Vector Algebra
Transcript of Vector Algebra
Vector Calculus ENEL2FT Field Theory 1
ENEL2FTFIELD THEORY
REFERENCES
1. M.N. Sadiku: Elements of Electromagnetics, Oxford University Press, 1995, ISBN 0-19-510368-8.
2. N.N. Rao: Elements of Engineering Electromagnetics, Prectice-Hall, 1991, ISBN:0-13-251604-7.
3. P. Lorrain, D. Corson: Electromagnetic Fields and Waves, W.H. Freeman & Co, 1970, ISBN: 0-7167-0330-0.
4. David T. Thomas: Engineering Electromagnetics, Pergamon Press, ISBN: 08-016778-0.
Vector Calculus ENEL2FT Field Theory 2
VECTOR CALCULUS VECTOR ALGEBRA Electromagnetics deals with the study of electric and magnetic
fields. Therefore one needs to understand the concepts of a field. Electric and Magnetic Fields are vector quantities and their
behaviour is governed by a set of laws known as Maxwell’s equations.
The mathematical formulation of Maxwell’s equations and their subsequent application require the understanding of the basic rules pertinent to mathematical manipulations involving vector quantities.
We first introduce simple rules of vector algebra without the manipulation of the coordinate system; thereafter, we introduce the Cartesian, cylindrical, and spherical coordinate systems.
After introducing the vector algebraic rules, we introduce the concepts of scalar and vector fields, static as well as time-varying.
Vector Calculus ENEL2FT Field Theory 3
VECTOR CALCULUS VECTOR ALGEBRA In the study of elementary physics, we come across several
quantities such as mass, temperature, velocity, acceleration, force, and charge.
Some of these quantities have associated with them not only a magnitude, but also a direction in space whereas others are characterized by magnitude only.
The former class of quantities are known as vectors, and the latter class are known as scalars.
Mass, temperature, and charge are scalars, whereas velocity, acceleration, and force are vectors.
Other examples are voltage and current for scalars , and electric and magnetic fields for vectors.
Vector Calculus ENEL2FT Field Theory 4
VECTOR CALCULUS VECTOR ALGEBRA Graphically, a vector, A, is represented by a straight line with an arrowhead
pointing in the direction of A and having a length proportional to the magnitude of A.
If the top of the page represents North, then vectors A and B are directed eastward, with the magnitude of B being twice that of A.
Vector C is directed towards the northeast and has a magnitude three times that of A. Vector D is directed towards the southwest, and has a magnitude equal to that of C. Thus C and D are equal in magnitude but opposite in phase.
A
B
CD
Vector Calculus ENEL2FT Field Theory 5
VECTOR CALCULUS VECTOR ALGEBRA - THE UNIT VECTOR Since a vector may have, in general, an arbitrary orientation in three-
dimensional space, we need to define a set of three reference directions at each and every point in space in terms of which we can describe vectors drawn at that point.
Thus if we define three mutually orthogonal reference directions as shown below, and direct unit vectors along the three directions as shown, where a unit vector has magnitude of unity.
i1
i2
i3
Set of three orthogonal unit vectors in a right-handed coordinate system.
Vector Calculus ENEL2FT Field Theory 6
VECTOR CALCULUS VECTOR ALGEBRA A vector of magnitude different from unity along any reference
directions can be represented in terms of the unit vector along that direction.
Thus 4i1 represents a vector of magnitude 4 units in the direction of i1, 6i2 represents a vector of magnitude 6 units in the direction of i2, and -2i3 represents a vector of magnitude 2 units in the direction opposite to that of i3.
Thus we define vector A as the sum of 4i1+6i2. That is:
The magnitude of vector A is given by:
21ˆ6ˆ4 iiA
211.764 22 A
Vector Calculus ENEL2FT Field Theory 7
VECTOR CALCULUS VECTOR ALGEBRA If vector B is defined as:
then the magnitude of B is:
In general, a vector A is said to have components A1, A2, and A3 along the directions 1, 2, and 3 is written as:
Now consider three vectors A,B, and C given by:
321 2ˆ6ˆ4 iiiB
4833.72642ˆ6ˆ4 222321 iiiB
332211ˆˆ iAiAiAA
332211
332211
332211
ˆˆ
ˆˆ
ˆˆ
iCiCiCC
iBiBiBB
iAiAiAA
Vector Calculus ENEL2FT Field Theory 8
VECTOR CALCULUS VECTOR ALGEBRA - Vector Addition, Subtraction, Multiplication Then the sum of vectors A and B, (A+B), is given by:
Vector subtraction is a special case of vector addition; thus:
The multiplication of a vector, A, by a scalar m, is the same as repeated addition of the vector:
33322111
332211332211
ˆˆˆ
ˆˆˆˆ
2 iBAiBAiBA
iBiBiBiAiAiABA
33322111
332211332211
ˆˆˆ
ˆˆˆˆ
2 iCBiCBiCB
iCiCiCiBiBiBCB
332211
332211
ˆˆ
ˆˆ
imAimAimA
iAiAiAmAm
Vector Calculus ENEL2FT Field Theory 9
VECTOR CALCULUS VECTOR ALGEBRA The magnitude of vector A is given by:
The unit vector along A , iA, has a magnitude equal to unity, but its direction is the same as that of A. Thus:
Two vectors A and B are equal if and only if the corresponding components of A and B are equal. That is:
23
22
21332211
ˆˆ AAAiAiAiAA
33
22
11 i
A
Ai
A
Ai
A
A
A
AiA
33;22;11
;332211332211ˆˆˆˆ
BABABA
iBiBiBiAiAiABA
Vector Calculus ENEL2FT Field Theory 10
VECTOR CALCULUS VECTOR ALGEBRA - SCALAR OR DOT PRODUCT The scalar or dot product of two vectors A and B is a scalar quantity defined
as:
Here is the angle between A and B. For mutually orthogonal unit vectors i1, i2, and i3, we have:
Thus we have the dot product between A and B as:
BA
BABABA
.coscos. 1
1.;0.;0.
0.;1.;0.
0.;0.;1.
33233
32222
32
1
1
1111
iiiiii
iiiiii
iiiiii
33211
33.3322.3311.33
33.2222.2211.2233.1122.1111.11
332211332211
2
ˆˆˆˆˆˆ
ˆˆˆˆˆˆˆˆˆˆˆˆ
ˆˆ.ˆˆ.
BABABA
iBiAiBiAiBiA
iBiAiBiAiBiAiBiAiBiAiBiA
iBiBiBiAiAiABA
Vector Calculus ENEL2FT Field Theory 11
VECTOR CALCULUS VECTOR ALGEBRA - VECTOR OR CROSS PRODUCT The vector or cross product of two vectors, A and B, is a vector quantity
whose magnitude is equal to the product of the magnitudes of A and B and the sine of the smaller angle between A and B whose direction is normal to the plane containing A and B.
For mutually orthogonal unit vectors i1, i2, and i3, we have:
Note that the cross-product is not commutative, and also the distributive property holds for the cross product:
NiBABxA ˆsin
0;;
;0;
;;0
3312323
1322232
2332
1
1
1111
xiiixiiixii
ixiixiiixii
ixiiixiixii
CxABxACBxA
BxAiABAxB N
sin
Vector Calculus ENEL2FT Field Theory 12
VECTOR CALCULUS VECTOR ALGEBRA - VECTOR OR CROSS PRODUCT Using the above properties, we obtain:
This can be expressed in determinant form in the manner:
The cross product is useful in obtaining the unit vector normal to the plane containing the two vectors A and B:
122131132332
123213132312231321
333322331133
332222221122331122111111
332211332211
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆˆˆˆˆˆˆ
ˆˆˆˆ
BABABABABABA
iBAiBAiBAiBAiBAiBA
iBiAiBiAiBiA
iBiAiBiAiBiAiBiAiBiAiBiA
iBiBiBxiAiAiABxA
xxx
xxxxxx
321
321
321ˆˆˆ
BBB
AAA
iii
BxA
BxA
BxAiN
Vector Calculus ENEL2FT Field Theory 13
VECTOR CALCULUS VECTOR ALGEBRA - TRIPLE PRODUCTS The scalar triple product involves three vectors in a dot product operation
and a cross product operation, such as, A.BxC. It is not necessary to include parentheses since this quantity can be
evaluated in only one manner - by evaluating BxC first, and then dotting the resulting vector with A.
We therefore have,
Since the value of the determinant on the right side remains unchanged if the rows are interchanged in a cylindrical manner, we have
321
321
321
321
321
321
332211
.
ˆˆˆ
.ˆˆˆ.
CCC
BBB
AAA
CxBA
CCC
BBB
iii
iAiAiACxBA
BxACAxCBCxBA
...
Vector Calculus ENEL2FT Field Theory 14
VECTOR CALCULUS VECTOR ALGEBRA - TRIPLE PRODUCTS The triple cross product involves three vectors in two cross product
operations. Caution must however be exercised in evaluating a triple cross
product since the order of evaluation is important; that is:
As an example, let us equate the three vectors to unit vectors as follows:
Therefore in a vector triple product, the parentheses are so important and must be included.
CxBxACxBxA
0ˆ0ˆˆˆ
ˆˆˆˆˆˆ
ˆ;ˆ;ˆ
2211
231211
211
ixixixiCxBxA
iixiixixiCxBxA
iCiBiA
Vector Calculus ENEL2FT Field Theory 15
VECTOR CALCULUS CARTESIAN COORDINATE SYSTEM So far, we have expressed a vector at a point in space in terms of its
component vectors along a set of three mutually orthogonal directions defined by three mutually orthogonal unit vectors at that point.
However, in order to relate vectors at one point in space to vectors at another point in space, we must define the set of three reference directions at each and every point in space. Thus we need a coordinate system.
Although there are several different coordinate systems, we are normally concerned with only three of these, namely, the Cartesian, cylindrical, and spherical coordinate systems.
The Cartesian coordinate system, also known as the rectangular coordinate system, is the simplest of the three since it permits the geometry to be simple, yet sufficient to study many of the elements of engineering electromagnetics.
Vector Calculus ENEL2FT Field Theory 16
VECTOR CALCULUS CARTESIAN COORDINATE SYSTEM The Cartesian coordinate system is defined by a set of three
mutually orthogonal vectors, x,y, and z, as shown below.
The coordinate axes are denoted as the x-, y-, and z-axes. The directions in which values of x, y, and z increase along the
respective coordinate axes are indicated by the arrowheads. Note that the positive x-, y-, and z-directions are chosen such that
they form a right-handed system.
i1=x
i2=y
i3=z
Vector Calculus ENEL2FT Field Theory 17
VECTOR CALCULUS CARTESIAN COORDINATE SYSTEM Therefore we have:
For a right-handed coordinate system, we have:
Consider two points, P1(x1,y1,z1), and P2 (x2,y2,z2) in the rectangular coordinate system. The position vector, r1, drawn from the origin to pint P1 and position vector r2 drawn from the origin to P2 are given by:
The resultant vector, R12, is given by:
ziyixi ˆˆ;ˆˆ;ˆˆ321
yxxzxzxyzyxx ˆˆˆ;ˆˆˆ;ˆˆˆ
zzyyxxr
zzyyxxr
ˆˆˆ
ˆˆˆ
222
111
2
1
zzzyyyxxxrrR ˆˆˆ 1212121212
Vector Calculus ENEL2FT Field Theory 18
VECTOR CALCULUS CARTESIAN COORDINATE SYSTEM
We can obtain the unit vector along the line drawn from P1 to P2 to be:
As an example, if P1 is (1,-2,0) and P2 is (4,2,5), then:
x
y
z
P1(x1,y1,z1)
P2(x2,y2,z2)
r1 r2
R12
2/12
122
122
12
121212
12
1212
ˆˆˆˆzzyyxx
zzzyyyxxx
R
Ri
Vector Calculus ENEL2FT Field Theory 19
VECTOR CALCULUS CARTESIAN COORDINATE SYSTEM
In our study of electromagnetic fields, we have to work with line, surface, and volume integrals.
These involve differential lengths, surfaces, and volumes obtained by incrementing the coordinates by infinitesimal amounts.
Since in the Cartesian coordinate system the three coordinates represent lengths, the differential length elements obtained by incrementing one coordinate at a time, keeping the other two constant, are, for the x-, y-, and z-coordinates respectively:
zyxi
zyxR
ˆ5ˆ4ˆ325
1ˆ
ˆ5ˆ4ˆ3
12
12
zdzandydyxdx ˆ,ˆ,ˆ
Vector Calculus ENEL2FT Field Theory 20
VECTOR CALCULUS DIFFERENTIAL LENGTH VECTOR
The differential length vector, dl, is the vector drawn from a point P(x,y,z) to a neighboring point Q(x+dx,y+dy,z+dz) obtained by incrementing the coordinates of P by infinitesimal amounts.
Thus it is the vector sum of the three differential elements as follows:
x
y
z
P(x,y,z)
Q(x+dx,y+dy,z+dz)
r1 r2
dl
zdzydyxdxld ˆˆˆ
Vector Calculus ENEL2FT Field Theory 21
VECTOR CALCULUS DIFFERENTIAL LENGTH VECTOR The differential lengths, dx, dy, and dz are, however, not independent
of each other since in the evaluation of line integrals, the integration is performed along a specified path on which the points P and Q lie.
As an example, consider the curve
Let us obtain the expression for the differential length vector dl along the curve at the point (1,1,1) and having the projection dz on the z axis. Then:
Note that x=1, y=1, z=1.
2zyx
dzzyxzdzydzxdzld
zdzydyxdxld
zdzdydx
ˆˆ2ˆ2ˆˆ2ˆ2
ˆˆˆ
2
Vector Calculus ENEL2FT Field Theory 22
VECTOR CALCULUS DIFFERENTIAL LENGTH VECTOR Differential length vectors are useful for finding the unit vector normal
to a surface at a point on that surface. This is done by considering two differential length vectors at the point
under consideration and tangential to the two curves on the surface then using the cross-product operation, which gives a vector that is normal to the crossed vectors.
Thus the unit vector normal to the surface is given by:
dl1
dl2
Surface
Curve 1
Curve 2
21
21ˆlxdld
lxdldin
Vector Calculus ENEL2FT Field Theory 23
VECTOR CALCULUS DIFFERENTIAL SURFACE VECTOR Two differential length vectors, dl1 and dl2 originating at a point define
a differential surface whose area dS is that of the parallelogram having dl1 and dl2 as two of its adjacent sides, as shown below:
From simple geometry and the definition of cross-product of two vectors, it can be seen that:
dS
dl1
dl2
in
2121 sin lxdlddldldS
Vector Calculus ENEL2FT Field Theory 24
VECTOR CALCULUS DIFFERENTIAL SURFACE VECTOR In the evaluation of surface integrals, it is convenient to define a
differential surface vector dS whose magnitude is the area dS and whose direction is normal to the differential surface.
Thus recognizing that the normal vector can be directed to either side of the surface, we have:
If we apply these equations to differential surface vectors in Cartesian coordinates, we obtain:
nidSlxdldSd ˆ21
zdxdyyxdyxdxtconszplaneFor
ydzdxxxdxzdztconsyplaneFor
xdydzzxdzydytconsxplaneFor
ˆˆˆ:tan
ˆˆˆ:tan
ˆˆˆ:tan
Vector Calculus ENEL2FT Field Theory 25
VECTOR CALCULUS DIFFERENTIAL VOLUME Three different length vectors, dl1, dl2, and dl3 originating at a point
define a differential volume dv which is that of the parallelepiped having dl1, dl2, and dl3 as three of its contiguous edges, as shown below.
It can be seen that:
dl1
dl2
dl3
dv
321
213
21
213
21321
.
..
ˆ.
)).((
lxdldlddv
lxdldldlxdld
lxdldldlxdldildlxdld
ipedparallelepofheightipedparallelepofareabasedv
n
Vector Calculus ENEL2FT Field Theory 26
VECTOR CALCULUS CYLINDRICAL COORDINATE SYSTEM Just like the Cartesian coordinate system is defined by a set of three
mutually orthogonal surfaces, the cylindrical coordinate system also involves a set of three mutually orthogonal surfaces.
For the cylindrical coordinate system, the three one of the planes is z=constant
x
y
z
P(r,,z)
r
Vector Calculus ENEL2FT Field Theory 27
VECTOR CALCULUS CYLINDRICAL COORDINATE SYSTEM One of these planes is the same as the z=constant plane in the
Cartesian coordinate system. The second plane contains the z-axis and makes an angle with a
reference plane, chosen to be the x-z plane of the Cartesian coordinate system. This plane is called the =constant plane.
The cylindrical coordinate system has the z-axis as its axis. But since the radial distance r from the z-axis to points on the cylindrical surface is constant, this surface is defined by r=constant.
Thus the three orthogonal surfaces defining the cylindrical coordinate system are: r=constant; =constant; and z=constant.
Only two of the coordinates (r and z) are distances; the third coordinate () is an angle.
We note that the entire space is spanned by varying r from 0 to ; z from - to +; and from 0 to 2.
Vector Calculus ENEL2FT Field Theory 28
VECTOR CALCULUS CYLINDRICAL COORDINATE SYSTEM
To obtain the expressions for the differential lengths, surfaces, and volumes in the cylindrical coordinate system, we now consider two points, P(r,,z) and Q(r+dr, +d, and z+dz) where Q is obtained by incrementing infinitesimally each coordinate from its value at P.
The three orthogonal surfaces intersecting at P, and the three orthogonal surfaces intersecting at Q, define a box which can be considered to be rectangular since dr,d, and dz are infinitesimal.
dz
rdr
dx
y
z
P
Q
Vector Calculus ENEL2FT Field Theory 29
VECTOR CALCULUS CYLINDRICAL COORDINATE SYSTEM The three differential length elements forming the contiguous sides of
the box are: The differential length vector dl from P to Q is thus given by:
The differential surface vectors defined by the pairs of the differential length elements are:
Finally, the differential volume dv is the volume of the box:
zdzrdrdr ˆ,ˆ,ˆ
zdzrdrdrld ˆˆˆ
zrdrdxrdrdr
drdzrxdrzdz
rdzrdzxdzrd
ˆˆˆ
;ˆˆˆ
;ˆˆˆ
dzrdrddzrddrdv
Vector Calculus ENEL2FT Field Theory 30
VECTOR CALCULUS SPHERICAL COORDINATE SYSTEM For the spherical coordinate system, the three mutually orthogonal
surfaces are a sphere, a cone, and a plane.
The three orthogonal surfaces defining the spherical coordinates of a point are:
r
x
y
z
tcons
tcons
tconsr
tan
tan
tan
Vector Calculus ENEL2FT Field Theory 31
VECTOR CALCULUS SPHERICAL COORDINATE SYSTEM The differential length elements and the differential length vector dl are
given by:
The differential surface vectors defined by pairs of differential length elements are:
The differential volume formed by the three differential lengths is:
ˆsinˆˆ
ˆsin,ˆ,ˆ
drrdrdrld
drrdrdr
ˆˆˆ
ˆsinˆˆsin
ˆsinˆsinˆ 2
rdrdxrdrdr
drdrrxdrdr
rddrdxrrd
ddrdrdrrddrdv sinsin 2
Vector Calculus ENEL2FT Field Theory 32
VECTOR CALCULUS CONVERSIONS BETWEEN THE COORDINATE SYSTEMS In the study of electromagnetics, it is useful to be able to convert from
one coordinate system to another, particularly from the Cartesian to the cylindrical system and vice-versa, and from the spherical system to the Cartesian system and vice-versa.
If rc is r in cylindrical coordinate, and rs is the designation of r in spherical coordinates, then we have the following conversions:
x
y
z
yxzyxr
zzx
yyxr
rzryrx
zzryrx
s
c
sss
cc
122
1222
122
tantan
tan
cossinsin;cossin
;sin;cos
Vector Calculus ENEL2FT Field Theory 33
VECTOR CALCULUS CONVERSIONS BETWEEN THE COORDINATE SYSTEMS Next consider the conversion of vectors from one coordinate system to
another. To do this, we need to express each of the unit vectors of the first
coordinate system in terms of its components along the unit vectors in the second coordinate system.
From the definition of the dot product of two vectors, the component of a unit vector along another unit vector, that is, the cosine of the angle between the two unit vectors, is simply the dot product of the two unit vectors.
For the sets of unit vectors in the cylindrical and Cartesian coordinate systems, we have:
1ˆ.ˆ0ˆ.ˆ0ˆ.ˆ
0ˆ.ˆcosˆ.ˆsinˆ.ˆ0ˆ.ˆsinˆ.ˆcosˆ.ˆ
zzyzxz
zyx
zryrxr ccc
Vector Calculus ENEL2FT Field Theory 34
VECTOR CALCULUS CONVERSIONS BETWEEN THE COORDINATE SYSTEMS Similarly, for the set of unit vectors in the spherical and Cartesian
coordinate systems, we obtain the dot products as follows:
Therefore when given a vector in spherical or cylindrical coordinates, it is possible to convert it into Cartesian coordinates, and vice-versa.
This is particularly so when solving electromagnetic radiation problems.
0ˆ.ˆcosˆ.ˆsinˆ.ˆsinˆ.ˆsincosˆ.ˆcoscosˆ.ˆcosˆ.ˆsinsinˆ.ˆcossinˆ.ˆ
zyx
zyx
zryrxr sss
Vector Calculus ENEL2FT Field Theory 35
VECTOR CALCULUS VECTOR DERIVATIVES AND INTEGRALS For a scalar function, F(t), we have:
Now suppose that F(t) were one component of a vector function, say Ax. Since each component would be a new scalar function, it follows that:
Suppose, instead, we asked for the partial derivative of vector A with respect to x? This asks for the change in A as we move along the x direction. This becomes:
t
tFttF
dt
dFt
)()(lim0
dt
dAz
dt
dAy
dt
dAx
dt
Ad zyx ˆˆˆ
x
Az
x
Ay
x
Ax
x
A zyx
ˆˆˆ
Vector Calculus ENEL2FT Field Theory 36
VECTOR CALCULUS VECTOR DERIVATIVES AND INTEGRALS The definition of a partial derivative is identical to the definition of an
ordinary derivative:
The only difference is that the function F(x,y) has now two independent variables, x and y.
Many such functions of two or more independent variables exist. For example, the height of a point above sea level depends on the position on the earth and requires two variables, latitudes (x) and longitude (y), to describe that position.
The partial derivative with respect to y is also defined as:
x
yxFyxxF
x
Fx
),(),(lim0
y
sFyyxF
y
Fy
)(),(lim0
Vector Calculus ENEL2FT Field Theory 37
VECTOR CALCULUS DIRECTIONAL DERIVATIVES The partial derivatives of F(x,y) with respect to x and y are both special
cases of a more general derivative, the directional derivative. Consider the same function, F(x,y), but now instead of partial derivative
with respect to x or y, we compute the derivative in a direction s, as shown below:
We wish to determine the partial derivative with respect to s:
x
y
st
s
sFssF
s
Fs
)()(lim0
Vector Calculus ENEL2FT Field Theory 38
VECTOR CALCULUS DIRECTIONAL DERIVATIVES The variables s,t, are orthogonal and related to x and y by the
equations:
Also recall the chain rule of differentiation from ordinary calculus:
Looking at the coordinate transformations, we find that:
cossin
sincos
tsy
tsx
y
F
s
y
x
F
s
x
s
F
sincos
;sin;cos
y
F
x
F
s
Fs
y
s
x
Vector Calculus ENEL2FT Field Theory 39
VECTOR CALCULUS DIRECTIONAL DERIVATIVES - THE GRADIENT What would be the maximum directional derivative of F(x,y) at the point
(x,y)? This is determined by setting the derivative of the directional derivative with respect to s equal to zero.
This would be denoted by F, the gradient of F, given by:
Here the del operator, , is defined as:
Thus the gradient is a vector operator, with the del operator, , operating on a scalar, F(x,y,z).
z
Fz
y
Fy
x
FxF
ˆˆˆ
zz
yy
xx
ˆˆˆ
Vector Calculus ENEL2FT Field Theory 40
VECTOR CALCULUS THE DIVERGENCE AND CURL OF A VECTOR Like the dot product of two vectors, the divergence of a vector field is
a scalar function, which, in rectangular coordinates, is given by:
The vector derivative or curl of a vector is defined in rectangular coordinates as:
z
A
y
A
x
AA zyx
.
zyx
zyx
AAAzyx
zyx
Ax
AzAyAxxz
zy
yx
xAx
ˆˆˆ
ˆˆˆˆˆˆ
Vector Calculus ENEL2FT Field Theory 41
VECTOR CALCULUS SOME VECTOR IDENTITIES Some useful vector identities are given below: 1. The Laplacian is defined as:
2. The Curl of the Gradient of a scalar:
3. The divergence of the curl of a vector:
4. The curl of the curl of a vector:
2
2
2
2
2
22 .
z
F
y
F
x
FFF
0 Fx
0. Ax
AAAxx
2.