Complete Mathematics Masterclass: College & University Level

Complete Mathematics Masterclass: College & University Level

English | MP4 | AVC 1280×720 | AAC 44KHz 2ch | 250 lectures (24h 42m) | 6.12 GB

All the Applied Mathematics Needed in Science, Programming, Engineering, IT & Business – Theory, Examples & Exercises

This course is for everyone who wants to learn applied mathematics on a college and university level!
It is a complete course containing all relevant topics like Calculus, Algebra, Statistics & Stochastics.

Advanced mathematics is relevant in many fields: Programming & IT, Engineering, Science (Physics, Chemistry, Biology, Pharmacy, …), Business & Economics. This course will teach you all you have to know in 24 hours.

You are kindly invited to join this carefully prepared course in which we derive the following concepts from scratch. I will present examples and give you exercises (incl. solutions) for all topics.

College-level mathematics (10 hours)

  • Limits of functions
  • Derivatives & Integrals in 1 dimension
  • Vectors in cartesian coordinates
  • Stochastic & Probability distributions

University-level mathematics (14 hours)

  • Sequences & Series
  • Taylor expansions
  • Complex numbers
  • Derivatives & Integrals in multiple dimensions
  • Alternative coordinate systems
  • Differential equations
  • Matrix algebra
  • Fourier transforms & Delta distribution

What you’ll learn

  • Deepen your understanding of school mathematics & go beyond: All mathematical concepts that you need & learn at university
  • College-level: Functions, Limits, Derivatives, Integrals, Probabilities & Vectors
  • University-level: Derivatives & Integrals in 3 dimensions, Differential equations & Vectors in different coordinate systems
  • University-level: Complex numbers, Sequences, Series, Taylor expansions, Matrices, Eigensystems & Fourier transform
Table of Contents

[Pilot Section] If you just finished school What you know & what you will learn
1 Overview of the course
2 Overview image
3 Section intro
4 Overview of this pilot section What you know and what you don’t know yet
5 What exactly is a function
6 Linear functions
7 Quadratic functions and solving quadratic equations
8 Factorizing polynomials using their roots & Outlook Complex numbers
9 Exponential function How is it defined exactly
10 Vector algebra in polar coordinate system
11 Vector rotation using matrices
12 Section outro
13 Download the slides of this section

[Part 1] College-level What you are expected to know at university
14 Overview of the first part of this course
15 Overview image

Limits of functions
16 Section intro
17 Limits Dealing with infinity
18 Limits versus function values
19 Polynomial fractions
20 Asymptotic behavior
21 Useful rules
22 Your turn! About exercises
23 Exercises
24 Solutions of Tasks 1 & 2 – Limits of functions
25 Solutions of Tasks 3 & 4 – Limits of functions
26 Add on Two advanced lectures & results that we need later on
27 Squeeze theorem Limit of sin(x)x for x to 0
28 L’Hospital’s rule (L’Hôpital’s rule)
29 Section outro
30 Download the slides of this section

Derivatives in one dimension
31 Section intro
32 What is a derivative
33 Derivative of constant, linear and quadratic functions
34 Sum rule for derivatives
35 Derivative of polynomials
36 Product rule for derivatives
37 Derivative of the 1 x function
38 Chain rule for derivatives
39 Quotient rule for derivatives
40 Derivative of the inverse function
41 Derivative of root functions
42 Derivative of power functions
43 Derivative of exponential & logarithm functions
44 Derivative of trigonometric functions Sine & Cosine
45 Outlook Derivative of trigonometric functions by series expansion
46 Higher derivatives
47 Extrema of functions & Inflection points
48 Curve sketching
49 Summary of the rules for derivatives
50 Your turn! About exercises
51 Exercises
52 Solutions Task 1 – Derivatives
53 Solutions Tasks 2 & 3 – Derivatives
54 Add on How to calculate derivatives numerically
55 Numerics with Python Calculating derivatives
56 Download the slides of this section

Integrals in one dimension
57 Section intro
58 What is an integral
59 Integration by parts
60 Integration by substitution
61 Alternative way of using integration by substitution
62 Summary of integration rules
63 Integration & Limits – Improper integrals
64 Integration versus differentiation
65 Integration of cosine square & sine square
66 Exploiting symmetry
67 Exercises
68 Solutions Task 1 – Integrals
69 Solutions Task 2 – Integrals
70 Add on How to calculate integrals numerically
71 Numerics with Python Calculating integrals
72 Download the slides of this section

Vectors in Cartesian coordinates
73 Section intro
74 What is a vector
75 Basic vector operations
76 Dot product or scalar product
77 Cross product or vector product
78 Triple product
79 Lines in parametric form
80 Lines & Points Calculating the distance
81 Lines & Lines Identical, parallel, intersecting or skew lines
82 Planes in parametric form
83 Planes in coordinate form
84 Planes in Hesse normal form
85 Planes & Points Calculating the distance
86 Planes & Lines Included, parallel or intersecting
87 Planes & Planes Identical, parallel or intersecting line
88 Exercises
89 Solutions Task 1 – Vectors Two points
90 Solutions Task 2 – Vectors Two lines
91 Solutions Task 3 – Vector algebra Planes
92 Solutions Task 4 – Vector identities
93 Solutions Task 5 – Vectors Distance between two lines
94 Section outro
95 Download the slides of this section

Stochastic & Probability distributions
96 Section intro
97 Probability & Tree diagrams for coin flip experiments
98 Event & Counter event in a dice experiment
99 Expectation values for coin, dice & urn problems
100 Calculating probabilities Urn problems
101 Binomial distribution
102 Discussion of the binomial distribution
103 Normal distribution (Gaussian distribution)
104 Poisson distribution
105 Exercises
106 Solutions Task 1 – Probabilities
107 Solutions Task 2 – Probabilities
108 Solutions Task 3 – Probabilities
109 Section outro
110 Download the slides of this section

[Part 2] University-level What is new at university
111 Overview of the second part of this course
112 Overview image

Sequences & Series
113 Section intro
114 Sequences
115 Limits of sequences
116 Series – Harmonic & Geometric series
117 Series & Relation to improper integrals
118 Your turn! About exercises
119 Exercises
120 Solutions Task 1 – Sequences
121 Solutions Task 2- Series
122 Download the slides of this section

Taylor expansion & Series representation of Exponential & Trigonometric function
123 Section intro
124 What is a Taylor expansion
125 Proof for Taylor series
126 Examples of Taylor series Polynomials
127 Examples of Taylor series Logarithmic function
128 Examples of Taylor series 1(1-x) function
129 Series representation of the exponential function
130 Series representation of sine & cosine
131 Exercises
132 Solutions Task 1 – Taylor expansion
133 Solutions Task 2 – Taylor expansion
134 Solutions Task 3 – Taylor expansion
135 Solutions Task 4 – Taylor expansion
136 Section outro
137 Download the slides of this section

Complex numbers
138 Section intro
139 What are complex numbers
140 Addition, subtraction & complex plane
141 Multiplication & division of complex numbers
142 Exponentials and polar representation of complex numbers
143 Exercises
144 Solutions – Complex numbers
145 Factorization of polynomials Fundamental theorem of Algebra
146 Section outro
147 Download the slides of this section

Differential calculus II Multiple dimensions
148 Section intro
149 Multidimensional functions
150 Partial derivatives & Directional derivatives
151 Total derivatives & Chain rule
152 Nabla operator Gradient
153 Nabla operator Divergence
154 Nabla operator Curl
155 Laplace operator
156 Nabla Useful relations
157 Exercises
158 Solutions – Derivatives in multiple dimensions
159 Finding local extrema in multiple dimensions
160 Taylor expansion of multidimensional functions
161 Section outro
162 Download the slides of this section

Integral calculus II Multiple dimensions & Alternative coordinate systems
163 Section intro
164 Part 1 Integration in Cartesian coordinates
165 Revision Integrals in one dimension
166 Integrals in two dimensions
167 Order of integration
168 Multidimensional integrals
169 Volume of a pyramid Integration in three dimensions by parametrization
170 Arc length – Length of a curve
171 Arc length Examples
172 Line integrals Scalar functions
173 Line integrals Vector functions
174 Difficult example Volume of a sphere in Cartesian coordinates
175 Part 2 Integration in spherical, cylindrical & polar coordinate systems
176 Alternative coordinate systems Why do we need them
177 Polar coordinates – Alternative coordinate systems 13
178 Cylindrical coordinates – Alternative coordinate systems 23
179 Spherical coordinates – Alternative coordinate systems 33
180 Volume and surface elements in spherical coordinates
181 Integration in spherical coordinates Volume & Surface area of a sphere
182 Volume and surface elements in cylindrical coordinates
183 Integration in cylindrical coordinates Solids of revolution
184 Surface elements in polar coordinates
185 Integration in polar coordinates Gaussian integral
186 Exercises
187 Solutions Task 1 – Integrals in multiple dimensions
188 Solutions Task 2 – Integrals in multiple dimensions
189 Solutions Task 3&4 – Integrals in multiple dimensions
190 Section outro
191 Download the slides of this section

Differential equations
192 Section intro
193 What are differential equations
194 Classification of differential equations Ordinary (ODE) versus partial (PDE)
195 Classification of ordinary differential equations
196 Trivial case Direct integration
197 Homogeneous linear differential equations, superposition & Exponential ansatz
198 Example for a linear ODE & using the exponential ansatz The harmonic oscillator
199 Inhomogeneous linear differential equations Homogeneous plus specific solution
200 Example for solving an inhomogeneous linear differential equation
201 Integration factor method – Linear ODE with non-constant coefficients
202 Example Integration factor method
203 Bernoulli equations Non-linear ODE
204 Example Bernoulli equations
205 Separation of variables – Nonlinear ordinary differential equations
206 Exercises
207 Solutions Tasks 1&2 – Differential equations
208 Solutions Tasks 3&4 – Differential equations
209 Solutions Task 5 – Differential equations
210 Numerical methods – Euler method
211 Numerically solving higher order differential equations
212 Add on Solving differential equations numerically
213 Numerics with Python Solving the differential equations of coupled oscillators
214 Partial differential equations The heat equation
215 Download the slides of this section

Matrices & Eigensystems
216 Section intro
217 What is a matrix
218 Matrix addition & subtraction
219 Matrix multiplication
220 Example Matrix multiplication
221 Transpose matrix
222 Calculating the determinant of a matrix
223 Example Calculating the determinant of a matrix
224 Calculating the inverse matrix
225 Example Calculating the inverse matrix
226 Inverse matrix Determinant method
227 Eigensystems Eigenvalues & Eigenvectors of a matrix
228 Example Calculating eigenvalues & eigenvectors
229 Trace of a matrix
230 Special matrices Symmetric & Hermitian matrices
231 Special matrices Unitary matrices
232 Special matrices Rotational matrices
233 Positive definite, negative definite & indefinite matrices
234 Exercises
235 Solutions Task 1&2 – Matrices
236 Solutions Task 3 – Matrices
237 Solutions Task 4&5 – Matrices
238 Add on Calculating the eigenvalues of a matrix numerically
239 Numerics with Python Determining the eigenfrequencies of 3 coupled oscillators
240 Download the slides of this section

Fourier transform & Delta distribution
241 Section intro
242 What is a Fourier transform
243 Example Fourier transform a Gaussian function
244 Example Fourier transform a harmonic function
245 Delta distribution
246 Harmonic functions & Delta distribution
247 Add on Calculating the Fourier transform numerically
248 Numerics with Python Eigenfrequencies of 3 oscillators using Fourier transform
249 Download the slides of this section
250 Goodbye!

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