[Instructor James Sousa] Welcome to a lesson
on Conservative Vector Fields.
The goals of this video are to
Define a conservative vector field,
Define the curl of a vector field,
and then Determine if a given vector field is conservative.
We will be talking more about conservative vector fields
when we begin to evaluate line integrals.
If we know that a vector field is conservative,
it will make evaluating a line integral much easier.
The integral of a conservative vector field is path
independent and only depends on the endpoints.
But this video will focus on determining if
the given vector field is conservative.
A vector field F is called conservative
if it is the gradient of some scalar function
such that the vector field is equal to the gradient of f.
And the function of little f is called the
potential function of the vector field big F.
Before we determine the test to determine if a given
vector field is conservative
we first need to talk about the curl of a vector field.
And I did make another video that addresses
just the curl of a vector field.
The curl of the vector field F is equal to
the differential operator cross with the vector field F.
So to evaluate this cross product, we're going to evaluate
this three by three determinant
where the first row would be i, j, k.
The second row would be the
partial derivative operators.
The third row would come from the components
of the vector field big F.
So if we evaluate this three by three determinant
we end up with another vector field
where the x component, y component, and z component
ends up being the difference of these partial derivatives.
And the Test for a Conservative Vector Field is
if the curl of F of x, y, z is equal to the zero vector,
the vector field is conservative.
So we go back to that definition of a curl.
If this is equal to the zero vector,
then each of these partial derivatives would have
to be equal to each other
to result in a zero for each component.
So the shortcut rule for the test to determine
if a vector field is conservative,
we could just make sure that
all these partial derivatives here are equal to each other.
Now if the given vector field is in R two
or two dimensional, that three by three determinant
is simplified, and we only have to check to see
if the partial of g with respects to x is equal
to the partial of f with respects to y.
Let's go and take a look at our examples.
We want to determine if the given
vector field is conservative.
Notice this vector field is in R two or two dimensional.
So we only have to determine if the partial of g
with respects to x is equal
to the partial of f with respects to y.
So f is equal to y squared cosine x.
And g is equal to sine of x y.
So we'll first test the partial of g with respects to x
treating y as a constant.
So we'd have cosine x y.
And then if y is a constant, the derivative of x y
would just be y.
So we have y cosine x y.
And now for the partial of f with respects to y
treating x as a constant,
we'll have two y times cosine x.
And obviously these are not equal to each other.
And therefore this vector field is Not Conservative.
Let's go and take a look at our next example.
Here we have another two dimensional vector field.
So again, to see if it's conservative we only have to check
to see if these two partial derivatives are equal
to each other.
Let's rewrite f as
x times the quantity y squared
minus x squared to the negative one half.
And g as
negative y
times the quantity y squared minus x squared
to the negative one half.
Now let's go and check our partial derivatives.
We first want the partial of g with respects to x.
So we'll treat y as a constant, so we'll have negative y
times a negative one half times the quantity
y squared minus x squared to the negative three halves
times the derivative of y squared minus x squared
with respects to x, that's going to give us negative two x.
Let's go ahead and simplify this.
See these two, simplify out.
We have three negatives so that's going to be negative x y
all over
y squared minus x squared
to the positive three halves power.
And now we'll check the partial of f with respects to y.
So we're treating x as a constant.
We're going to have x times negative one half
times y squared minus x squared to the negative three halves
times the derivative of y squared minus x squared
with respects to y.
That's going to give us two y.
Let's go ahead and simplify this.
Again the two's simplify out.
So we have a negative x y
divided by
y squared minus x squared to the three halves power.
So these partial derivatives are equal to each other,
therefore this vector field is conservative.
Let's take a look at a three dimensional vector field, now.
So we need to calculate the curl of this vector field
and see if it's equal to the zero vector
which would be the same as determining if these
three sets of partial derivatives are equal to each other.
So we have f would be equal to y natural log z.
g would be equal to x natural log z.
And h is equal to
x y divided by z.
Let's go and start by determining
these two partial derivatives.
So the partial of h with respects to y
would be x divided by z
and the partial of g with respects to z
would be x divided by z, as well.
So far so good, now let's go ahead and check
these two partial derivatives.
Partial of h with respects to x,
well that would be y divided by z.
And the partial of f with respects to z
well, here's f.
Well that would by y times one over z
or y over z.
So this also checks.
Now we have one more.
The partial of g with respects to x
and the partial of f with respects to y.
So the partial of g with respects to x would be
natural log z.
And the partial of f with respects to y
would also be natural log z.
So all these check,
and therefore the given vector field is conservative.
That's it for this video.
I hope you found it helpful.