ENGR 1320 Final Review - Math• Major Topics:

– Trigonometry– Vectors

• Dot product• Cross product

– Matrices• Matrix operations• Matrix equations• Gaussian Elimination

– Complex numbers• Polar coordinates • Exponential form

– Polynomials• Curve fitting• Roots

– Calculus

Trigonometry

• We use 3 functions over and over again: sin(θ), cos(θ), tan(θ)

• sin(θ) = y/r• cos(θ) = x/r• tan(θ) = y/x

• Example: Find the y component of this triangle:

x

y

r

θ

x

y10

pi/3

Vectors

• Vectors represent a quantity in physical space with magnitude and direction– Knowing the magnitude and angle,

trigonometry gives us the x and y components

– It works the other way too: given the x and y components, we can find the magnitude and angle• Magnitude from pythagorian theorem• Angle from arc (or inverse) tangent

• Example: What is the magnitude of this vector v

x component

y componentmagnitude

θ

10

5vθ

Vector Notation

• When representing vectors, we can either specify their magnitude and direction, or write them in components. The component method is generally more useful. We use unit vectors i and j to signify the x and y directions, respectively. So a vector that is three units in the x direction and 4 in the y direction would be written: v = 3i + 4j

• Question: What is the magnitude of this vector? The angle with the x-axis?

Vector addition

• If we have 2 vectors v1 and v2, we can add them together by adding their components:

This is the ‘tip to tail’ method

Vector Operations

• There are 2 ways of multiplying vectors– Dot product

• +

– Cross product

• See Vectors in MathCAD.pptx• Example:

What are the dot and cross products of these two vectors?

Matrices

• A matrix is a collection of values in structure.• Special matrix operations:– TransposeSee matrix math– DeterminantSee Determinants and Adjoints– InverseSee matrix inverse

Matrix Equations

• We looked at several ways to solve the equation Ax = b.– Matrix inverse (similar to dividing by a matrix)– Gaussian elimination (similar to solving

simultaneous algebraic equations)See Gaussian Elimination and More Gaussian EliminationExample: solve using the matrix inverse and gaussian elimination methods

Complex numbers

• We often find real applications that contain complex numbers in engineering– Vibrations in mechanics– Stability in control systems

• We define the complex number i to be .– Question: what is i2? I3?

• This result often arises in solving the quadratic formula:– What are the roots of See Introduction to Complex numbers

Complex numbers • We can write complex numbers in 3 ways:

– Components• Specify the real and imaginary components

– Polar• Use trigonometry to convert into angles and magnitudes on the complex plane

(Argand diagram)• See More complex numbers

– Exponential• Taking the angle and magnitude from polar form, write the complex number as

an exponential• See Exponential Form

• Example: find the roots of and write them in the 3 different forms

Polynomials

• A polynomial is an expression that follows the form:

– This polynomial is nth order• How many roots does this polynomial have?• We used mathCAD and Matlab to find roots of

polynomials of higher order than 2.• See Polynomials, Polynomials in MathCAD

Calculus

• We looked at a few basic concepts from calculus– Derivative

• The slope of a curve at any point

– Integral• The area under the curve at any point

• We won’t be using MathCAD on the exam, so you will not be asked to solve equations with derivatives or integrals, but you might be asked questions on these general concepts.

Study Strategy

• Exam problems will be similar to homeworks• Several problems have been revisited in this

class:– Electric circuits, truss equations, etc…

• Look over the first 2 exams for representative problems (particularly the 1st for math-related problems)