Welcome to a lesson
on determining the potential function
for a conservative vector field.
The reason that determining
a potential function is very useful,
is based upon the fundamental theorem of line integrals.
This theorem states that
if the vector field F is conservative,
then the line integral between any two points
is the difference in values of the potential function,
little f as we see here,
at the two points regardless of the path.
So, this theorem can simplify the process
for evaluating a line integral
if we know the vector field is conservative
and if we can determine the potential function.
So, remember if F is a conservative vector field,
then the components are the partial derivatives of F,
with respects to x, y, and z as we see here.
So, to determine the potential function to help us
evaluate that line integral,
we'll have to integrate each partial derivative
with respects to x, y, and z, as we see here.
And then we'll compare the results to determine
the potential function f.
So, let's go ahead and give it a try.
Again, we're already given that this
is a conservative vector field.
I do have another video that shows how to
test for a conservative vector field.
But since we know this is conservative,
we know that the partial derivative of f
with respects to x is going to be equal to 3x+2y.
And then the partial derivative of f with respects to y
would be equal to 2x-3y.
And so the potential function would be the
original function f of x y.
So if we integrate 3x+2y,
with respects to x,
we should recover the x part of the original function
and if we integrate 2x-3y with respects to y,
we should recover the y part of the original function.
So let's go ahead and give it a try.
Here we have 3 times x squared over 2,
or three halves x squared,
plus 2xy, plus--
now our constant of integration could be a function
in terms of y because we're only recovering the x part
of the original function.
So we'll let the constant of integration be c of y.
Now, let's go ahead and integrate 2x-3y with respects to y.
So we'd have 2xy.
Here we'd have minus 3 times y squared over 2,
or 3 halves y squared.
And in this integral, we're only recovering the y part
of the original function,
so our constant of integration could be a function of x.
Now that we've recovered both the x and the y part
of the original function,
we want to compare these two anti-derivatives
to determine what the actual function would be.
So, f of xy must contain the term three halves x squared.
It must also contain the term 2xy.
And notice this term does occur in both anti-derivatives.
And looking at our second anti-derivative,
we also have a y squared term that we have to include,
so well have minus three halves y squared.
And we still could be missing a constant,
so we'll call it plus c.
This would be our potential function,
that we would use to evaluate a line integral,
if we know the vector field was conservative.
As we know from Calc 1 we could leave off a constant
because that would be eliminated.
Let's go ahead and take a look at a second example.
Here we have a three-dimensional vector field,
and since we're told that it is conservative,
we know the partial derivative of f with respects to x
would be z squared plus 2xy.
The partial derivative of f with respects to y
would be x squared plus 2,
and the partial derivative of f with respects to z
would be 2xz-1.
And now we'll need to set up three integrals
to recover the x, y, and z parts
of the potential function.
So we'll first have the integral of z squared plus 2xy,
with respects to x,
and then we'll have the integral of x squared plus 2,
with respects to y,
and then lastly we'll have the integral of
2xz-1 with respects to z.
So we'll find all three of these anti-derivatives
and then make sure the potential function
has all of the possible terms.
Integrating in respects to x will have xz squared,
integrating 2xy with respects to x,
that'll give us 2 times x squared over 2 times y,
so the 2s would simplify out,
we'd have x squared y,
plus our constant of integration can be a function
in terms of y and z,
so we'll call it c of y, z.
Integrating in respects to y here
will have x squared y,
plus 2y,
and then our constant of integration can be
a function in terms of x and z.
Integrating in respects to z here,
we're going to have 2x times z squared over 2,
or xz squared minus z
plus a function in terms of x and y.
So, our potential function, f of xyz
must contain all of the terms we see here,
we'd just have to watch out for any duplication.
Let's go ahead and start with x squared y,
I see an x squared y term here and here,
so that term must be part of the original function.
Here we have an xz squared term,
and there's another one here,
so let's include that term.
And we must also include 2y minus z.
And then of course our constant of integration,
we'll just call that c.
And again, the significance of this
is if we know the vector field is conservative,
then the line integral between any 2 points
is the difference in values of the potential function,
which we just found,
at these two points regardless of the path.
So, the potential function can make evaluating
a line integral in this case much simpler.
I hope you found this helpful,
thank you for watching.