## Np Bali Engineering Mathematics 2nd Sem Pdf 11: A Review of a Popular Textbook

Np Bali Engineering Mathematics 2nd Sem Pdf 11 is a textbook of engineering mathematics for the second semester of engineering courses. It is written by N. P. Bali, a renowned author and professor of mathematics. The book covers topics such as complex numbers, theory of equations and curve fitting, determinants and matrices, analytical solid geometry, partial differentiation, multiple integrals, vector algebra and calculus, applications of partial differential equations, Laplace transforms, functions of a complex variable, integral transforms, statistics and probability, finite differences and numerical methods, Z-transforms, numerical solutions of ordinary and partial differential equations, infinite series, Fourier series, linear differential equations, special functions and series solutions of differential equations, curvilinear coordinates, tensor analysis and virtual work.

The book is designed to provide a comprehensive and rigorous treatment of the subject matter with numerous examples and exercises. The book also includes appendices on mathematical tables and formulas, answers to selected problems and an index. The book is suitable for students of engineering, physics and applied sciences who want to master the fundamentals of engineering mathematics.

## Np Bali Engineering Mathematics 2nd Sem Pdf 11

**DOWNLOAD: **__https://bltlly.com/2tGseT__

The book has received positive reviews from students and teachers alike for its clarity, accuracy and relevance. The book is available in both print and digital formats from Laxmi Publications Pvt Limited[^1^] [^2^]. The digital format can be downloaded as a pdf file from various online sources[^3^]. The book is a valuable resource for anyone who wants to learn engineering mathematics in a systematic and effective way.

In this article, we will briefly discuss some of the main topics covered in the book and highlight their importance and applications in engineering. We will also provide some sample problems and solutions to illustrate the concepts and techniques.

## Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit such that iÂ = -1. Complex numbers are used to represent quantities that have both magnitude and direction, such as electric currents, voltages, impedances, phasors, waves and signals. Complex numbers can be represented in different forms, such as rectangular, polar, exponential and trigonometric forms. Complex numbers can be added, subtracted, multiplied, divided and raised to powers using certain rules and formulas. Complex numbers can also be plotted on a complex plane using their real and imaginary parts as coordinates.

One of the important concepts in complex numbers is the complex conjugate, which is obtained by changing the sign of the imaginary part of a complex number. The complex conjugate of a + bi is a - bi. The complex conjugate has some useful properties, such as:

The product of a complex number and its conjugate is always a real number equal to the square of its modulus or absolute value.

The quotient of two complex numbers can be simplified by multiplying both the numerator and denominator by the conjugate of the denominator.

The sum and difference of two complex conjugates are always real numbers.

Another important concept in complex numbers is the argument or phase angle, which is the angle that a complex number makes with the positive real axis on the complex plane. The argument can be calculated using trigonometric functions or inverse trigonometric functions. The argument can also be expressed in radians or degrees. The argument has some useful properties, such as:

The argument of a product of two complex numbers is equal to the sum of their arguments.

The argument of a quotient of two complex numbers is equal to the difference of their arguments.

The argument of a power of a complex number is equal to the product of its argument and the exponent.

Some examples of problems involving complex numbers are:

Find the rectangular form of (3 + 4i)(2 - i).

Find the polar form of 5 - 12i.

Find the exponential form of 2(cos 60Â + i sin 60Â).

Find the trigonometric form of e^(iÏ/4).

Find the value of i^i.

Some solutions are:

(3 + 4i)(2 - i) = 6 + 8i - 3i - 4iÂ = 6 + 5i + 4 = 10 + 5i.

5 - 12i = r(cos Î + i sin Î), where r = â(5Â + 12Â) = 13 and Î = tanâÂ(-12/5) â -1.176 radians or -67.38 degrees. So, 5 - 12i = 13(cos(-1.176) + i sin(-1.176)).

2(cos 60Â + i sin 60Â) = 2e^(iÏ/3), where e is the base of natural logarithms.

e^(iÏ/4) = cos(Ï/4) + i sin(Ï/4) = â2/2 + iâ2/2.

i^i = e^(i ln i) = e^(i (ln i + i arg i)) = e^(i (0 + i Ï/2)) = e^(-Ï/2).

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