Mathematics for Engineers. Volume 2: Integral Calculus, Taylor and Fourier Series, Calculus for Multivariable Functions, 1st Order Differential Equations, Laplace Transform

Table of Contents
8 Integral Calculus
	8.1 Integration
	8.2 Fundamental Theorem of Calculus
	8.3 Rules of Integral Calculus
	8.4 Integration Methods
		8.4.1 Integration by Parts
		8.4.2 Integration by Substitution
		8.4.3 Decomposition into Partial Fraction
	8.5 Improper Integrals
	8.6 Applications of Integral Calculus
		8.6.1 Area calculations
		8.6.2 Kinematics
		8.6.3 Electrodynamics
		8.6.4 Work Integrals
		8.6.5 Linear and Quadratic Mean Values
		8.6.6 Center of Gravity of a Plane Surface
		8.6.7 Averaging property
		8.6.8 Arc length
		8.6.9 Curvature
		8.6.10 Volume and Surface Area of Bodies of Revolution
	8.7 Problems on Integral Calculus
9 Taylor Series
	9.1 Series of Numbers
		9.1.1 Majorant Criterion
		9.1.2 Quotient criterion
		9.1.3 Leibniz Criterion
	9.2 Power Series
	9.3 Taylor Series
	9.4 Applications
	9.5 Complex Functions
		9.5.1 Complex Power Series
		9.5.2 Euler's Formula
		9.5.3 Properties of the Complex Exponential Function
		9.5.4 Complex Hyperbolic Functions
		9.5.5 Differentiation and Integration
	9.6 Problems on Series and Taylor Series
10 Dierential Calculus for Multivariate Functions
	10.1 Multi-Variable Functions
		10.1.1 Introduction and Examples
		10.1.2 Representing Functions with Two Variables
	10.2 Continuity
	10.3 Differential Calculus
		10.3.1 Partial Derivative
		10.3.2 Total Differentiability
		10.3.3 Gradient and Directional Derivative
		10.3.4 Taylor's Theorem
	10.4 Applications of Differential Calculus
		10.4.1 The Differential as Linear Approximation
		10.4.2 Error Calculation
		10.4.3 Local Extrema for Functions with Two Variables
		10.4.4 Relative Extremes for Functions with Multiple Variables
		10.4.5 Compensation of Measurement Errors; Regression Line
	10.5 Problems on Differential Calculus
11 Positive Definite Matrices
	11.1 Eigenvalues and Eigenvectors
	11.2 Calculation of the Eigenvalues of a Matrix
	11.3 Calculation of the Eigenvectors
	11.4 Symmetric Matrices
	11.5 Positive Definite Matrices
	11.6 Problems on Positive Definite Matrices
12 Integral Calculus for Multi-Variable Functions
	12.1 Double Integrals (Domain Integrals)
		12.1.1 Definition
		12.1.2 Calculating Double Integrals
		12.1.3 Domain Decomposition
		12.1.4 Applications
	12.2 Triple Integrals
		12.2.1 Definition and Calculation of Triple Integrals
		12.2.2 Applications
	12.3 Problems on Integral Calculus
13 First-Order Differential Equations
	13.1 Introduction Problems
	13.2 Solving Homogeneous Differential Equations
	13.3 Solving Inhomogeneous Differential Equations
	13.4 Linear Differential Equations with Constant Coecients
		13.4.1 Homogeneous Solution
		13.4.2 Particular Solution
	13.5 First-Order Non-linear Differential Equations
		13.5.1 Differential Equation with Separable Variables
		13.5.2 Solving Differential Equations by Substitution
		13.5.3 Power Series Approach
	13.6 Numerical Solution of 1st Order DEq
		13.6.1 Directional Fields
		13.6.2 Euler's Line Traction Method
	13.7 Problems on First-Order Differential Equations
14 Laplace Transform
	14.1 Laplace Transform
	14.2 Inverse Laplace Transform
	14.3 Two Basic Properties
		14.3.1 Linearity
		14.3.2 Laplace Transform of the Derivative
	14.4 Methods of the Inverse Transform
	14.5 Applications of the Laplace Transform
	14.6 Properties of the Laplace Transform
		14.6.1 Shift Theorem
		14.6.2 Damping Theorem
		14.6.3 Similarity Theorem
		14.6.4 Convolution Theorem
		14.6.5 Limit Theorem
	14.7 Problems on the Laplace Transform
Index of Volume 1
Index
Additional Material