Vector Analysis 3: Green’s, Stokes’s, and Gauss’s
Theorems
Thomas Banchoff and Associates
June 17, 2003
1
Introduction
In this final laboratory, we will be treating Green’s theorem and two of its generalizations, the theorems of Gauss and Stokes. These theorems relate vector fields and
integrals - Green’s theorem for vectors in two dimensions, and the other theorems
for vector fields in three dimensions.
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Green’s Theorem
Green’s Theorem relates the line integral over a closed path to a double integral
over the region bounded by that path. Studying Green’s theorem will tie together
concepts you have learned about double integration and line integrals.
Specifically, Green’s Theorem states that for a region
D and a vector field V (x, y) = (p(x, y), q(x, y)), the double integral
ZZ
qx (x, y) − py (x, y)dydx
over D is equal to the line integral
Z
pdx + qdy
over the boundary ∂D of D, for any counter-clockwise oriented parametrization of
the path ∂D.
By counter-clockwise orientation, we mean that as the parameter increases and
the corresponding point moves along the curve, the region will lie on its left-hand
side. This is a very important condition for Green’s Theorem to be correct. To
generate a counter-clockwise parametrization from a clockwise one, you can simply
switch the limits of integration so the curve is traversed in the opposite direction.
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In other words, if we’re given a line integral along a closedR curve C (The curve
must be closed; Green’s Theorem does not apply otherwise.) C pdx + qdy, we can
create a function of two variables based on the vector field (p(x, y), q(x, y)) by letting
f (x, y) = qx (x, y) − py (x,
RRy). The double integral of this function over the region
enclosed in the curve C R qx (x, y) − py (x, y)dydx is then equal to the original line
integral.
This provides a convenient way for evaluating line integrals that may be difficult
in line-integral form. We can rewrite them as double integrals which may be easier
to compute.
Green’s Theorem
Enter the functions in the c(x) and d(x) TypeIns. These will define a region
in the plane. WARNING: Make sure that c(x) ≤ d(x) for all points in your x
domain. We give no guarantees as to accuracy of results (or fnord not crashing) if
this condition does not hold. The boundary of this region consists of the lower and
upper curves c(x) and d(x), and also the two vertical curves where y lies between
c(x) and d(x) at the edges of the x domain (degenerate if c(x) = d(x)).
Also enter a two dimensional vector field in the V TypeIn.
In the Graph of Qx-Py over Region window, the graph of qx −py will be displayed
along with approximating prisms. The represents the surface integral part of Green’s
Theorem. The total volume of the prisms is shown in the Signed vol of prisms over
region Printer.
The Curve with Graph of Integrand window shows a piece of the vector field
with the boundary of the region defined as above. The height function over the
curve is shown in green, with vertical yellow lines representing the partition of its
area. This represents the line integral part of Green’s Theorem. The value of the
line integral is displayed in the Value of the integral on boundary Printer.
Green’s Theorem states that these two values are equal. Compare them to
verify the theorem. Any small discrepancies between the values are a result of our
approximations, not of any inadequacies of the theorem.
Remember, the double integral
ZZ
qx (x, y) − py (x, y)dydx
over D is equal to the line integral
Z
(pdx + qdy)dt
over the boundary ∂D of D, for any parametrization of the path ∂D in terms of a
parameter t].
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This demo illustrates Green’s Theorem for a region between two functions of x
in the plane. Green’s Theorem will work for any bounded region in the plane, but
for simplicity we limit the demonstration to this type ofregion.
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Green’s Theorem and Conservative Vector Fields
Green’s Theorem is more than just an alternate way to evaluate line integrals. It
has many profound implications. One of these involves conservative vector fields.
The curl of a vector field V = (p(x, y), q(x, y), 0) is (0, 0, qx − py ). Its magnitude
is given by
|(0, 0, qx − py )| = (0, 0, qx − py ) · (0, 0, 1) = qx − py .
This is exactly the integrand in Green’s Theorem.
Recall that we showed in lab 7 that a vector field is conservative (i.e. a gradient)
if and only if its curl is identically the zero vector. Therefore, by Green’s Theorem,
if we want to integrate any line integral along a closed curve C over a conservative
vector field V , we find that
Z
ZZ
V · ds =
(qx − py )dydx
C
R
ZZ
=
0dydx
R
= 0
where R is the region bounded by C.
In other words, the line integral along a closed curve over a conservative vector
field is always zero.
Why is this true? Consider a man walking on a surface given by a function of
two variables F (x, y). The path he is taking is the curve C(t). We would like to
find how much work he has to do to get from one point on the surface to another.
Clearly, he will be working hardest when climbing in the steepest direction up, and
gravity will be working for him when he is walking in the steepest direction down.
The gradient of the function ∇F gives the direction of maximum increase of the
function. The amount of work that he is doing is most when he is walking parallel
to the gradient vector and zero when he is walking perpendicular to it. Therefore
his work is given by C 0 (t) · ∇F (x, y).
If he walks for a while and arrives at his starting point (C(t) is closed), we
expect, by the law of Conservation of Energy (hence the name conservative field),
that he will have done no total work. (We count the work that gravity does as
negative work). This is why the line integral is zero.
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You might also realize that, by the same physical law, even if the curve were
not closed, the amount of work he would do would depend only upon his starting
height and his ending height, not upon the actual path that he followed. This is in
fact the case. For any gradient vector field ∇F (x, y),
Z
t=b
Z
t=b
∇F (x, y) · ds =
t=a
Fx dx + Fy dy = F (x(b), y(b)) − F (x(a), y(a)).
t=a
This property is called independence of path.
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Flux and Circulation
Recall the quantities associated with vector fields called curl and divergence from
lab 7. We said that the curl of a vector field is another vector field including the
components of the vectors in the original vector field that point neither towards
nor away from the origin. The divergence, on the other hand, was said to be a
scalar valued function measuring the part of the vectors pointing either directly
away from or directly towards it. Like these two quantities, we will develop new
quanties associated with a curve in the vector field, called circulation and flux.
Suppose that our vector field V (x, y) represents the flow of water on the surface
of a certain lake. We can mark off a portion of the lake with a closed curve C(t)
and call the part of the surface of the lake on the inside of the curve R. In this
system, we might like to measure how much water is produced (for example, by an
underground spring or a waterfall) or lost (for example, into a drain) in R. We might
also like to measure how much of the water in R never leaves it but just continually
flows in circles. These two measures are respectively the flux and the circulation of
the vector field V over the curve C.
One thing we notice is that both flux and circulation can be calculated based
on the behavior of the vector field along the curve, regardless of how it acts inside
the region. If we find how much water crosses C from the inside to the outside and
then subtract the amount that crosses C from the outside to the inside, we have
found the total amount of water that leaves the region without entering it. This
is the flux. On the other hand, if we find how much water goes along C but flows
neither into, nor out of, the region, we have found the circulation.
R
The circulation, therefore, is simply the value of the line integral C V · T ds,
where T is, as usual, the unit tangent vector to the curve.
To find the flux, we do not want to take the line integral of V ·T , which measures
how much of the flow is tangent to, or in the direction of, the curve, but a different
line integral measuring how much of the flow is perpendicular to the curve, so we
can see how much enters and exits the region.
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For this purpose, we define a new vector, the outer unit normal vector, given by
1
N (t) = p
(x0 (t))2
+ (y 0 (t))2
(y 0 (t), −x0 (t))
. Notice that this vector is always perpendicular
R to T (t).
The flux is therefore equal to Rthe value of RR
C V · N ds.
Green’s Theorem states that C V · T ds = (curl V ) · (0, 0, 1)dydx, where the
double integral is over the region R bounded by C. That is, the circulation of V
around the curve C is equal to the integral over the region R of the third component
~ . This is to be expected since the curl measures how much the vector
of the curl of V
field rotates around the origin. When we take the integral, we are in effect adding
all the vectors in the curl to one another. If we think of the vectors pointing
clockwise as being negative, then they will cancel with some of the vectors pointing
counterclockwise. The resulting sum will therefore be the difference between the
counterclockwise and clockwise components of the vectors in V , or the total amount
of rotation in the region, which is what circulation measures.
Circulation and Green’s Theorem
This demonstration allows you to compare F · T , F · T ds
dt , and Py − Qx .
The Curve and Vector Field window shows a vector field F and an ellipse C in
the plane. At sampled points of the ellipse the tangential component F · T of F , a
vector along the tangent with length the magnitude of the dot product, is displayed.
In the Graph of (F.T)*ds/dt vs t window the graph of F · T ds
dt as a function of t
is displayed. A third display window shows the graph of the function Py − Qx , the
third component of the curl of
F over the domain in the plane. Green’s Theorem states that the value of the
line integral of
F · T ds
dt , which is the circulation, is equal to the signed volume under the graph
of the signed magnitude of the curl over
R.
This demonstration shows Green’s Theorem applied to circulation.
Exercise 1
Study the graphs for a constant vector field.
Exercise 2
Study the graphs for the position vector field
F (x, y) = (x, y) .
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Exercise 3
Study the graphs for the field obtained by rotating the position vector ninety
degrees in the clockwise direction, that is,
F (x, y) = (−y, x) .
Exercise 4
Study the graphs for the field obtained by rotating the position vector ninety
degrees in the clockwise direction and then dividing by the square of the length,
that is,
F (x, y) = (−y,x)
. You can enter this field in fnord as
x2 +y 2
[x,y]-¿(1/x*x+y*y)*[-y,x] if (x!=0 and y!=0) else [0,0]
Substituting N for T in the formulation of Green’s Theorem and making all
corresponding
changes
R
R R will show that it equivalently states that
divV dydx , where the double integral is over the region
C V · N ds =
R bounded by
C . That is, the flux of
V across
C is equal to the integral over the region
R of the divergence of
V . This, too, is to be expected. The integral takes the sum of the divergence
over the entire region. Therefore it subtracts the parts of the vectors in the field
pointing towards the origin from those pointing away. In a continuous vector field,
these should be equal unless water is generated somewhere or lost somewhere, in
which case it measures how much. This is the flux.
Flux and Green’s Theorem
This demonstration allows you to compare F · N ,
F · N ds
dt , and
Px + Qy .
The Curve and Vector Field
window shows a vector field
F and an ellipse
C in the plane. At sampled points of the ellipse the normal component
F · N of
F , a vector along the outward-pointing normal with length the magnitude of
the dot product, is displayed. In the
Graph of (F.N)*ds/dt vs t
window the graph of
F · N ds
dt as a function of
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t is displayed. A third display window shows the graph of the function
Px + Qy , the divergence of
F over the domain in the plane. Green’s Theorem states that the value of the
line integral of
F · N ds
dt , which is the flux, is equal to the signed volume under the graph of the
divergence over
R.
This demonstration shows Green’s Theorem applied to flux.
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The Divergence Theorem
Two theorems extend these interpretations of Green’s Theorem to three dimensions.
To study them, we first need to extend our idea of a line integral to three dimensions.
A parametrized curve in three space is given by the vector-valued function C(t) =
(x(t), y(t), z(t)). This definition acts analogously to our previous definition of a
parametrized curve in the plane (cf. lab 8). Associated with each t = t0 is a point
C(t0 ) = (x(t0 ), y(t0 ), z(t0 )), which is a point on the space curve.
C 0 (t)
This curve has a unit tangent vector given by T (t) = |C
0 (t)| . We define the unit
0
normal vector to the curve by |TT 0 (t)
(t)| .
The line integral over a three dimensional vector field V (x, y, z) along this curve
is given by
Z
Z b
V · T ds =
V (C(t)) · C 0 (t)dt
C
a
Here, ds = |C 0 (t)|dt = (x0 (t))2 + y 0 (t))2 + z 0 (t))2 )dt. This will be of use in the
next section on Stokes’s Theorem.
Another extension of line integrals to three-space is the surface integral. For any
function f (x, y) of two variables, the vector valued function F (x, y) = (x, y, f (x, y))
defines a surface which we will call S. A normal vector to a surface at a point is
one that is perpendicular to the tangent plane to the surface at that point. In other
words, it points directly out of the surface.
Since we know that the cross product of two vectors is perpendicular to each, if
we can find two vectors in the tangent plane, their cross product will give a vector
normal to the surface. Fortunately, we do know two such vectors, those given by the
partial derivatives of the vector-valued function, which are actually used to define
the tangent plane (cf. lab 2).
We define the unit normal vector to a surface F (x, y) (not to be confused with
p
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the normal to a curve) by
N (x, y) =
(1, 0, fx (x, y)) × (0, 1, fy (x, y))
Fx (x, y) × Fy (x, y)
= p
|Fx (x, y) × Fy (x, y)|
1 + (fx (x, y))2 + (fy (x, y))2
The extension of ds, the arc-length element, is dS, the surface-area element, given
by
dS = |Fx (x, y) × Fy (x, y)|dyds = 1 + (fx (x, y))2 + (fy (x, y))2 dydx
Now we have enough information to define the surface integral over a three
dimensional vector field V (x, y, z) along the surface F (x, y) = (x, y, (f (x, y)). It is
Z Z
S
Z bZ
Z Z
V ·dS =
V ·N dS =
S
d
V (F (x(t), y(t))·(Fx (x(t), y(t))×Fy (x(t), y(t)))dt.
a
c
If V (x, y, z) = (p(x, y, z), q(x, y, z), r(x, y, z)), then this integral may also be written
in the form
Z Z
p(x, y, z)dydz + q(x, y, z)dzdx + r(x, y, z)dxdy
S
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Optional Material
We may also compute surface integrals along more generally defined surfaces given
by parameters in curvilinear coordinate systems.
The main advantage of defining a surface parametrically instead of as a function of
two variables is exactly that we are not limited to functions. A sphere, for example,
is not a function in any direction (i.e. we cannot pick a direction such that every
line in that direction will intersect the sphere at most once.)
Given a pair of independent variables, u and v, and a domain, a ≤ u ≤ b,
c ≤ v ≤ d, we can define a parametric surface S in three-space by F (u, v) =
(x(u, v), y(u, v), z(u, v)). For any point (u0 , v0 ) in the domain, F (u0 , v0 ) will be a
point in three-space and the union of these points will be a surface.
We define the unit normal vector to this surface as
N (u, v) =
Fu × Fv
|Fu × Fv |
This vector will be perpendicular to the tangent plane of the surface at any given
point.
The surface integral over a vector field V along this surface is given by
Z Z
V · N dS.
S
Here dS, the surface area element, is given by |N (u, v)|dvdu.
You should note that if we set x(u, v) = u, y(u, v) = v, and z(u, v) = f (u, v) where
f is some function of two variables, our surface F is reduced to a function graph.
General Parametrized Surfaces
A parametrized surface may be selected in the Function Control window. If the Use
Predefined Function
CheckBox is on, a predefined function will be used. You can select one with the
Function Selector Slider. Each value this Slider takes on is associated with a different
surface. The name of the selected surface is shown in a small window. If the
Use Predefined Function CheckBox is off, you may enter your own function in the
Function Type In Type-In.
The domain, enetered in the Domain Control window, is displayed in the The Domain window.
The parametrized surface is displayed in the Surface window.
This is a demonstration of general parametrized surfaces. Try out different functions
and, for each, vary the domain to see how the surface changes.
One way to generalize Green’s Theorem to vector fields in three-dimensional
space is known Gauss’s Theorem, or the Divergence Theorem.
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Given a vector field V (x, y, z) = (p(x, y, z), q(x, y, z), r(x, y, z)) defined over a
domain in three-space, and a surface S given by F (x, y) = (x, y, f (x, y)), with
a ≤ x ≤ b and c ≤ y ≤ d, the divergence theorem states that
ZZ
ZZ Z
V · N dS =
divV dzdydx
W
, where the double integral is over the surface
S,
dS represents an area element on that surface,
W is the three-dimensional region bounded by
S, and N represents the unit normal vector of S.
This is a direct extension of the version of Green’s Theorem using the normal
vector to the curve. Gauss’s Theorem equates the integral of the divergence over
the region in space with the surface integral along its boundary. Both measure the
flux of the vector field across the boundary, where the meaning of flux is analogous
to the two-dimensional case.
The Divergence Theorem
This is a demonstration of the divergence theorem. It shows the surface of
a sphere, the center and radius of which we can manipulate with a Type-In and
a Slider, respectively. The sphere is displayed in yellow. The vector field can be
displayed in the domain, and we show the push-off of the sphere along the vector field
V in red. We also show the push-off along the normal vectors scaled by the normal
component of the field V, in green. Several integrals are calculated and printed. The
volume of the sphere, the volume inside the surface created by pushing off along the
field vectors, and the volume inside the surface created by pushing off along the
normal vectors a distance equal to the normal component of V are shown in the
input window.
The two integrals that we are most interested in are displayed at the bottom of
the input window. The integral of
div(F )dV over the sphere and the integral of
F · N dS over the sphere are printed. The divergence theorem says that these
last two values are equal.
The volume integral is simply the integral of the divergence function over the
region inside the sphere. The surface integral is calculated by multiplying the areas
of the small quadrilaterals into which the surface of the sphere is divided by the
lengths of the components of the vector field in the direction of the normals to the
surface and taking the sum of all of these quantities.
Note: There might be an inconsistency between the values of the Volume Integral
of Divergence and the Surface Integral of Normal Component of the Field. This is
because of errors in numerical calculation and not because they are different values.
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They should, however, be fairly close.
This is a demonstration of the divergence theorem limited, for simplicity, to the
case where the surface is a sphere.
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Stokes’s Theorem
The other important extension of Green’s Theorem to three dimensions is Stokes’s
Theorem. It states that for a surface
S given by F (x, y) = (x, y, f (x, y)) bounded by a curve
C(t) = (x(t), y(t), z(t)) and a vector field
V (x, y, z) = (p(x, y, z), q(x, y, z), r(x, y, z)) , all in three-space, the value of the
line integral of
V · T with respect to arclength along the boundary curve
C is equal to the integral over the region of the normal component of the curl of
V with respect to area on the surface. More specifically, it states that
Z
Z Z
V · T ds =
(curlV ) · N dS
C
S
, where the double integral is over the surface
S , and
dS is an area element for the surface, and
N is the outward-pointing unit normal of that surface. Another way of saying
this is that the circulation of
V around
C is equal to the flux of the curl of
V through any surface bounded by
C . It may be difficult to comprehend exactly what this means in terms of the
behavior of the vector field, but the remarkable thing is that the circulation along
C can be calculated by determining the flux through any surface bounded by
C . Since we have our choice of surface, we usually use something simple like
part of a sphere or a region in the plane.
You should realize that Green’s Theorem is really just the projection of Stokes’s
Theorem into the plane. If we take our surface S to be the xy-plane then it is given
by F (x, y) = (x, y, 0). The unit normal to the surface is then
Fx (x, y) × Fy (x, y)
= (0, 0, 1).
|Fx (x, y) × Fy (x, y)|
The surface area element is then simply given by dS = 1dydx = dydx.
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The curve , too, we will take to be in the plane. Stokes’s Theorem gives
Z
Z Z
V · T ds =
(curlV ) · (0, 0, 1)dydx,
C
S
which is a version of Green’s Theorem.
Stokes’s Theorem
This is a demonstration of Stokes’s Theorem. The important quantities to observe are given in the Integral of Tangential Component and Integral of Normal
Comp of Curl
Printers. Stokes’s Theorem states that these are equal.
The first of these is a line integral, whe geometric interpretation of which is
shown in the V.T vs s window. The area of the region shown in red is the value
of the integral. The height function is generated by going around the curve and
calculating the quantity V · T , where V is the vector field and T is the unit tangent
to the curve. Thus it is the tangential component of the curve in the vector field.
The second quantity is a surface integral. To see what it measures, turn on the
Do Ellipse and Surface , Curl(V).N Push-Off Vecs and
Curl(V).N Push-Off Surf CheckBoxes adn turn off all others. Also set the
Amount of Surface Slider to its maximum.
The orange surface is the actual surface, divided into quadrilaterals. The blue
lines represent vectors that are generated by taking the normal vector N to the
surfce at each point, and moving along it away from the surface an amount equal
to (curlV ) · N . The green surface is just the union of the heads of those vectors, so
they are easier to see. To calculate the surface integral, we take the area of each of
the quadrilaterals on the surface and multiply it by the length of the vector with its
tail at the lower left corner of the quadrilateral. We then add all of these quantities
together.
Note: Neither of the integrals being calculated in this demonstration are perfect
and thus will have some error depending on the way in which they were calculated.
Thus, they are only approximations and shouldn’t be seen as exact.
This demo is somewhat complicated, but if you take it step-by-step it can help
you see what is happening with Stokes’s Theorem.
In this demo we will be dealing with a curve
C in space, a vector field
V in space, and a surface
S (also in space) bounded by the curve
C. First, let’s note how we obtain the curve
C. To obtain this curve, we first define an ellipse in the plane (the demo lets you
enter the center of the ellipse and the ratio of its major to its minor axis). Then,
we take a function graph
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f that maps R2 7→ R. The image of the ellipse under
f will be a curve in 3-space: that is our
C. This does not allow
C to be any possible curve in 3-space, but rather only curves whose projection
into the
xy-plane is an ellipse. However, it gives us a fairly wide set of curves to work
with: playing with
f and with the shape and center of the ellipse will give you a large variety of
curves!
Now, we can also use
f to define our surface
S. Specifically, if we take the region in the
xy-plane bounded by our ellipse, and apply
f to it, the image will be a surface in space bounded by
C. Note that by varying
f , while keeping the same ellipse, you are choosing different surfaces bounded
by the same
C. Stokes’s Theorem holds for any surface bounded by
C, so it will hold for any of these various surfaces.
We now have our
C and our
S, and you define
V . Thus, we have the basics for investigating Stokes’s Theorem.
The first thing that we will look at is the tangential component of
V along
C. At any point
C(t) on
C, this is a vector in the direction of the tangent
T (t) to
C(t) with magnitude equal to the absolute value of
V (t) · T (t) (if this expression is negative, then the direction is opposite to the
direction of the
T (t)). This expression - the signed magnitude of the tangential component - is
what we integrate along the curve in the line integral part of Stokes’s Theorem.
The next thing we will look at is the curl of
V along
C. (Remember, the curl of a vector field is another vector field.) For Stokes’s
Theorem, the curl is only an intermediate step. We have to do something else to it.
What we have to do with the curl is look at its component along the normal
vector to the surface
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S (remember, this surface is just the image of
f applied to the region bounded by the ellipse in the plane. It’s a plain old
function graph and therefore has a normal vector at each point). This component
is a vector in the direction of the normal vector
N with magnitude equal to the absolute value of
curl(V ) · N (again, if this expression is negative it reverses the direction of the
vector). It is that expression (without the absolute value) that that we integrate
as the surface integral part of Stokes’s Theorem. It is really a measure of the area
between the original space curve and the space curve offset along the vector field by
a distance equal to curl(V ) · N .
So, what does Stokes’s Theorem say? That the integral over
C of
V · T ds is equal to the integral over
S of
V · N dS. To put it another way, the circulation of
V around
C is equal to the flux of the curl of
V through the surface bounded by
C.
Stokes’s Theorem states that the circulation of F around the curve C is the flux
of the curl of F through the surface bounded by C.
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