Chapter 1

Mathematical Preliminaries

The main objective of this chapter is to provide a brief and necessary background of the following main topics of mathematics:

• fundamentals of scalar and vector coordinate systems
• operations with del
• determinants and matrices
• series and identities
• trigonometric functions
• differentiation and integration formulae
• integral theorems and so on
• points/formulae to remember, solved problems, objective questions and exercise problems.

Mathematics is the backbone of science and engineering. For all analytical or computational purposes, mathematical background is essential. Mathematical modeling of systems is a common practice. The design and analysis of any antenna problem is possible only with mathematical concepts. No engineering system can be designed without using mathematics. Moreover, the solutions of antenna problems are simplified by mathematical approach.

In view of this, basics of mathematics is presented in this chapter.

1.1 FUNDAMENTALS OF SCALARS AND VECTORS

In this book, a scalar is represented by simple letters like A, B and a vector is represented by bold-faced letters like A, B. Unit vectors are represented by small bold-faced letters like a, b.

The vector A is expressed in two forms:

1. A = (A_x, A_y, A_z). A_x, A_y, A_z are known as the components of vector A.
2. A = A_x a_x + A_y a_y + A_z a_z . a_x , a_y , a_z are unit vectors along the coordinate axes.

The magnitude of A is written as A,

that is,     A = |A|

The unit vector of A is a and it is given by

                

The sum and difference of two vectors are given by

 

The dot product is denoted by

            A . B or B . A

            A . B = B . A = AB cos q

                      = A_x B_x + A_y B_y + A_z B_z

Here θ is the angle between the vectors A and B.

 

The cross product is denoted by A × B.

 

where a_n is the unit vector perpendicular to A and B

or                

	= ax [AyBz − AzBy] + ay [AzBx − AxBz] + az [AxBy − AyBx]
where	A = Axax + Ayay + Azaz
and	B = Bxax + Byay + Bzaz

 

1.2 COORDINATE SYSTEMS

Coordinate systems, basically, are of three types:

1. Cartesian coordinate system
2. Cylindrical coordinate system
3. Spherical coordinate system

Cartesian Coordinate System

Here a point, P is represented by P (x, y, z). The variables are x, y and z.

A point obtained by the intersection of three planes given by

 

The unit of x, y and z is metre.

The Cartesian coordinates are represented in Fig. 1.1.

 

The unit vectors along the coordinate axes are represented by a_x, a_y and a_z. Their magnitude is unity and they are in the increasing directions of x, y and z axes respectively.

Properties of unit vectors

 

Cylindrical Coordinate System

 

The unit of ρ is metre.

The unit of ϕ is degree or radian.

The unit of z is metre.

 

In this book, the term Cylindrical Coordinate system is used to indicate circular cylindrical coordinate system. However, a point in cylindrical coordinate system is shown in Fig. 1.2. The coordinate ρ is the radius of the cylinder. ϕ is measured from x-axis and z is the same as in Cartesian system.

Here a_ρ , a_ϕ, a_z represent the unit vectors along the coordinates ρ, ϕ, z. Their magnitude is unity and they are in the increasing directions of ρ, ϕ, z respectively.

It is obvious that increase in ρ results in cylinders of higher radius. ϕ increases in anti-clockwise direction. z is the same as in Cartesian system.

The relations between x, y, z and ρ, ϕ, z:

                x = ρ cos ϕ

                y = ρ sin ϕ

                z = z

 

and		0 ≤ ρ < ∞
	ϕ = tan−1 x,	0 ≤ ϕ < 2π
	z = z,	0 ≤ z < ∞

Dot products of a_x, a_y and a_z with a_ρ, a_ϕ and a_z are given by

            a_x.a_ρ = cos ϕ

            a_x. a_ϕ = − sin ϕ

            a_y. a_ρ = sin ϕ

            a_y. a_ϕ = cos ϕ

            a_z. a_ρ = 0

            a_z. a_ϕ = 0

The unit vectors of cylindrical coordinates in terms of Cartesian coordinates are given by

            a_ρ = cos ϕ a_x + sin ϕ a_y

            a_ϕ = − sin ϕ a_x + cos ϕ a_y

            a_z = a_z

A vector, A = (A_x a_x + A_y a_y + A_z a_z) is expressed in cylindrical coordinates as

A = (A_ρ, A_ϕ, A_z) = [(A_x cos ϕ + A_y sin ϕ), (− A_x sin ϕ + A_y cos ϕ), A_z]

 

That is	Aρ = Ax cos ϕ + Ay sin ϕ
	Aϕ = − Ax sin ϕ + Ay cos ϕ
	Az = Az

A is also written as

A = (A_x cos ϕ + A_y sin ϕ) a_ρ + (− A_x sin ϕ + A_y cos ϕ) a_ϕ + A_z a_z.

Spherical Coordinate System

 

A point is obtained by the intersection of three surfaces, namely,

        a spherical surface, r = k (constant), metre

        a cone, θ = α (constant), radian, and

        a plane, ϕ = β (constant), radian

All these three surfaces are mutually perpendicular to each other.

These are said to be orthogonal.

a_r, a_θ, a_ϕ represent unit vectors along the coordinate axes. Their magnitude is unity and they are in the increasing directions of r, θ and ϕ axes.

A point in spherical coordinate system is shown in Fig. 1.3.

Increase of r results in spheres of larger radii. θ increases in the clockwise direction and ϕ increases in the anti-clockwise direction.

The variables of Cartesian and spherical coordinates are related by

 

x = r sin θ cos ϕ,	− ∞ < x < ∞
y = r sin θ sin ϕ,	− ∞ < y < ∞
z = r cos θ,	− ∞ < z < ∞

and

The relations between the variables of cylindrical and spherical coordinates are given by

The unit vectors of spherical coordinates in terms of Cartesian coordinates are given by

    a_r = sin θ cos ϕ a_x + sin θ sin ϕ a_y + cos θ a_z

    a_θ = cos θ cos ϕ a_x + cos θ sin ϕ a_y − sin θ a_z

    a_ϕ = − sin ϕ a_x + cos ϕ a_y

A vector,

 

A	=	(Ax, Ay, Az), is expressed in spherical coordinates as
A	=	(Ar, Aθ, Aϕ)
	=	[(Ax sin θ cos ϕ + Ay sin θ sin ϕ + Az cos θ), (Ax cos θ cos ϕ + Ay cos θ + sin ϕ − Az sin θ), (− Ax sin ϕ + Ay cos ϕ)]

The dot products of a_x, a_y and a_z with a_r, a_θ and a_ϕ are given by

            a_x . a_r = sin θ cos ϕ

            a_x . a_θ = cos θ cos ϕ

            a_x . a_ϕ = − sin ϕ

            a_y . a_r = sin θ sin ϕ

            a_y . a_θ = cos θ sin ϕ

            a_y . a_ϕ = cos ϕ

            a_z . a_r = cos θ

            a_z . a_θ = − sin θ

            a_z . a_ϕ = 0

Here,

    A_r = A_x sin θ cos ϕ + A_y sin θ sin ϕ + A_z cos θ

    A_θ = A_x cos θ cos ϕ + A_y cos θ sin ϕ − A_z sin θ

    A_ϕ = − A_x sin ϕ + A_y cos ϕ

The point A (x, y, z) = A (ρ, ϕ, z) = A (r, θ, ϕ) in Cartesian, cylindrical and spherical coordinate systems is shown in a single Fig. 1.4 to understand the concept at a glance.

The relations for differential length, area and volume are given in Table 1.1.

 

Relations between the polar coordinates (ρ, ϕ) and Cartesian coordinates (x, y) are:

1.3 DEL (∇) OPERATOR

Definition    The del or nabla is known as differential vector operator and is defined as

Del has units of 1/metre (1/m)

Del is operated in three ways.

1.4 GRADIENT OF A SCALAR, V (= ∇ V)

Gradient of a scalar is a vector and is defined as

Examples are gradient of temperature, gradient of electric potential and so on.

It gives the maximum space rate of change of the scalar. The scalar can be temperature, potential and so on.

1.5 DIVERGENCE OF A VECTOR, A (= ∇.A)

Divergence of a vector is a scalar and is defined as

Divergence means the spreading or diverging of a quantity from a point. It is applicable to vectors only. The divergence of a vector indicates the net flow of quantities like gas, fluid, vapour, electric and magnetic flux lines. In other words, it is a measure of the difference between outflow and inflow.

The divergence of a vector is positive if the net flow is outward.

It is negative if the net flow is inward.

The fluid is said to be incompressible if the divergence is zero, that is, ∇ . A = 0 is the condition of incompressibility.

Examples and Features of Divergence

1. Leaking air from a balloon yields positive divergence.
2. Rushing of air into the drum under the carriage of a train yields negative divergence.
3. Divergence of water or oil is almost zero and hence they are incompressible.
4. Divergence of electric flux density is equal to volume charge density,

   or,                     ∇ . D = ρ_v

    

5. Divergence of magnetic flux density is equal to zero,

   or,                     ∇ . B = 0

    

6. Divergence of gradient of scalar electric potential is equal to the laplacian of the scalar,

   or,                 ∇ . ∇V = ∇² V.

1.6 CURL OF A VECTOR (≡ ∇ × A)

Curl of a vector is a vector and is defined as

Curl A

As the curl of a vector represents rotation, it is also written as

            curl A = rot A = ∇ × A

It may be noted that curl (gradient of a scalar) = ∇ × (∇V) is zero.

This means that the gradient of fields is irrotational. Also div (curl) = 0.

Examples:

• When a leaf floats in sea water and its rotation is about the z-axis, curl of velocity V is in the z-direction. When (∇ × V)_z is positive, it represents rotation from x to y.
• For a rotating rigid body, the curl of velocity is in the direction of the axis of rotation. Its magnitude is equal to twice the angular speed of rotation.

In Cartesian coordinates,

In cylindrical coordinates,

is a vector.

In spherical coordinates,

Vector identities

 

B . B	=	B2
B . B*	=	B2
B + C	=	C + B
B . C	=	C . B
C × B	=	− B × C
(D + B) . C	=	D . C + B . C
(D + B) × C	=	D × C + B × C
D . B × C	=	B . C × D = C . D × B
D × (B × C)	=	(D . C) B − (D . B) C
(E × B) . (C × D)	=	E . B × (C × D)
	=	(E . C) (B . D) − (E . D) (B . C)
(E × B) × (C × D)	=	(E × B . D) C − (E × B . C) D
∇ (VmV)	=	Vm∇V + V∇Vm
∇ . (∇ × C)	=	0
∇ × ∇V	=	0
∇ (Vm + V)	=	∇Vm + ∇V
∇ . (C + B)	=	∇ . C + ∇ . B
∇ × (C + B)	=	∇ × C + ∇ × B
∇ . (VC)	=	C . ∇V + V∇ . C
∇ × (VC)	=	∇V × C + V∇ × C
∇ (C . B)	=	(C . ∇) B + (B . ∇) C + C × (∇ × B) + B × (∇ × C)
∇ . (C × B)	=	B . ∇ × C − C . ∇ × B
∇ × (C × B)	=	C(∇ . B) − B (∇ . C) + (B . ∇) C − (C . ∇) B
∇ × ∇ × C	=	∇ (∇ . C) − ∇2 C

1.7 LAPLACIAN OPERATOR (∇²)

Laplacian of a scalar electric potential (∇² V), is a scalar.

Laplacian of an electric field (∇² E), is a vector.

Laplacian of a scalar and a vector in different coordinate systems.

(Spherical)

1.8 DIRAC DELTA

Direct Delta is used to represent very short pulse of high amplitude. Sometimes it is also called unit impulse function.

At t = 0, it is represented by δ (t)

At t = t₀, it is represented by δ (t – t₀)

Dirac Delta is shown in Fig. 1.5.

1.9 DECIBEL AND NEPER CONCEPTS

Decibel (dB) is defined as ten times the common logarithm of the power ratio, that is,

If P₁ and P₂ are input and output powers respectively of an electric circuit, then dB is negative if P₂ > P₁. This indicates power loss. If P₁ > P₂, dB is positive and this indicates power gain.

Decibel has no dimensions and it is used to express the ratio of two powers, voltages, currents or sound intensities.

One Bel (B) is equal to ten decibels.

When input and output currents and voltages are known, we have

Neper (NP)    It has no dimensions and is used to express the ratio of powers in communications.

A Neper is defined as the natural logarithm of the square root of the power ratio, that is,

In terms of input and output currents or voltages, it is given by

and 1 Np = 8.686 dB.

1.10 COMPLEX NUMBERS

By definition, a complex number is an ordered pair represented by

                A ≡ (x, y)

where x is the real part and y is the imaginary part of A. Hence it is also represented as A = x + jy.

Here, is called imaginary unit.

Properties of Complex Numbers

1. Two complex numbers are equal if their real parts are equal and their imaginary parts are also equal.
2. If A₁ = x₁ + jy₁, A₂ = x₂ + jy₂ the sum of two complex numbers A₁ and A₂ is given by

    

   A = A₁ + A₂ = (x₁ + x₂) + j (y₁ + y₂)

    

3. Subtraction of A₂ from A₁ is given by

    

   B = A₁ − A₂ = (x₁ − x₂) + j (y₁ − y₂)

    

4. The product of A₁ and A₂ is

    

   C	=	A1 A2 = (x1 + jy1) (x2 + jy2)
   	=	x1 x2 + jx1 y2 + jx2 y1 − y1 y2
   	=	(x1 x2 − y1 y2) + j (x1 y2 + x2 y1)

    

5. The division of A₁ by A₂ is given by
   

Multiplying numerator and denominator by (x₂ – jy₂), we get

So

1.11 LOGARITHMIC SERIES AND IDENTITIES

This series converges for – 1 < x < 1.

It diverges for x = – 1

1.12 QUADRATIC EQUATIONS

A quadratic equation is represented by ax² + bx + c = 0 where a, b and c are constants. Its roots are given by

and

The roots may be real or imaginary and may be equal or unequal depending on whether the quantity (b² – 4ac) is positive, zero or negative.

1.13 CUBIC EQUATIONS

It is given by

 

By substituting for x, f (x) becomes

    y³ + dy + e = 0

Here,

If

the three values of y are

If , the trigonometric solution is found by evaluating the cube roots of complex quantities. Then the angle ϕ is evaluated from

The values of y will be

1.14 DETERMINANTS

It is said to be a second order determinant. Its value is given by

Example    The value of a determinant

Similarly, is called a determinant of the third order.

Example    The value of a third order determinant

Application of Determinants

They are widely used in solving simultaneous equations and are also useful in developing the theory of matrices.

Minor of a Determinant

In a determinant the minor of b₃ is given by and the minor of c₂ is given by

that is, the minor of an element in a determinant is a determinant obtained by deleting the row and the column which intersect at that element.

The Cofactor    The cofactor of an element in a determinant is its minor with a suitable sign. The sign in the i^th row and j^th column is given by (– 1)^i+j . Then the cofactor, B₃ of b₃ is (− 1)^{2 + 3} × minor of b₃.

Properties of Determinants

1. Value of the determinant remains the same after changing its rows into columns and columns into rows.
2. Value of the determinant remains the same if any two rows or columns are interchanged but the sign changes.
3. If two rows or columns are identical, the value of the determinant is zero.
4. If each element of a row or column is multiplied by a factor, the determinant is multiplied by the same factor.
5. If each element of a row consists of n terms, the determinant can be expressed as the sum of n determinants.
6. When equi-multiples of the corresponding elements of one or more parallel lines are added to each element of a line, the determinant value remains the same.

1.15 MATRICES

Examples

is a matrix containing two rows and one column.

Similarly, is a matrix containing two rows and three columns.

Here, a row means a horizontal line and a column means a vertical line.

Applications of Matrices

Matrices are widely used to solve linear systems of equations. They often appear as models of different problems. For example, in electrical circuits and numerical methods they are used for solving differential equations.

Types of Matrices

1. Row Matrix—Example [1 2 3].
2. Column Matrix—Example
3. Square Matrix—Example
4. Diagonal Matrix—Example
5. Unit Matrix—Example
6. Null Matrix—Example
7. Symmetric Matrix—Example
8. Skew Symmetric Matrix—Example
9. Upper Triangular Matrix—Example
10. Lower Triangular Matrix—Example

Properties of Matrices

1. Two matrices A and B are equal if and, only if, they are of the same order and each element of A is equal to the corresponding element of B.
2. The sum of A and B exists only when they are of the same order.
3. The difference of A and B exists only when they are of the same order.

   If

   
4. Addition of matrices is commutative, that is, A + B = B + A.
5. Addition and subtraction of matrices are associative, that is,

    

   (A + B) − C = A + (B − C)                  
   = B + (A − C)

    

6. The multiplication of a matrix, A by a scalar, K gives a matrix each element of which is K times the corresponding elements of A. The distributive law is valid for such a product,

   that is,             K (A + B) = KA + KB

    

7. All the laws of ordinary algebra hold good for the addition or subtraction of matrices and their multiplication by scalars.
8. Two matrices can be multiplied only when the number of columns in the first is equal to the number of rows in the second. Such matrices are said to be conformable.
9. AB ≠ BA

   Sometimes if AB exists, BA may not exist at all.

10. IA = AI = A if A is a square matrix which has the same order as that of I.
11. OA = AO = 0, where O is a null matrix.
12. If AB = 0, it does not mean that A or B is a null matrix.
13. Multiplication of matrices is associative, that is, (AB) C = A (BC) provided A, B are conformable for the product AB and B, C are conformable for the product BC.
14. Multiplication of matrices is distributive, that is, A (B + C) = AB + AC provided A, B are conformable for AB and A, C are conformable for AC.
15. If A is a square matrix, then A . A = A² .
16. The matrix obtained from a given matrix A by interchanging rows and columns is called Transpose of A denoted by A′.
17. For a symmetric matrix, A′ = –A
18. (AB)′ = B′. A′
19. Adjoint of a matrix A is the transposed matrix of cofactors of A.
20. If A is any matrix, then a matrix, B if it exists such that AB = BA = I, is called the Inverse of A. Both the matrix and its inverse must be non-singular. Inverse of a matrix, A is denoted by A^-1 so that A . A^-1 = A^-1. A = I.

    Also,                

21. Inverse of a matrix is unique.
22. (AB)¹ = B⁻¹ A⁻¹
23. A matrix is said to be of rank r when it has at least one non-zero minor of order r and every minor of order higher than r vanishes. It means that the rank of a matrix is the largest order of any non-vanishing minor of the matrix.

1.16 FACTORIAL

Factorial of n is defined as the product of the first n natural numbers.

Factorial of n is represented by n! or ∠n

 

Using Stirling’s approximation, n! is evaluated from

1.17 PERMUTATIONS

The number of permutations of n different things taken r at a time is given by np_r and

If p, q, r types of elements exist among n elements, then the number of permutations are given by

 

The number of permutations of n different things taken all at a time is np_n.

1.18 COMBINATIONS

The number of combinations or groups of n different elements taken r at a time is given by nc_r

where

1.19 BASIC SERIES

Binomial Series These are given by

Here the last term shown is the (r + 1)th term.

If n is a positive integer, the series is finite and the last term is bⁿ. If n is less than unity, the series is infinite and converges only when b < a. For any value of n, and convergent, when x² is less than unity, we have

1.20 EXPONENTIAL SERIES

This series converges for all values of x.

For x = 1,

1.21 SINE AND COSINE SERIES

If x is a variable,

1.22 SINH AND COSH SERIES

1.23 HYPERBOLIC FUNCTIONS

Hyperbolic sine of
Hyperbolic cosine of
Hyperbolic tan of
Hyperbolic cot of

Similarly,

It may be noted that

                sinh 0 = 0, cosh 0 = 0, and tanh 0 = 0

      cosh x + sinh x = e^x

      cosh x + sinh x = e^−x

    cosh² x − sinh² x = 1

1.24 SINE, COSINE, TAN AND COT FUNCTIONS

For all values of θ,

If

and            

Similarly,

                  tan jx = j tanh x

                 sinh jx = j sin x

                cosh jx = cos x

                 tanh jx = j tan x

                sin (−x) = −sin x

               cos (−x) = cos x

               1 degree = 0.01745 radian

                1 radian = 57.29577 degrees

      sin² x + cos² x = 1

            sin (x + y) = sin x cos y + cos x sin y

            sin (x − y) = sin x cos y − cos x sin y

           cos (x + y) = cos x cos y − sin x sin y

           cos (x − y) = cos x cos y + sin x sin y

                   sin 2x = 2 sin x cos x

                  cos 2x = cos² x − sin² x

1.25 SOME SPECIAL FUNCTIONS

Gamma function

Γ (α) is defined as

This is true if α > 0

Beta function

Error function

Bessel function

 

Considering circular waveguides, solutions of the axial component is written conveniently in terms of Bessel function. For transverse magnetic wave, the solution is

 

for transverse electric wave

 

 

where	Jn (Kc r) = Bessel function of the first kind
	r = radius of the guide
	Kc = cut-off wave number

       A, B, A¹, B¹, = constants

The solutions for the Bessel function are obtained for certain values of K_c. These values of K_c are known as eigen values. If K_c is to produce solution of the Bessel function, (K_c r) must be the roots of the Bessel function. Then

            Jn (K_c r) = 0

The propagation parametres for nm^th mode of TM waves are

Phase constant,

where            

Also,            

 

where	pnm = the roots of the Bessel function
	K = free space wave number

The roots of the Bessel function for TM mode are shown in Table 1.2.

 

The roots of the Bessel function for TE mode are shown in Table 1.3.

 

The propagation parametres for TE_nm mode are

where                

So,

The variation of the first three orders of the Bessel function and their roots are shown in Fig. 1.6.

Fresnel integral

Sine integral

Cosine integral

Exponential integral

Logarithmic integral

1.26 PARTIAL DERIVATIVE

Assume z = f (x, y) is a real function of two independent real variables, x and y. If x is kept constant, say x = x₁ and y is a variable, then f (x₁, y) depends on y only. If the derivative of f (x, y₁) with y for a value y = y₁ exists, then the value of this derivative is called partial derivative of f (x, y) with respect to y at the point (x₁, y₁). It is represented by

1.27 SOME DIFFERENTIATION FORMULAE

Assume u, v to be functions of x and let a, n be the constants. Then,

1.28 SOME USEFUL INTEGRATION FORMULAE

1.29 RADIAN AND STERADIAN

This is shown in Fig. 1.7. It is well known that the circumference of a circle is 2πr and there are 2π radians in a full circle.

The differential area ds on the surface of a sphere of radius r is

                ds = r² sin θ dθ dϕ (m²)

The element of solid angle dΩ of the sphere is given by

This is illustrated in Fig. 1.8.

1.30 INTEGRAL THEOREMS

Stokes’s Theorem

 

Divergence Theorem

If a field vector like D and its first derivatives are continuous at all points in a region of volume V bounded by a closed elementary surface dS, then Divergence theorem states that

This means a surface integral is converted into volume integral and vice versa by Divergence theorem.

Problem 1.1    If a vector A = a_x + 2a_y + 3a_z , find its magnitude and direction.

Solution    A = a_x + 2a_y + 3a_z ,

The magnitude of

Its direction is given by unit vector,

Problem 1.2    If A = 2a_x + 5a_y + 6a_z and B = a_x – 3a_y + 6a_z, find A + B and A – B.

Solution

Problem 1.3    If A = a_x + a_y + 2a_z and B = 2a_x + a_y + a_z, find A.B.

Solution

            A.B = A_xB_x + A_yB_y + A_zB_z

                   = 1.2 + 1.1 + 2.1 = 2 + 1 + 2 = 5

Problem 1.4    Given A = 2a_x + a_y + 2a_z and B = a_x + 2a_y + a_z, find A × B.

Solution

Problem 1.5    For A = (1,3,4) and B = (1, 0, 2), find A .B.

Solution

 

Here	Ax = 1,	Ay = 3,	Az = 4
	Bx = 1,	By = 0,	Bz = 2

 

	A.B	=	AxBx + AyBy + AzBz
		=	1 × 1 + 3 × 0 + 4 × 2
		=	1 + 0 + 8 = 9

Problem 1.6    In Cartesian coordinates, a point is described by P (1, 2, 4). What are the orthogonal planes whose intersection give this point?

Solution    The planes which intersect at the point P (1, 2, 4) are:

                            x = 1

                            y = 2 and

                            z = 4

Problem 1.7    Represent the point P (0.5, 0.2, 0.1) in terms of unit vectors along the coordinate axes.

Solution    At the point, P (0.5, 0.2, 0.1), the coordinates are x = 0.5, y = 0.2 and z = 0.1. Therefore, P is represented by a vector

 

Problem 1.8    How is a point P (2.0 m, π/4, 1.0 m) obtained in cylindrical coordinates?

Solution    The point P (2.0 m, π/4, 1.0 m) is obtained by the intersection of a cylindrical surface, ρ = 2.0 m a plane, ϕ = π/4 and a plane, z = 1.0 m.

Problem 1.9    A point in Cartesian coordinates is given by P (1, 2, 3). Express it in cylindrical coordinates.

Solution    The coordinates of P in Cartesian system are

 

The coordinates of P in cylindrical system are given by

In cylindrical coordinates P is (2.236, 63.43°, 3).

Problem 1.10    If a vector, A is 4a _x + 2a_y + a_z, express it in cylindrical coordinate system.

Solution                A = 4a_x + 2a_y + a_z

We have

that is,

 

	Aρ	=	3.57 + 0.894 = 4.46
	Aϕ	=	−Ax sin ϕ + Ay cos ϕ
		=	− 4 × 0.447 + 2 × 0.894
		=	− 1.789 + 1.789
		=	0
	Az	=	1.0

A in cylindrical coordinates is

 

Problem 1.11    How is a point P (1.0m, 10°, 30°) obtained in spherical coordinates?

Solution    The point P (1.0m, 10°, 30°) is obtained by the intersection of a sphere, r = 1.0m a cone, θ= 10° and a plane, ϕ = 30°.

Problem 1.12    If a point in Cartesian coordinates is given by P (1, 2, 3), express it in spherical coordinates.

Solution    The Cartesian coordinates of P are

 

The coordinates of P in spherical system are given by

Problem 1.13    A scalar function, V is given by V = xyz². Find the gradient of V.

Solution    Grad

Problem 1.14    If a vector, B = 4xy²a_x + 2y³a_y + xyza_z, find the divergence of B.

Solution

Problem 1.15    Given a vector, A = 3xa_x + ya_y + 5za_z, find the curl of A.

Solution

Problem 1.16    If the scalar potential is given by V = x² – y² – z² volts, find the Laplacian of V.

Solution                V = x² – y² – z² volts

Problem 1.17    If power gain, A_p is 22, find its value in decibels.

Solution    Power gain = 22

 

Ap in	dB = 10 log10 22
	= 13.424 dB

Problem 1.18    When the ratio of output and input voltages of an amplifier is 95, find its gain in dB.

Solution         A_V = voltage gain = 95

 

AV in dB	= 20 log10 95
	= 39.55 dB

Problem 1.19    If the power gain in a communication system is 16, what is its value in nepers?

Solution    A_p = power gain = 16

Problem 1.20    If the current gain in a circuit is 34, what is its value in nepers?

Solution    A_I Current gain      = 34

 

AI in Np	= loge 34
	= 3.526

Problem 1.21    If a complex number, A is given by 2 + j4, find its magnitude and phase.

Solution    A = 2 + j4

Magnitude of
Phase of A,

Problem 1.22    If a complex number is given by A = 1 + j3, find its complex conjugate, magnitude of A and its phase.

Solution    The complex conjugate of A is

 

	A * = 1 − j3
The magnitude of
The phase of

Problem 1.23    If the amplitude of a complex number, A is 5.0 and its phase is 45°, find its real and imaginary parts.

Solution    The given complex number

 

	A = 5.0 ∠45°
Its magnitude,	ρ = 5.0
Its phase,	ϕ = 45°
So the real part of	A = ρ cos ϕ = 5 cos 45° = 3.535
The imaginary part of	A = ρ sin ° = 3.535
A is represented as 3.535 + j3.535.

Problem 1.24    If A₁ = 2 + j3 and A₂ = 4 + j5, find the sum of A₁ and A₂.

Solution

             A₁ = 2 + j3, A₂ = 4 + j5

             A = A₁ + A₂ = 2 + j3 + 4 + j5 = 6 + j8

Problem 1.25    If A₁ = j6 and A₂ = 1 – j2, find A₁ – A₂.

Solution

                     A₁ = j6, A₂ = 1 − j2

            A₁ − A₂ = j6 − 1+ j2 = −1 + j8

Problem 1.26    Given two complex numbers, A = 0.4 + j5, B = 2 + j3, find the product of A and B.

Solution        A = 0.4 + j5

                         B = 2 + j3

The product,

Problem 1.27    Given two complex numbers, A = 10 + j6 and B = 2 – j3, find the ratio of A and B.

Solution

Problem 1.28    Find the roots of the quadratic equation x² + 2x + 4 = 0.

Solution    For x² + 2x+4 =0, the roots are

Problem 1.29    For the determinant, find the minor of 8 and 9.

Solution    The minor of the element, 8 is

The minor of the element, 9 is

Problem 1.30    For the determinant, find the cofactor of 6 and 5.

Solution The    minor of 6 is

Its cofactor

The minor of 5 is

Its cofactor

Problem 1.31    Find the factorial of 4 and 6.

Solution    Factorial of    4 = 4 × 3 × 2 × 1 = 24

Factorial of           6 = 6 × 5 × 4 × 3 × 2 × 1 = 720

 

1. The wavelength of an EM wave depends on its velocity.	(Yes/No)
2. The velocity of an EM wave depends on the medium in which it propagates.	(Yes/No)
3. The unit vectors along the Cartesian coordinate axes are not mutually perpendicular to each other.	(Yes/No)
4. In cylindrical coordinates, ρ has the unit of degrees.    (Yes/No)
5. In cylindrical coordinates, ρ has the units of metres.	(Yes/No)
6. In spherical coordinates, θ increases in anticlockwise direction.	(Yes/No)
7. In cylindrical coordinates, ϕ increases in anticlockwise direction.	(Yes/No)
8. In spherical coordinates, θ increases in clockwise direction.	(Yes/No)
9. Del is a scalar operator.	(Yes/No)
10. ∇2 is a vector operator.	(Yes/No)
11. ∇2 can be operated on scalar and vector.	(Yes/No)
12. sin jx is equal to j sinh x.	(Yes/No)
13. sin jx is equal to j sin x.	(Yes/No)
14. cosh jx is cos x.	(Yes/No)
15. tanh jx is j tan x.	(Yes/No)
16. Fresnel integrals C(∞) = S(∞).	(Yes/No)
17. Fresnel integral,	(Yes/No)
18. Error function, erf (∞) = 1.	(Yes/No)
19. In cylindrical coordinates, ϕ = a constant is a plane.	(Yes/No)
20. In spherical coordinates, θ = a constant is a cone.	(Yes/No)
21. ax.aρ = sin ϕ.	(Yes/No)
22. ax.aϕ = sinϕ.	(Yes/No)
23. ay.aρ = 0.	(Yes/No)
24. az.aρ = 0.	(Yes/No)
25. az.aϕ = 0.	(Yes/No)
26. ay.aϕ = cos ϕ.	(Yes/No)
27. The gradient of a vector does not exist.	(Yes/No)
28. Divergence of a scalar does not exist.	(Yes/No)
29. Curl of a scalar exists.	(Yes/No)
30. Del (∇) has no units.	(Yes/No)
31. Divergence of water or oil is almost zero.	(Yes/No)
32. Divergence of gradient of scalar electric potential is the Laplacian of the potential.	(Yes/No)
33. j = (0, 1).	(Yes/No)
34. Complex conjugate of A = 2 + j5 is 2 – j5.	(Yes/No)
35. A complex number B = (x, y) is nothing but (x + jy).	(Yes/No)
36.	(Yes/No)
37. 1 Neper = 8.686 dB	(Yes/No)
38. loge x = 0.434 log10 x	(Yes/No)
39. A . A = A . A* = A2.	(Yes/No)
40. Radar X-band is ________.
41. The frequency range of radar mm-band is ________.
42. The frequency range UHF is ________.
43. The wavelength range of VHF band is ________.
44. In cylindrical coordinates, ρ is the ________.
45. In spherical coordinates, r is ________.
46. dv in cylindrical coordinates is ________.
47. dL in spherical coordinates is ________.
48. Dot product of az and aϕ is ________.
49. Dot product of ax and ar is ________.
50. Dot product of ax and ay is ________.
51. In spherical coordinates, θ varies between ________.
52. In spherical coordinates, ϕ varies between ________.
53. The divergence of an incompressible fluid is ________.
54. Del has ________ units.
55. A point is obtained by the intersection of ________ in spherical coordinates.
56. 1 NP is equal to ________.
57. 1 dB is defined as ________.
58. 1 NP is defined as ________.
59. 0! is ________.
60. For x << 1, (1 + xit)–1 = ________.
61. For x<< 1, (1 + x)+1 = ________.
62. sinh 0 is ________.
63. tanh 0 is ________.
64. cosh 0 is ________.
65. Gamma function, is ________.

1. Find the magnitude and direction for the vectors A = 5a_x + 3a_y + 2a_y and B = 4a_y + 0.5a_z.
2. If A = (2, 4, 1) and B = (0, 0.1, 5), find A + B, A – B.
3. If A = 6a_x + 3a_z and B = 2a_y + 5a_z, find A.B.
4. When A = (2, 0, 7), B = (3, 2, 0), what is A × B??
5. If a point in Cartesian coordinates is given by (2, 0, 4), express it in cylindrical and spherical coordinate systems.
6. The electric scalar potential is given by V = 3x + 5y² + 4z³. Find its gradient.
7. For a vector A = 12a_x + 2a_y + 3a_z, find the divergence and curl.
8. If a voltage gain is 22 dB, express it in nepers.
9. Find the roots of the quadratic equation 5x² + 2x + 4 = 0.
10. What are the roots of the cubic equation given by x³ – 2x² + 4x-8 = 0?
11. Find the value of the determinant .
12. Find the minor of 0 in Problem 11.
13. What is the cofactor of 4 and 3 in the determinant
14. Find the value of (1 + x)^1/2 if x = 0.01, using the series expansion.
15. Find sin x if x = 0.2, using the series expansion.