one.
If we start here, we go down to y equals negative three.
Let's take a look at some negative values of x now.
When x equals negative one along this vertical line here,
every vector would have an x component of zero,
but now the y component would be positive one.
So it's very similar to ...
Except now they'll be in the opposite direction.
So if we started up here at y equals three,
the vector zero, one would look like this.
At y equals one it would look like this and so on.
And again, I'm purposely spacing these vectors apart
so they don't run into each other.
And when x equals negative two,
each vector would be zero comma two.
So if I start here at y equals two
along the line x equals negative two,
zero, two would look like this.
And I'll jump down to here.
Zero, two would look like this
and zero, two would look like this.
So you can see that when x is positive,
the vector field is moving downward
and when x is negative, it's moving upward.
And as x moves to the right, the magnitude increases
as well as if it moves to the left, the magnitude increases.
Using some software, the same vector field
would look like this.
And let's go and take a look at one more.
Here we have the vector field F of x, y equals x comma y.
And for this one we are going to plot vectors
that have the same magnitude.
So the magnitude of this vector would be the square root
of x squared plus y squared.
We're going to let this equal some constant, c.
Pick some values of c which would tell us the magnitude
of the vectors and then we'll also sketch the level curve
which would be where the initial points
of those vectors would occur.
What I mean by that is if we let c equal one,
that means the magnitude of each vector would be one
and those vectors would occur along the level curve
where x squared plus y squared equals c squared
or one squared or in this case just one.
That's a circle with a radius of one centered
at the origin and all along here each vector
would have a magnitude of one.
So if I start here at the point zero, one.
If x is zero and y is one, this would be our vector.
If x is one and y is zero, this would be our vector.
And at zero, negative one this would be the vector.
And at negative one, zero this would be the vector.
Again, subbing in the coordinates into the formula
for our vector field.
Now let's let c equal two.
If c equals two, that means each vector
would have a magnitude of two and it would occur
along the level curve where x squared plus y squared
equals two squared or four.
Well that's a circle with radius two centered at the origin.
So now we'll plot vectors along this level curve
and each of them will have a magnitude of two.
So if I pick the point right here,
where a 45-degree angle would intersect that circle,
I know that the two legs of the reference triangle
would be equal which is what we want
because the x and the y components are equal to each other.
So I know that I would go out two units here,
along a 45-degree angle.
And the same thing for each 45-degree reference angle
in each quadrant.
The vector would form a 45-degree reference angle.
And each vector has a length of two units.
And we can pick another value for c,
but we're going to go ahead and stop here
and take a look at the computer-generated vector field.
And here it is.
And you can see if we were to sketch the level curves,
the vectors with the same magnitude have
their initial points on a circle centered at the origin.
I hope you found this introduction helpful.
Thank you for watching.