Field Theory Vector Algebra

ENE L2FTF IE LD THEO RY 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, PrecticeHall, 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.

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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.

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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.

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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

C B

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VECTOR ALGEBRA - THE UNIT VECTOR Since a vector may have, in general, an arbitrary orientation in threedimensional 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.

i3 i2 i1 Vector Calculus

Set of three orthogonal unit vectors in a right-handed coordinate system.

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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: A = 4iˆ1 + 6iˆ2  A is 2given The magnitude of vector by: A = 4 + 6 2 = 7.211

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VECTOR ALGEBRA If vector B is defined as:  B = 4iˆ1 + 6iˆ2 − 2i3 then the magnitude of B is:  B = 4iˆ1 + 6iˆ2 − 2i3 = 4 2 + 6 2 + 2 2 = 7.4833 In general, a vector A is said to have components A1, A2, and A3 along the directions 1, 2, and 3 is written as:  A = A1iˆ1 + A2iˆ2 + A3i3 Now consider three  vectors A,B, and C given by: A = A1iˆ1 + A2iˆ2 + A3i3  B = B1iˆ1 + B2iˆ2 + B3i3  C = C1iˆ1 + C2iˆ2 + C3i3

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VECTOR ALGEBRA - Vector Addition, Subtraction, Multiplication Then the sum of vectors A and B, (A+B), is given by:   A + B = ( A1iˆ1 + A2iˆ2 + A3i3 ) + ( B1iˆ1 + B2iˆ2 + B3i3 ) = ( A1 + B1 ) iˆ1 + ( A2 + B2 ) iˆ2 + ( A3 + B3 ) iˆ3 Vector subtraction is a special case of vector addition; thus:   B − C = ( B1iˆ1 + B2iˆ2 + B3i3 ) − ( C1iˆ1 + C2iˆ2 + C3i3 ) = ( B1 − C1 ) iˆ1 + ( B2 − C2 ) iˆ2 + ( B3 − C3 ) iˆ3 The multiplication of a vector, A, by a scalar m, is the same as repeated addition of the vector:  mA = m( A1iˆ1 + A2iˆ2 + A3i3 ) = mA1iˆ1 + mA2iˆ2 + mA3i3

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VECTOR ALGEBRA The magnitude of vector A is given by:  A = A1iˆ1 + A2iˆ2 + A3i3 = A12 + A22 + A32 The unit vector along A , iA, has a magnitude equal to unity, but its direction is the same as that of A. Thus:  A3 A A1 A2    i A = = i1 + i2 +  i3 A A A A Two vectors A and B are equal if and only if the corresponding components of A and B are equal. That is:   A = B ⇒ A1iˆ1 + A2iˆ2 + A3i3 = B1iˆ1 + B2iˆ2 + B3i3 ; ∴

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VECTOR ALGEBRA - SCALAR OR DOT PRODUCT The scalar or dot product of two vectors A and B is a scalar quantity defined  as:     

A.B A.B = A B cosα ⇒ α = cos −1      AB  

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Here α is the angle between A and B. For mutually orthogonal unit vectors i1, i2, and i3, we have:

i1.i1 = 1; i1.i2 = 0; i1.i3 = 0

i2 .i1 = 0; i2 .i2 = 1; i2 .i3 = 0 i3 .i1 = 0; i3 .i2 = 0; i3 .i3 = 1 ■

Thus we have  the dot product between A and B as:

A.B = ( A1iˆ1 + A2iˆ2 + A3i3 ).( B1iˆ1 + B2iˆ2 + B3i3 ) = A1iˆ1 . B1iˆ1 + A1iˆ1 . B2iˆ2 + A1iˆ1 . B3iˆ3 + A2iˆ2 . B1iˆ1 + A2iˆ2 . B2iˆ2 + A2iˆ2 . B3iˆ3 +

A3iˆ3 . B1iˆ1 + A3iˆ3 . B2iˆ2 + A3iˆ3 . B3iˆ3

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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 α between A and B whose direction is normal to the plane containing A and B.     AxB = A B sin αiˆN For mutually orthogonal unit vectors i1, i2, and i3, we have:

i1 xi1 = 0; i1 xi2 = i3 ; i1 xi3 = −i2

i2 xi1 = −i3 ; i2 xi2 = 0; i2 xi3 = i1 i3 xi1 = i2 ; i3 xi2 = −i1; i3 xi3 = 0 ■

Note that the cross-product is not commutative, and also the distributive property holds for the cross product:

      BxA = B A sin α ( − iN ) = − AxB        Ax ( B + C ) = AxB + AxC

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VECTOR ALGEBRA - VECTOR OR CROSS PRODUCT Using the above properties, we obtain:

  AxB = ( A1iˆ1 + A2iˆ2 + A3i3 ) x( B1iˆ1 + B2iˆ2 + B3i3 ) = A1iˆ1 xB1iˆ1 + A1iˆ1 xB2iˆ2 + A1iˆ1 xB3iˆ3 + A2iˆ2 xB1iˆ1 + A2iˆ2 xB2iˆ2 + A2iˆ2 xB3iˆ3 A3iˆ3 xB1iˆ1 + A3iˆ3 xB2iˆ2 + A3iˆ3 xB3iˆ3 = A1 B2iˆ3 − A1 B3iˆ2 − A2 B1iˆ3 + A2 B3iˆ1 + A3 B1iˆ2 − A3 B2iˆ1 +

= ■

( A2 B3 − A3 B2 ) + ( A3 B1 − A1B3 ) + ( A1B2 − A2 B1 )

This can be expressed in determinant form in the manner:

iˆ1   AxB = A1 B1

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iˆ3 A3 B3

The cross product is useful in obtaining the unit vector normal to the plane containing the two vectors A  and  B:

AxB iN =   AxB

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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,

iˆ1 iˆ2   A.BxC = ( A1iˆ1 + A2iˆ2 + A3iˆ3 ). B1 B2 C1 C2

A1 A2   ⇒ A.BxC = B1 B2 C1 C2 ■

iˆ3 B3 C3

A3 B3 C3

Since the value of the determinant on the right side remains unchanged if the rows are interchanged in a cylindrical manner, we have

      A.BxC = B.CxA = C. AxB

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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:       Ax ( BxC ) ≠ ( AxB ) xC As an example, let us equate the three vectors to unit vectors as follows:    ˆ ˆ A = i1 ; B = i1 ; C = iˆ2    ⇒ Ax( BxC ) = iˆ1 x( iˆ1 xiˆ2 ) = iˆ1 xiˆ3 = −iˆ2    ( AxB ) xC = ( iˆ1 xiˆ1 ) xiˆ2 = 0 xiˆ2 = 0 Therefore in a vector triple product, the parentheses are so important and must be included.

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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.

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CARTESIAN COORDINATE SYSTEM The Cartesian coordinate system is defined by a set of three mutually orthogonal vectors, x,y, and z, as shown below.

i3=z i2=y i1=x ■ ■

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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.

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CARTESIAN COORDINATE SYSTEM Therefore we have: iˆ1 = xˆ; iˆ2 = yˆ ; iˆ3 = zˆ

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For a right-handed coordinate system, we have: xˆxyˆ = zˆ; yˆ xzˆ = xˆ; zˆxxˆ = yˆ

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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: r1 = x1 xˆ + y1 yˆ + z1 zˆ  r2 = x2 xˆ + y2 yˆ + z 2 zˆ The resultantvector, R12, is given by:   R12 = r2 − r1 = ( x2 − x1 ) xˆ + ( y2 − y1 ) yˆ + ( z 2 − z1 ) zˆ

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CARTESIAN COORDINATE SYSTEM

P1(x1,y1,z1)

R12

z r1

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We can obtain the unit vector along the line drawn from P1 to P2 to be:  ( x2 − x1 ) xˆ + ( y2 − y1 ) yˆ + ( z2 − z1 ) zˆ R iˆ12 = 12 = 1/ 2 R12 ( x2 − x1 ) 2 + ( y2 − y1 ) 2 + ( z2 − z1 ) 2

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As an example, if P1 is (1,-2,0) and P2 is (4,2,5), then:

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CARTESIAN COORDINATE SYSTEM  R12 = 3 xˆ + 4 yˆ + 5 zˆ 1 ( 3xˆ + 4 yˆ + 5 zˆ ) iˆ12 = 5 2 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: dxxˆ , dyyˆ , and dzzˆ

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DIFFERENTIAL LENGTH VECTOR P(x,y,z)

dl

z r1

Q(x+dx,y+dy,z+dz)

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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:  dl = dxxˆ + dyyˆ + dzzˆ

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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 x = y = z2

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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: dx = dy = 2 zdz  ∴ dl = dxxˆ + dyyˆ + dzzˆ  dl = 2dzxˆ + 2dzyˆ + dzzˆ = ( 2 xˆ + 2 yˆ + zˆ ) dz Note that x=1, y=1, z=1.

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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. dl2 Curve 1 Curve 2

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dl1 Surface

Thus the unit vector normal to thesurface is given by:  dl xdl iˆn = 1 2 dl1 xdl2

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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:

in

dl2 dS α dl1

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From simple geometry and the definition of cross-product of two vectors, it can be seen that:   dS = dl1dl2 sin α = dl1 xdl2

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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:    dS = ± dl1 xdl2 = ± dSiˆn If we apply these equations to differential surface vectors in Cartesian coordinates, we obtain: For plane x = cons tan t : ± dyyˆ xdzzˆ = ± dydzxˆ For plane y = cons tan t : ± dzzˆxdxxˆ = ± dzdxyˆ For plane z = cons tan t : ± dxxˆxdyyˆ = ± dxdyzˆ

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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.

dv dl3

dl2 dl1

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It can be seen that: dv = (base area of parallelepiped ).(height of parallelepiped )         dl3 .dl1 xdl2    ˆ = dl1 xdl2 dl3 .in = dl1 xdl2   = dl3 .dl1 xdl2 dl1 xdl2    ⇒ dv = dl1.dl2 xdl3

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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.

z P(r,φ,z)

y φ

r

x ■

For the cylindrical coordinate system, the three one of the planes is z=constant

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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 φ with a reference plane, chosen to be the x-z plane of the Cartesian coordinate system. This plane is called the φ=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; φ=constant; and z=constant. Only two of the coordinates (r and z) are distances; the third coordinate (φ) is an angle. We note that the entire space is spanned by varying r from 0 to ∞; z from -∞ to +∞; and φ from 0 to 2π.

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CYLINDRICAL COORDINATE SYSTEM Q

z dz

P y x

φ

dφ r

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dr To obtain the expressions for the differential lengths, surfaces, and volumes in the cylindrical coordinate system, we now consider two points, P(r,φ,z) and Q(r+dr, φ+dφ, 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,dφ, and dz are infinitesimal.

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CYLINDRICAL COORDINATE SYSTEM The three differential length elements forming the contiguous sides of the box are: drrˆ, rdφφˆ, dzzˆ The differential length vector dl from P to Q is thus given by:  dl = drrˆ + rdφφˆ + dzzˆ The differential surface vectors defined by the pairs of the differential length elements are: rdφφˆxdzzˆ = rdφdzrˆ; dzzˆxdrrˆ = drdzφˆ; drrˆxrdφφˆ = rdrdφzˆ Finally, the differential volume dv is the volume of the box: dv = ( dr )( rdφ )( dz ) = rdrdφdz

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SPHERICAL COORDINATE SYSTEM For the spherical coordinate system, the three mutually orthogonal surfaces are a sphere, a cone, and a plane. z r θ y x

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φ

The three orthogonal surfaces defining the spherical coordinates of a point are: r = cons tan t θ = cons tan t φ = cons tan t

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SPHERICAL COORDINATE SYSTEM The differential length elements and the differential length vector dl are given by: drrˆ, rdθθˆ, r sin θdφφˆ  ∴ dl = drrˆ + rdθθˆ + r sin θdφφˆ The differential surface vectors defined by pairs of differential length elements are: rdθθˆxr sin θdφφˆ = r 2 sin θdθdφrˆ r sin θdφφˆxdrrˆ = r sin θdrdφθˆ drrˆxrdθθˆ = rdrdθφˆ

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The differential volume formed by the three differential lengths is: dv = ( dr )( rdθ )( r sin θdφ ) = r 2 sin θdrdθdφ

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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 = rc cos φ ; x = rs sin θ cos φ ; rc = x 2 + y 2 rs = x 2 + y 2 + z 2

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y = rc sin φ ; z=z y = rs sin θ sin φ z = rs cosθ y φ = tan −1   z=z  x 2   2 −1  x + y  −1  y  θ = tan  φ = tan    z  x  

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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: rˆc .xˆ = cos φ rˆc . yˆ = sin φ rˆc .zˆ = 0 φˆ.xˆ = − sin φ φˆ. yˆ = cos φ φˆ.zˆ = 0 zˆ.xˆ = 0 zˆ. yˆ = 0 zˆ.zˆ = 1

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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: rˆs .xˆ = sin θ cos φ rˆs . yˆ = sin θ sin φ rˆs .zˆ = cosθ θˆ.xˆ = cosθ cos φ θˆ. yˆ = cosθ sin φ θˆ.zˆ = − sin θ φˆ.xˆ = − sin φ φˆ. yˆ = cos φ φˆ.zˆ = 0

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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.

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VECTOR DERIVATIVES AND INTEGRALS For a scalar function, F(t), we have: dF  F (t + ∆t ) − F (t )  = lim   dt ∆t →0  ∆t

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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: dAy dA dAx dA = xˆ + yˆ + zˆ z dt dt dt dt 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: ∂Ay ∂A ∂Ax ∂A = xˆ + yˆ + zˆ z ∂x ∂x ∂x ∂x

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VECTOR DERIVATIVES AND INTEGRALS The definition of a partial derivative is identical to the definition of an ordinary derivative: ∂F  F ( x + ∆x, y ) − F ( x, y )  = lim   ∂x ∆x→0  ∆x 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: ∂F  F ( x, y + ∆y ) − F ( s )  = lim   ∂y ∆y →0  ∆y 

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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: y t

s θ

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x

We wish to determine the partial derivative with respect to s: ∂F  F ( s + ∆s ) − F ( s )  = lim   ∂s ∆s→0  ∆s

Vector Calculus

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VECTOR CALCUL US ■ ■

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DIRECTIONAL DERIVATIVES The variables s,t, are orthogonal and related to x and y by the equations: x = s cosθ − t sin θ y = s sin θ + t cosθ Also recall the chain rule of differentiation from ordinary calculus: ∂F ∂x ∂F ∂y ∂F = + ∂s ∂s ∂x ∂s ∂y Looking at the coordinate transformations, we find that: ∂x ∂y = cosθ ; = sin θ ; ∂s ∂s ∂F ∂F ∂F ∴ = cosθ + sin θ ∂s ∂x ∂y

Vector Calculus

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VE CT OR CA LCULUS ■ ■

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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:  ∂F ∂F ∂F ∇F = xˆ + yˆ + zˆ ∂x ∂y ∂z

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Here the del operator, ∇, is defined as:  ∂ ∂ ∂ ∇ = xˆ + yˆ + zˆ ∂x ∂y ∂z

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Thus the gradient is a vector operator, with the del operator, ∇, operating on a scalar, F(x,y,z).

Vector Calculus

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VECTOR CALCUL US ■ ■

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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:   ∂Ax ∂Ay ∂Az ∇. A = + + ∂x ∂y ∂z The vector derivative or curl of a vector is defined in rectangular coordinates as:    ∂ ∂ ∂ ∇xA =  xˆ + yˆ + zˆ  x ( xˆAx + yˆ Ay + zˆAz ) ∂y ∂z   ∂x xˆ   ∂ ⇒ ∇xA = ∂x Ax

Vector Calculus

yˆ ∂ ∂y Ay

zˆ ∂ ∂z Az

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VECTOR CALCUL US ■ ■ ■

SOME VECTOR IDENTITIES Some useful vector identities are given below: 1. The Laplacian is defined as:  ∂2F ∂2F ∂2F ∇ F = ∇.∇F = 2 + 2 + 2 ∂x ∂y ∂z 2. The Curl of the Gradient of a scalar:   ∇x∇F = 0 3. The divergence of the curl of a vector:   ∇.∇xA = 0 4. The curl of the curl of a vector:        2 ∇x∇xA = ∇( ∇. A) − ∇ A 2

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