Complete Mathematics Masterclass: College & University Level
Loại khoá học: Teaching & Academics
All the Applied Mathematics Needed in Science, Programming, Engineering, IT & Business - Theory, Examples & Exercises
Mô tả
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
Why me?
My name is Börge Göbel and I am a postdoc working as a scientist in theoretical physics.
I have refined my advisor skills as a tutor of Bachelor, Master and PhD students in theoretical physics and have other successful courses here on Udemy.
I always had a passion for the mathematical side of science. Still today, I use the concepts of this course on a daily basis when I am programing on the PC or when I have to solve mathematical problems analytically on a sheet of paper.
I hope you are excited and I kindly welcome you to our course!
Bạn sẽ học được gì
Yêu cầu
Nội dung khoá học
16 sections
Overview of the course
04:51
Overview image
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Section intro
02:20
Overview of this pilot section: What you know and what you don't know yet
05:50
What exactly is a function?
07:49
Linear functions
07:12
Quadratic functions and solving quadratic equations
13:55
Factorizing polynomials using their roots & Outlook: Complex numbers
14:42
Exponential function: How is it defined exactly?
13:03
Vector algebra in polar coordinate system
09:32
Vector rotation using matrices
05:39
Section outro
00:52
Download the slides of this section
00:04
Overview of the first part of this course
01:22
Overview image
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Section intro
01:02
Limits: Dealing with infinity
08:02
Limits versus function values
08:13
Polynomial fractions
09:16
Asymptotic behavior
07:13
Useful rules
01:52
Your turn! About exercises
01:21
Exercises
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Check your solutions
12 questions
Solutions of Tasks 1 & 2 - Limits of functions
05:06
Solutions of Tasks 3 & 4 - Limits of functions
13:04
Add on: Two advanced lectures & results that we need later on
01:55
Squeeze theorem: Limit of sin(x)/x for x to 0
09:43
L'Hospital's rule (L'Hôpital's rule)
05:14
Section outro
00:16
Download the slides of this section
00:04
Section intro
02:46
What is a derivative?
07:16
Derivative of constant, linear and quadratic functions
08:45
Sum rule for derivatives
04:02
Derivative of polynomials
12:31
Product rule for derivatives
08:59
Derivative of the 1 / x function
04:00
Chain rule for derivatives
10:46
Quotient rule for derivatives
06:18
Derivative of the inverse function
04:38
Derivative of root functions
05:13
Derivative of power functions
03:50
Derivative of exponential & logarithm functions
05:47
Derivative of trigonometric functions: Sine & Cosine
04:45
Outlook: Derivative of trigonometric functions by series expansion
08:14
Higher derivatives
03:38
Extrema of functions & Inflection points
13:42
Curve sketching
07:41
Summary of the rules for derivatives
02:22
Your turn! About exercises
00:28
Exercises
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Check your solutions
6 questions
Solutions Task 1 - Derivatives
12:07
Solutions Tasks 2 & 3 - Derivatives
07:26
Add on: How to calculate derivatives numerically
03:07
Numerics with Python: Calculating derivatives
19:15
Download the slides of this section
00:04
Section intro
01:48
What is an integral?
10:25
Integration by parts
09:57
Integration by substitution
09:27
Alternative way of using integration by substitution
06:19
Summary of integration rules
04:00
Integration & Limits - Improper integrals
06:23
Integration versus differentiation
05:03
Integration of cosine square & sine square
05:06
Exploiting symmetry
06:17
Exercises
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6 questions
Solutions Task 1 - Integrals
09:02
Solutions Task 2 - Integrals
10:05
Add on: How to calculate integrals numerically
01:25
Numerics with Python: Calculating integrals
15:08
Download the slides of this section
00:04
Section intro
01:34
What is a vector?
05:02
Basic vector operations
07:56
Dot product or scalar product
06:20
Cross product or vector product
07:11
Triple product
04:42
Lines in parametric form
06:53
Lines & Points: Calculating the distance
10:46
Lines & Lines: Identical, parallel, intersecting or skew lines
12:44
Planes in parametric form
04:53
Planes in coordinate form
07:35
Planes in Hesse normal form
08:27
Planes & Points: Calculating the distance
08:28
Planes & Lines: Included, parallel or intersecting
03:56
Planes & Planes: Identical, parallel or intersecting line
06:50
Exercises
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9 questions
Solutions Task 1 - Vectors: Two points
04:19
Solutions Task 2 - Vectors: Two lines
16:21
Solutions Task 3 - Vector algebra: Planes
10:03
Solutions Task 4 - Vector identities
07:22
Solutions Task 5 - Vectors: Distance between two lines
08:48
Section outro
00:29
Download the slides of this section
00:04
Section intro
00:48
Probability & Tree diagrams for coin flip experiments
05:14
Event & Counter event in a dice experiment
06:46
Expectation values for coin, dice & urn problems
09:55
Calculating probabilities: Urn problems
11:24
Binomial distribution
06:09
Discussion of the binomial distribution
09:27
Normal distribution (Gaussian distribution)
04:11
Poisson distribution
03:13
Exercises
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Check your solutions
8 questions
Solutions Task 1 - Probabilities
03:52
Solutions Task 2 - Probabilities
07:51
Solutions Task 3 - Probabilities
05:21
Section outro
00:19
Download the slides of this section
00:04
Overview of the second part of this course
04:37
Overview image
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Section intro
02:50
Sequences
09:39
Limits of sequences
07:50
Series - Harmonic & Geometric series
08:41
Series & Relation to improper integrals
04:53
Your turn! About exercises
00:49
Exercises
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Check your solutions!
8 questions
Solutions Task 1 - Sequences
05:01
Solutions Task 2 - Series
05:51
Download the slides of this section
00:04
Section intro
01:22
What is a Taylor expansion?
07:41
Proof for Taylor series
05:50
Examples of Taylor series: Polynomials
07:37
Examples of Taylor series: Logarithmic function
12:48
Examples of Taylor series: 1/(1-x) function
08:03
Series representation of the exponential function
06:27
Series representation of sine & cosine
05:08
Exercises
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Solutions Task 1 - Taylor expansion
12:44
Solutions Task 2 - Taylor expansion
03:30
Solutions Task 3 - Taylor expansion
07:29
Solutions Task 4 - Taylor expansion
03:41
Section outro
00:57
Download the slides of this section
00:04
Section intro
00:48
What are complex numbers?
09:25
Addition, subtraction & complex plane
13:08
Multiplication & division of complex numbers
09:37
Exponentials and polar representation of complex numbers
18:20
Exercises
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Check your solutions
11 questions
Solutions - Complex numbers
14:11
Factorization of polynomials: Fundamental theorem of Algebra
05:44
Section outro
00:15
Download the slides of this section
00:04
Section intro
01:19
Multidimensional functions
06:11
Partial derivatives & Directional derivatives
12:45
Total derivatives & Chain rule
03:56
Nabla operator: Gradient
04:52
Nabla operator: Divergence
10:37
Nabla operator: Curl
09:58
Laplace operator
01:26
Nabla: Useful relations
08:40
Exercises
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Solutions - Derivatives in multiple dimensions
15:33
Finding local extrema in multiple dimensions
11:33
Taylor expansion of multidimensional functions
06:49
Section outro
00:35
Download the slides of this section
00:04
Section intro
01:03
Part 1: Integration in Cartesian coordinates
00:12
Revision: Integrals in one dimension
03:59
Integrals in two dimensions
03:04
Order of integration
08:30
Multidimensional integrals
03:13
Volume of a pyramid: Integration in three dimensions by parametrization
07:32
Arc length - Length of a curve
04:54
Arc length: Examples
06:20
Line integrals: Scalar functions
07:13
Line integrals: Vector functions
03:43
Difficult example: Volume of a sphere in Cartesian coordinates
12:02
Part 2: Integration in spherical, cylindrical & polar coordinate systems
00:05
Alternative coordinate systems: Why do we need them?
03:23
Polar coordinates - Alternative coordinate systems 1/3
03:31
Cylindrical coordinates - Alternative coordinate systems 2/3
03:33
Spherical coordinates - Alternative coordinate systems 3/3
03:32
Volume and surface elements in spherical coordinates
07:49
Integration in spherical coordinates: Volume & Surface area of a sphere
07:00
Volume and surface elements in cylindrical coordinates
04:15
Integration in cylindrical coordinates: Solids of revolution
06:29
Surface elements in polar coordinates
02:58
Integration in polar coordinates: Gaussian integral
10:25
Exercises
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Solutions Task 1 - Integrals in multiple dimensions
04:33
Solutions Task 2 - Integrals in multiple dimensions
03:43
Solutions Task 3&4 - Integrals in multiple dimensions
16:25
Section outro
00:29
Download the slides of this section
00:04
Section intro
02:01
What are differential equations?
12:59
Classification of differential equations: Ordinary (ODE) versus partial (PDE)
04:58
Classification of ordinary differential equations
06:26
Trivial case: Direct integration
03:48
Homogeneous linear differential equations, superposition & Exponential ansatz
12:25
Example for a linear ODE & using the exponential ansatz: The harmonic oscillator
09:15
Inhomogeneous linear differential equations: Homogeneous plus specific solution
03:32
Example for solving an inhomogeneous linear differential equation
04:27
Integration factor method - Linear ODE with non-constant coefficients
07:14
Example: Integration factor method
05:34
Bernoulli equations: Non-linear ODE
05:06
Example: Bernoulli equations
08:21
Separation of variables - Nonlinear ordinary differential equations
07:33
Exercises
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Solutions Tasks 1&2 - Differential equations
07:16
Solutions Tasks 3&4 - Differential equations
14:38
Solutions Task 5 - Differential equations
17:30
Numerical methods - Euler method
07:14
Numerically solving higher order differential equations
05:11
Add on: Solving differential equations numerically
01:12
Numerics with Python: Solving the differential equations of coupled oscillators
17:28
Partial differential equations: The heat equation
11:17
Download the slides of this section
00:04
Section intro
01:49
What is a matrix?
10:16
Matrix addition & subtraction
02:11
Matrix multiplication
05:43
Example: Matrix multiplication
08:10
Transpose matrix
04:16
Calculating the determinant of a matrix
07:33
Example: Calculating the determinant of a matrix
08:34
Calculating the inverse matrix
04:20
Example: Calculating the inverse matrix
09:22
Inverse matrix: Determinant method
05:22
Eigensystems: Eigenvalues & Eigenvectors of a matrix
11:39
Example: Calculating eigenvalues & eigenvectors
11:54
Trace of a matrix
01:24
Special matrices: Symmetric & Hermitian matrices
09:02
Special matrices: Unitary matrices
02:53
Special matrices: Rotation matrices
04:08
Positive definite, negative definite & indefinite matrices
05:16
Exercises
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Solutions Task 1&2 - Matrices
12:55
Solutions Task 3 - Matrices
09:13
Solutions Task 4&5 - Matrices
07:53
Add on: Calculating the eigenvalues of a matrix numerically
01:30
Numerics with Python: Determining the eigenfrequencies of 3 coupled oscillators
19:07
Download the slides of this section
00:04
Section intro
01:43
What is a Fourier transform?
07:16
Example: Fourier transform a Gaussian function
07:34
Example: Fourier transform a harmonic function
08:17
Delta distribution
07:35
Harmonic functions & Delta distribution
07:51
Add on: Calculating the Fourier transform numerically
00:38
Numerics with Python: Eigenfrequencies of 3 oscillators using Fourier transform
08:26
Download the slides of this section
00:04
Goodbye!
00:50
Congratulations! Bonus Content!
00:32
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