## 1. Mathematical Tools

• We use math in almost every problem we solve. As a result the more relevant topics of mathematics are summarized here.

• This is not intended for learning, but for reference.

1.1 Introduction

• This section has been greatly enhanced, and tailored to meet our engineering requirements.

• The section outlined here is not intended to teach the elements of mathematics, but it is designed to be a quick reference guide to support the engineer required to use techniques that may not have been used recently.

• For those planning to write the first ABET Fundamentals of Engineering exam, the following topics are commonly on the exam.

straight line equations: slop and perpendicular

conics, circles, ellipses, etc.

matrices, determinants, adjoint, inverse, cofactors, multiplication

limits, L’Hospital’s rule, small angle approximation

integration of areas

complex numbers, polar form, conjugate, addition of polar forms

maxima, minima and inflection points

first order differential equations: guessing and separation

second order differential equation: linear, homogeneous, non-homogeneous, second order

triangles, sine, cosine, etc.

integration: by parts and separation

solving equations using inverse matrices, Cramer’s rule, substitution

eigenvalues, eigenvectors

dot and cross products, areas of parallelograms, angles and triple product

divergence and curl: solenoidal and conservative fields

centroids

integration of volumes

integration using Laplace transforms

probability: permutations and combinations

mean, standard deviation, mode, etc.

log properties

Taylor series

partial fractions

basic coordinate transformations: cartesian, cylindrical, spherical

trig identities

derivative: basics, natural log, small angles approx., chain rule, partial fractions

#### 1.1.1 Constants and Other Stuff

• A good place to start a short list of mathematical relationships is with greek letters

• The constants listed are amount some of the main ones, other values can be derived through calculation using modern calculators or computers. The values are typically given with more than 15 places of accuracy so that they can be used for double precision calculations.

#### 1.1.2 Basic Operations

• These operations are generally universal, and are described in sufficient detail for our use.

• Basic properties include,

1.1.2.1 - Factorial

• A compact representation of a series of increasing multiples.

#### 1.1.3 Exponents and Logarithms

• The basic properties of exponents are so important they demand some sort of mention

• Logarithms also have a few basic properties of use,

• All logarithms observe a basic set of rules for their application,

#### 1.1.4 Polynomial Expansions

• Binomial expansion for polynomials,

1.2 Functions

#### 1.2.1 Discrete and Continuous Probability Distributions

• The Binomial distribution is,

• The Poisson distribution is,

• The Hypergeometric distribution is,

• The Normal distribution is,

#### 1.2.2 Basic Polynomials

• The quadratic equation appears in almost every engineering discipline, therefore is of great importance.

• Cubic equations also appear on a regular basic, and as a result should also be considered.

• On a few occasions a quartic equation will also have to be solved. This can be done by first reducing the equation to a quadratic,

#### 1.2.3 Partial Fractions

• The next is a flowchart for partial fraction expansions.

• The partial fraction expansion for,

• Consider the example below where the order of the numerator is larger than the denominator.

• When the order of the denominator terms is greater than 1 it requires an expanded partial fraction form, as shown below.

• We can solve the previous problem using the algebra technique.

#### 1.2.4 Summation and Series

• The notation is equivalent to assuming and are integers and . The index variable is a placeholder whose name does not matter.

• Operations on summations:

• Some common summations:

for both real and complex .

for both real and complex . For , the summation does not converge.

1.3 Spatial Relationships

#### 1.3.1 Trigonometry

• The basic trigonometry functions are,

• Graphs of these functions are given below,

• NOTE: Keep in mind when finding these trig values, that any value that does not lie in the right hand quadrants of Cartesian space, may need additions of ±90° or ±180°.

• Now a group of trigonometric relationships will be given. These are often best used when attempting to manipulate equations.

• These can also be related to complex exponents,

#### 1.3.2 Hyperbolic Functions

• The basic definitions are given below,

• some of the basic relationships are,

• Some of the more advanced relationships are,

• Some of the relationships between the hyperbolic, and normal trigonometry functions are,

1.3.2.1 - Problems

Problem 1.1 Find all of the missing side lengths and corner angles on the two triangles below,

#### 1.3.3 Geometry

• A set of the basic 2D and 3D geometric primitives are given, and the notation used is described below,

• A general class of geometries are conics. This for is shown below, and can be used to represent many of the simple shapes represented by a polynomial.

#### 1.3.4 Planes, Lines, etc.

• The most fundamental mathematical geometry is a line. The basic relationships are given below,

• If we assume a line is between two points in space, and that at one end we have a local reference frame, there are some basic relationships that can be derived.

• The relationships for a plane are,

1.4 Coordinate Systems

#### 1.4.1 Complex Numbers

• In this section, as in all others, ‘j’ will be the preferred notation for the complex number, this is to help minimize confusion with the ‘i’ used for current in electrical engineering.

• The basic algebraic properties of these numbers are,

• We can also show complex numbers graphically. These representations lead to alternative representations. If it in not obvious above, please consider the notation above uses a Cartesian notation, but a polar notation can also be very useful when doing large calculations.

• We can also do calculations using polar notation (this is well suited to multiplication and division, whereas Cartesian notation is easier for addition and subtraction),

• Note that DeMoivre’s theorem can be used to find exponents (including roots) of complex numbers

• Euler’s formula:

• From the above, the following useful identities arise:

#### 1.4.2 Cylindrical Coordinates

• Basically, these coordinates appear as if the Cartesian box has been replaced with a cylinder,

#### 1.4.3 Spherical Coordinates

• This system replaces the Cartesian box with a sphere,

1.5 Matrices and Vectors

#### 1.5.1 Vectors

• Vectors are often drawn with arrows, as shown below,

• Cartesian notation is also a common form of usage.

• Vectors can be added and subtracted, numerically and graphically,

#### 1.5.2 Dot (Scalar) Product

• We can use a dot product to find the angle between two vectors

• We can use a dot product to project one vector onto another vector.

• We can consider the basic properties of the dot product and units vectors.

#### 1.5.3 Cross Product

• First, consider an example,

• The basic properties of the cross product are,

• When using a left/right handed coordinate system,

• The properties of the cross products are,

#### 1.5.5 Matrices

• Matrices allow simple equations that drive a large number of repetitive calculations: as a result they are found in many computer applications.

• A matrix has the form seen below,

• Matrix operations are available for many of the basic algebraic expressions, examples are given below. There are also many restrictions: many of these are indicated.

• The eigenvalue of a matrix is found using,

#### 1.5.6 Solving Linear Equations with Matrices

• We can solve systems of equations using the inverse matrix,

• We can solve systems of equations using Cramer’s rule (with determinants),

#### 1.5.7 Problems

Problem 1.2 Perform the matrix operations below.

Problem 1.3 Perform the vector operations below,

Problem 1.4 Solve the following equations using any technique,

1.6 Calculus

• NOTE: Calculus is very useful when looking at real systems. Many students are turned off by the topic because they "don’t get it". But, the secret to calculus is to remember that there is no single "truth": it is more a loose collection of tricks and techniques. Each one has to be learned separately, and when needed you must remember it, or know where to look.

#### 1.6.1 Single Variable Functions

1.6.1.1 - Differentiation

• The basic principles of differentiation are,

• Differentiation rules specific to basic trigonometry and logarithm functions

• L’Hospital’s rule can be used when evaluating limits that go to infinity.

• Some techniques used for finding derivatives are,

1.6.1.2 - Integration

• Some basic properties of integrals include,

• Some of the trigonometric integrals are,

• Some other integrals of use that are basically functions of x are,

• Integrals using the natural logarithm base ‘e’,

#### 1.6.2 Vector Calculus

• When dealing with large and/or time varying objects or phenomenon we must be able to describe the state at locations, and as a whole. To do this vectors are a very useful tool.

• Consider a basic function and how it may be represented with partial derivatives.

• Gauss’s or Green’s or divergence theorem is given below. Both sides give the flux across a surface, or out of a volume. This is very useful for dealing with magnetic fields.

• Stoke’s theorem is given below. Both sides give the flux across a surface, or out of a volume. This is very useful for dealing with magnetic fields.

#### 1.6.3 Differential Equations

• Solving differential equations is not very challenging, but there are a number of forms that need to be remembered.

• Another complication that often occurs is that the solution of the equations may vary depending upon boundary or initial conditions. An example of this is a mass spring combination. If they are initially at rest then they will stay at rest, but if there is some disturbance, then they will oscillate indefinitely.

• We can judge the order of these equations by the highest order derivative in the equation.

• Note: These equations are typically shown with derivatives only, when integrals occur they are typically eliminated by taking derivatives of the entire equation.

• Some of the terms used when describing differential equations are,

ordinary differential equations: if all the derivatives are of a single variable. In the example below ’x’ is the variable with derivatives.

first order differential equations: have only first order derivatives,

second order differential equations: have at least on second derivative,

higher order differential equations: have at least one derivative that is higher than second order.

partial differential equations: these equations have partial derivatives

• Note: when solving these equations it is common to hit blocks. In these cases backtrack and try another approach.

• linearity of a differential equation is determined by looking at the dependant variables in the equation. The equation is linear if they appear with an exponent other than 1.

1.6.3.1 - First Order Differential Equations

• These systems tend to have a relaxed or passive nature in real applications.

• Examples of these equations are given below,

• Typical methods for solving these equations include,

guessing then testing

separation

homogeneous

1.6.3.1.1 - Guessing

• In this technique we guess at a function that will satisfy the equation, and test it to see if it works.

• The previous example showed a general solution (i.e., the value of ’C’ was not found). We can also find a particular solution.

1.6.3.1.2 - Separable Equations

• In a separable equation the differential can be split so that it is on both sides of the equation. We then integrate to get the solution. This typically means there is only a single derivative term.

1.6.3.1.3 - Homogeneous Equations and Substitution

• These techniques depend upon finding some combination of the variables in the equation that can be replaced with another variable to simplify the equation. This technique requires a bit of guessing about what to substitute for, and when it is to be applied.

1.6.3.2 - Second Order Differential Equations

• These equations have at least one second order derivative.

• In engineering we will encounter a number of forms,

homogeneous

non-homogeneous

1.6.3.2.1 - Linear Homogeneous

• These equations will have a standard form,

• An example of a solution is,

1.6.3.2.2 - Non-homogeneous Linear Equations

• These equations have the general form,

• to solve these equations we need to find the homogeneous and particular solutions and then add the two solutions.

• Consider the example below,

1.6.3.3 - Higher Order Differential Equations 1.6.3.4 - Partial Differential Equations

• Partial difference equations become critical with many engineering applications involving flows, etc.

#### 1.6.4 Other Calculus Stuff

• The Taylor series expansion can be used to find polynomial approximations of functions.

1.7 Numerical Methods

• These techniques approximate system responses without doing integrations, etc.

#### 1.7.1 Approximation of Integrals and Derivatives from Sampled Data

• This form of integration is done numerically: this means by doing repeated calculations to solve the equation. Numerical techniques are not as elegant as solving differential equations, and will result in small errors. But these techniques make it possible to solve complex problems much faster.

• This method uses forward/backward differences to estimate derivatives or integrals from measured data.

#### 1.7.2 Euler First Order Integration

• We can also estimate the change resulting from a derivative using Euler’s equation for a first order difference equation.

#### 1.7.3 Taylor Series Integration

• Recall the basic Taylor series,

• When h=0 this is called a MacLaurin series.

• We can integrate a function by,

#### 1.7.4 Runge-Kutta Integration

• The equations below are for calculating a fourth order Runge-Kutta integration.

#### 1.7.5 Newton-Raphson to Find Roots

• When given an equation where an algebraic solution is not feasible, a numerical solution may be required. One simple technique uses an instantaneous slope of the function, and takes iterative steps towards a solution.

• The function f(x) is supplied by the user.

• This method can become divergent if the function has an inflection point near the root.

• The technique is also sensitive to the initial guess.

• This calculation should be repeated until the final solution is found.

1.8 Laplace Transforms

• The Laplace transform allows us to reverse time. And, as you recall from before the inverse of time is frequency. Because we are normally concerned with response, the Laplace transform is much more useful in system analysis.

• The basic Laplace transform equations is shown below,

#### 1.8.1 Laplace Transform Tables

• Basic Laplace Transforms for operational transformations are given below,

• A set of useful functional Laplace transforms are given below,

• Laplace transforms can be used to solve differential equations.

1.9 Fourier Series

• These series describe functions by their frequency spectrum content. For example a square wave can be approximated with a sum of a series of sine waves with varying magnitudes.

• The basic definition of the Fourier series is given below.

1.10 References

1.1 Spiegel, M. R., Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series, McGraw-Hill Book Company, 1968.