## Advanced Math for College Students

Join us on a journey through the fascinating realms of algebra, calculus, statistics, and beyond.

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Mastering Numerical Methods-From Theory to Practice Course

Welcome to this course on Numerical Methods and Series solutions of Differential Equations. This course is primarily intended for you if you are studying Math in College and if you are learning Engineering Math. Tips to help you understand Math better.

You will start with a brief introduction to Taylor Series method and how to use Taylor Series method to solve an equation upto 4 decimal places. As a modification of Taylor series method, you will learn modified Euler's theorem and how to use it to solve equations. As a side note, these formulas involve a lot of calculations at the problem solving stage.

Next, you will be introduced to Runge Kutta method of 4th order and how to use it in solving problems. 2 predictor and corrector methods are taught here, namely Milne' s Predictor method and Adam Bashforth methods. The numericals here involve several iterations and have been explained step by step.

In lesson 3, you will be introduced to Bessel's Differential equation and how to solve it. The solution is rather lengthy and has been explained keeping all steps in mind. This will lead you to the Bessel's function at the end . Note the use of Gamma functions in Bessel's function is shown.

Lesson 4 is on Bessel's function and it's properties. The orthogonality property of the Bessel's function is also proved leading to two cases, one of which leads to Lommel's Integral formula.

In Lesson 5, you will learn about Legendre Differential equation and how to solve it using the power series method. As a conclusion to this , you will learn about Legendre functions and how they are derived from Legendre Differential Equations. How Legendre Polynomials lead to Rodrigue's formula is also shown.

Lesson 6 is a problem solving session where you will learn to use Rodrigue's formula to solve problems.

The course concludes with an assignment which discusses possible questions that can be asked. Note that each of these questions have been discussed during the course.

You'll also learn what is nth order derivative of a function and how to evaluate it. An interesting and important topic.

Learn the method of finite differences and the forward and backward difference table and how to use it in problem solving.

An important point to keep in mind is that this course is highly theoretical and involves you to write these proofs to gain mastery.

Motivating you to learn Mathematics!

Harmonizing Mathematics-A Journey through Fourier Series

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Welcome to this course on Fourier Series. This is a part of Engineering Maths. I am Suman Mathews, math educator and tutor.

I have a double master's in Maths and I have taught Math for three decades to college and High school students. By taking this Course on Fourier Series, you will learn the basics of Analytical Mathematics.

In this course, you are introduced to the Euler's formula and basic formulas of integration which involve integrals of the product of the exponential and trigonometric functions. You learn how to write the Fourier series of a function. You learn about odd and even functions in different contexts, not just the traditional definition. Different aspects of odd and even functions are explained with examples.

The Fourier series of odd and even functions is explained. You will understand how to calculate the Fourier coefficients in different intervals for different functions. You understand the trick of solving identities using the Fourier series.There is a brief introduction to the role of periodic functions in evaluating the Fourier series.

You learn how to evaluate the Fourier series of the exponential, trigonometric, algebraic, modulus functions and more.The course concludes with half range Fourier Series

You will also learn how to calculate the cosine series and the sine series of a function. A number of problems are completely solved with steps clearly explained. The last assignment covers all the properties learnt in this course together with an understanding of the Fourier half series.

As a bonus, you will learn about Harmonic Analysis and how to calculate the 1st and 2nd harmonic for a function. Also learn how to calculate Harmonic coefficients for different types of intervals.

So let's start learning!

Course 3

Foundations of Mathematical Logic

I hope to help you understand Mathematical Logic, which is an easy, yet ,tricky part of Discrete Mathematics which is also part of College Mathematics. I can help you understand College Mathematics by breaking down complex topics into bite sized lectures.

Mathematical Logic is extensively used in Computer Science. The course introduces you to basic terminology of Logic. You'll learn about negation, conjunction, disjunction, implication, bi conditional and how to calculate the truth tables for each.

Learn about the laws of Logic next. You'll also come to understand the difference between a tautology and a contradiction. Realise how important the laws of logic are when it comes to problem solving.

You'll be introduced to the laws of Duality and the concept of NAND and NOR. How NAND and NOR are inverses of each other is also explained. The course proceeds to explain the laws of inference.

Understand the terms

Conjunctive Simplification

Disjunctive Amplification

Syllogism

Modus Pones

Modus Tollens and more.

Use these and more, to test the validity of a statement. You'll also be introduced to quantifiers, open statements and how to calculate the truth value of an open statement. You are introduced to the rules of Logic which connect universal and existential quantifiers.

Learn about Quantified statements and how to calculate their negation. Practice writing the negations of Quantified statements with a simple trick.

Master Logical Puzzles: Identify the Odd Man Out and Syllogism problems.

I hope this course helps you. Would you care to help any of your fellow students who need help in Logic, by sharing details of this course. Once again, thank you and hoping to see you in the course.

Looking forward to hearing from you!

Course 4

Beta, Gamma Functions and Laplace Transforms in Mathematics and Engineering

I welcome each of you to learning Beta and Gamma functions in Integral Calculus. You are in one of the best teaching platforms.

Beta and Gamma functions is extensively used in Calculus and in various branches of Mathematics. This course introduces you to Beta and Gamma functions, teaches you the important properties and prepares you for problem solving. As an add on, there is also an introduction to Laplace's Transforms.

Beta and Gamma functions are used in Bessel functions and Graph Theory to name a few.

You will start with the basic definition of Beta and Gamma functions, their important properties and how they are inter related. You'll learn the Duplication formula and it's applications.

Moving on to Gamma functions, you will be introduced to important concepts and properties . These involve a knowledge of integral Calculus, which is explained in detail here. You will then proceed to work out problems applying these concepts.

Problems are systematically explained with all the calculations. Towards the end of the course, I have introduced Laplace's transforms. You learn what is a Laplace transform and how to calculate it.

Learn simple formulas to help you evaluate Laplace Transforms. Learn how to evaluate Laplace transforms of product functions. Learn advanced problem solving techniques in Laplace Transforms.

Integrals in Two and Three Dimensions-Multivariable Calculus

f you want to learn more about integration in Multivariable Calculus, then join this course. If not, then you can benefit from the preview. My students have always found my teaching interesting.

Here, you will learn how to integrate double and triple integrals, Basic techniques of evaluating definite double and triple integrals are taught. You need to remember basic curves such as circle, parabola, ellipse, straight lines.

You will then proceed to evaluating double integrals over a specified region. Some of the regions explained include the first quadrant of an ellipse, a triangle bounded by the lines y = 0, y = x, y+x=2 . Given two curves, you need to note that is important to first find the point of intersection of the two curves.

Next , you will learn how to change the order of integration. How to change the limits when dx dy changes to dy dx and vice versa. How to change the limits for unbounded regions is also shown. One of the examples include changing the order of integration when the region is enclosed between a parabola and a straight line. Another example is changing the order of integration of a semicircle.

The session concludes with an assignment. You will learn how to evaluate the integral of an odd function within the limits -a to a. Integrating over a region bounded by the x axis, the ordinate x = 2a and the parabola x^2=4ay is also shown.

Bonus:You are introduced to polar coordinates and how to integrate using these. It is exciting to note how the limits change and how to calculate them. Note that polar coordinates are extremely important in Mathematics.

Learn about Vector fields and Line Integrals and how to evaluate line integrals over a curve. Also learn how to evaluate line integrals of parametric functions.

Tackle Green's Theorem-Problems and Solutions.

Mastering Partial Differential Equations-A Complete Course

Solving PDE's can be classified into the following parts

Introduction to Partial Differential Equations

You learn what is a homogeneous PDE, Both degree 1 and 2 are discussed here

Formation of PDE is taught , equation of the form f(u,v) =0 is discussed. Chain Rule is also taught when z is expresses as the sum of 2 functions. First order partial differential equations are discussed.

PDE math teaches you how to solve a non homogeneous partial differential equation by direct integration.

Steps to solve a homogeneous PDE is also discussed. An ideal partial differential equations solver.

What trends are occurring in Mathematics? The study of Partial Differential equations is definitely one which will take you long in your field of study.

This course introduces you to PDE, explains the difference between homogeneous and non homogeneous equations and how to solve each of them. You are also taught techniques on how to form a PDE. A simple course which promises you a depth of knowledge! A little knowledge of solving linear differential equations with constant coefficients is needed. This is discussed in my lecture on important tips and formulas.

There is an assignment at the end. All the formulas learnt in this course are also compiled together for you.

Don't miss out on partial differential equations examples.

Learn about polar coordinates and the relationship between rectangular and polar coordinates. Also learn basic formulas for these.

Also learn how to go beyond second order derivatives and evaluate nth order derivatives. You'll be introduced to non linear differential equations and how to solve them. Get a basic understanding of Exact Differential equations.

Explore Orthogonal Trajectories of a Curve.

Optimal Prefix Codes and Sampling Theory

Optimal Prefix Codes

A sequence consisting of only 0 and 1 is called a binary sequence. A binary sequence is used as a code for messages sent through certain channels. Now the question is, how do we decode the sequence correctly?

Given a binary sequence, a prefix code is a code, say, P, in which no sequence can be the prefix of any other sequence in that code. Prefix codes are represented by binary trees, as can be seen in the course.

We assign the symbol 0 to every edge that is directed towards the child in the left and the symbol 1 to every edge that is directed towards the child in the right. The concept of parent and child is explained during the course. My students just love it.

What is a weighted binary tree?

If we assign positive integers to the leaves of a binary tree, you get a weighted binary tree. A tree which carries the minimum weight is called an optimal binary tree.

If we assign a prefix code for the symbols representing the leaves so that we get an optimal binary tree, that prefix code is called the optimal prefix code. You'll learn a number of questions where you can construct an optimal prefix code for a string.

As a sequel to this course, you'll learn the sum and product rule. Note that sum stands for 'OR' and product stands for 'AND'.

Bonus-Understanding Sampling Theory. Learn how to calculate the expected mean and standard deviation and also learn how to use the Normal Distribution table.

Learn how to check if the given hypothesis is unbiased using probable limits and standard error.

Graph Theory Essentials-Trees and Rook Polynomials

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The course starts with the definition of trees, a cycle, a complete graph in relation to a tree. You learn that a connected graph with n vertices and n-1 edges is a tree. The relationship between number of vertices and number of edges in a tree is also shown.

Properties of a tree in Graph Theory such as sum of the degrees of vertices of a tree are discussed with proof. Problems on a tree are also shown.

Rooted trees and it's properties are discussed. What is a root? What is an m-ary tree? You will learn about the relationship between the leaves, internal vertices and number of vertices. Binary trees in Graph Theory is also introduced.

The Merge-Sort Method is taught which teaches you to sort numbers.

You will be introduced to a Forest as a collection of trees. Given 2 trees, and their edge set cardinality, you will learn how to find a relation between their vertex sets. You'll learn about pendant vertices.

If a tree has three vertices of degree two , four vertices of degree three , three vertices of degree four, you will learn how to find the number of pendant vertices in the tree and similar problems.

There is a formula handbook at the end. Learn about Rook polynomials and how to construct basic rook polynomials for 4,5,6,7 squares. Also learn how to apply Rook Polynomials in real life situations.

Learn the basics of Generating Functions.

Matrix Mathematics- Exploring Linear Algebra

You will also learn how to find two non singular matrices P and Q so that PAQ is the normal form. Next, you'll learn about Gaussian elimination method and Gauss Jordan method and the difference between the two. You'll learn what is an augmented matrix in this context.

As you proceed with the course, you'll learn about Gauss Siedel method and use it to solve a diagonally dominant system of equations. You'll also learn about diagonally dominant form and how to convert a set of equations to the diagonally dominant form. You'll also learn the traditional method of calculating inverse of a matrix using adjoint.

Next you'll learn about linear transformations in two or three variables and regular transformations. Learn about orthogonal transformations and how the matrix associated with an orthogonal transformation is called an orthogonal matrix. You'll dive into linear independence of vectors.

Learn about characteristic equation of a matrix and how to calculate eigenvalues and eigenvectors of a matrix using this. So enhance your knowledge with this course on Linear Algebra. Share this with your friends who may need this. Learn about Cayley Hamilton Theorem and diagonalisation of a matrix

You'll be introduced to linear transformations, orthonality of vectors and more. Learn how Vectors are used in Linear Algebra.

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Understanding the Mysteries of Graph Theory-An Introduction

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The course starts with a basic knowledge of Graph theory and some standard terms such as vertices and edges. You'll learn about connected graphs and solve problems based on these. Learn what are trails or circuits in graphs.

Moving on, you'll learn simple properties of graphs, such as the sum of the degrees of the vertices of a graph. You'll also learn what is a complete bipartite graph and how to calculate the total number of edges in it. The course progresses to isomorphic graphs and how to check for isomorphism in graphs.

Learn about in degree and out degree of vertices. An important concept which you'll learn next is Eulerian graphs and Eulerian circuits. Learn to determine when a connected graph has an Eulerian circuit or an Eulerian Trial. You'll also learn what are Hamiltonian graphs and how to solve problems on these.

You'll get a basic overview of regular graphs, complement of a graph, union and intersection of a graph. Also learn about ring sum of a graph and graph decomposition. Labeling the vertices and edges of a graph is also explained.

Learn how to write the Matrix representation of graphs and how to understand the incidence and adjacency matrix of a graph.

Also learn what are Digraphs and how to construct the incidence matrix for a digraph.