We're asked to consider the vector field F
to determine if the vector field is conservative.
If it is, find a function little f so that
the vector field F equals the gradient of little f,
which mean f of x comma y would be the potential function,
and then finally use the potential function to evaluate
the line integral along the curve c of F dot differential r,
where the curve c is defined here.
For a quick review, if we know that
a vector field is conservative, it'll make
evaluating a line integral much easier.
The integral of a conservative vector field
is path independent and only depends on the endpoints.
A vector field F is called conservative if it is the
gradient of some scalar function little f, such that
the vector field F equals the gradient of little f.
And again, F is called the potential
function of the vector field f.
The test for a conservative
vector field in a plane is stated here.
If P and Q have continuous partial derivatives
on an open disk R, then the vector field F
with components P comma Q is conservative if
and only if the partial of Q with respect to x
equals the partial of P with respect to y.
So if the given vector field is conservative,
then we'll be able to evaluate the line integral
along the curve c of F dot differential r
using the Fundamental Theorem of Line Integrals.
Let's first determine if the vector field F is conservative.
Well using our notes below, notice how P
is equal to four x plus three y,
and Q is equal to three x plus two y.
So if the given vector field is conservative,
then the partial of Q with respect to x
must equal the partial of P with respect to y.
So the partial of Q with respect to x is equal to the
derivative of three x plus two y with respect to x,
well the derivative of three x with respect to x
would be three, the derivative of two y with respect to x
would be zero because we're treating y as a constant,
and now the partial of P with respect to y would be equal
to the derivative of four x plus three y with respect to y,
well the derivative of four x with respect to y is zero,
the derivative of three y with respect to y is three.
So because the partial derivatives are equal to each other,
we now know that the given vector field is conservative.
So this first answer is yes,
the vector field is conservative.
Now let's work on determining the potential function.
Again, we now know the given vector field is conservative,
we want to find little f of x comma y such that
the vector field F equals the gradient of f.
Remember, the gradient of f would have an x component
that's a partial of f with respect to x,
and a y component that's a partial of f with respect to y,
which means four x plus three y must be equal
to the partial of f with respect to x,
and three x plus two y must be equal
to the partial of f with respect to y.
So to reconstruct the potential function
f of x comma y will integrate, we want to integrate
the x component with respect to x
and the y component with respect to y.
So here we'd have the integral of four x plus three y
integrated with respect to x, and here we'd have
the integral of three x plus two y with respect to y.
So this integral will recover the x part of f,
and this integral will recover the y part of f.
Integrating with respect to x, here we'd have
four times x squared divided by two
plus three y times x, or three x y.
Now here, we're only recovering the x part of the equation,
so we can still be missing y terms in a constant,
so we'll put plus g of y to represent a function of y.
Simplifying, we have two x squared plus three x y
plus g of y, and here we integrate with respect to y,
so the anti-derivative of three x with respect to y
would be three x times y, or three x y,
plus two times y squared divided by two.
Here, we're only recovering the y part of f, so we could
be missing a function of x,
so we'll put plus, let's say, h of x.
Simplifying, we have three x y plus y squared plus h of x.
Now by combining this anti-derivative
and this anti-derivative, we can recover
the potential function f of x comma y.
f of x comma y is going to be equal to,
well here he have two x squared,
notice how both anti-derivatives have the term three x y,
so we got plus three x y, and then we still have
a y squared term here, so plus y squared, and then
plus a constant of integration, let's call it k.
So notice how by combining these two anti-derivatives,
we recover or reconstruct
the potential function f of x comma y.
So going back to our first slide,
we now know the potential function f of x comma y
is equal to two x squared plus three x y plus y squared,
and the plus k is already here for us.
Now we'll use the Fundamental Theorem of Line Integrals
to evaluate this line integral.
The Fundamental Theorem of Line Integrals states
that if we let C be a piecewise smooth curve lying
in an open region R given by the vector function R of t,
if the vector field f is continuous and conservative,
then the line integral along the curve C
of F dot differential r is equal to f of
x of b comma y of b minus f of x of a comma y of a.
Remember, f is the potential function of the vector field F.
So essentially this is telling us that
if the vector field F is conservative,
then the line integral between any
two points is a difference in the values
of the potential function f at these points
regardless of the path used.
So again, here's the given vector field
and here's our potential function.
Our curve C is defined by r of t, shown here,
so notice how x of t equals t squared
and y of t is equal to t cubed.
And t is on the closed interval from zero to three.
And now using the Fundamental Theorem of Line Integrals,
because we know the vector field F is conservative,
this line integral is equal to little f,
the potential function, of x of three comma y of three
minus f of x of zero comma y of zero.
We'll notice how x of three is equal
to three squared, which equals nine, y of three
is equal to three cubed, which equals 27,
and both x of zero and y of zero are equal to zero.
So here we'd have f of nine comma 27
minus f of zero comma zero, so using the potential function,
we would have two times nine squared plus three times nine
times 27 plus 27 squared and then minus,
when both x and y are zero,
notice how all the terms would be zero.
So we'd have zero plus zero plus zero.
Simplifying, two times nine squared, that's 162,
plus, three times nine times 27 is equal to 729,
plus 27 cubed is also 729, which is equal to 1,620.
So notice how by determining the given vector field
was conservative, we were able to evaluate
this line integral by evaluating the potential function
at the endpoint, then the starting point,
and then find the difference of these two function values.
I hope you found this helpful.