Advanced Calculus of Higher Mathematics from Udemy

Learn Advanced Calculus of Higher Mathematics through animation. This course includes videos explanation starting right from introduction and basics, then takes graphical and numerical phase with formulas, verification and proofs both graphically and mathematically. At the end it carries plenty of solved numerical problems with the relevant examples. The lectures' videos are appealing, attractive, fancy (with some nice graphic designing), fast and take less time to walk you through the whole lecture. It's a prefect choice for students who feel boredom watching long lectures and wants things to finish them quickly with the maximum knowledge gain. So join me here and do it in a quick and easy way. This course covers the below list of topics:

Parseval's Identity of Fourier Series
- Introduction
- Basics and Equations,
- Mathematical Proofs
- Problem Solutions
Harmonic Analysis of Fourier Series
- Introduction and Basics
- Different orders of Harmonic series
- Problem Solutions
Complex Fourier Series
- Introduction and Basics
- Prerequisites
- Equations derivation
- Mathematical Proofs
- Problem Solutions
Fourier Transform
- Introduction and Basics
- Graphs
- Fourier Sine Transform
- Fourier Cosine Transform
- Convolution theorem
- Mathematical Proofs
- Problem Solutions
Z-Transform
- Introduction and Basics
- Region of convergence
- Properties of Z-Transform
- Equation Derivation
- Mathematical Proof
- Inverse Z-Transform
- Problem Solutions
Power Series
- Introduction and Basics
- Region of convergence
- Radius of convergence
- Interval of convergence
- Differentiation of Power series
- Integration of Power series
- Equation Derivation
- Mathematical Proofs
- Problem Solutions
Binomial Series
- Introduction and Basics
- Prerequisites
- Methods to solve binomial series
- Finite series
- Infinite series
- General Term
- Binomial series as power series
- Problem Solutions

What's inside

Learning objectives

Parseval's identity of fourier series: introduction,basics, equations, mathematical proof and problem solutions
Harmonic analysis: introduction, basics, different orders of harmonic series and problem solutions
Complex fourier series : basics, prerequisites, equations derivation & mathematical proof and problem solutions
Fourier transform: introduction,basics, graphs, fourier sine & cosine transform, convolution theorem,mathematical proof and problem solutions

Z-transform: introduction, basics, region of convergence, properties,equation derivation, mathematical proof, inverse z-transform and problem solutions
Power series: introduction, basics, region and radius of convergence, interval of convergence, differentiation & integration,equation derivation, mathematical proof and problem solutions
Binomial series: introduction, basics, prerequisites, methods to solve binomial series, finite series, infinite series, general term, binomial series as a power series, and problem solutions

Parseval's identity of fourier series: introduction,basics, equations, mathematical proof and problem solutions
Harmonic analysis: introduction, basics, different orders of harmonic series and problem solutions
Complex fourier series : basics, prerequisites, equations derivation & mathematical proof and problem solutions
Fourier transform: introduction,basics, graphs, fourier sine & cosine transform, convolution theorem,mathematical proof and problem solutions
Z-transform: introduction, basics, region of convergence, properties,equation derivation, mathematical proof, inverse z-transform and problem solutions
Power series: introduction, basics, region and radius of convergence, interval of convergence, differentiation & integration,equation derivation, mathematical proof and problem solutions
Binomial series: introduction, basics, prerequisites, methods to solve binomial series, finite series, infinite series, general term, binomial series as a power series, and problem solutions

Syllabus

Introduction

They will learn everything about Perseval's Identity of Fourier Series

What is Perseval's Identity of Fourier Series? where and why we use it?

Career center

Learners who complete Advanced Calculus of Higher Mathematics will develop knowledge and skills that may be useful to these careers:

Applied Mathematician

An Applied Mathematician uses mathematical principles and techniques to solve real-world problems across diverse fields such as science, engineering, and finance. This Advanced Calculus of Higher Mathematics course provides a direct and rigorous foundation for such a career. The comprehensive coverage of Fourier Series, Fourier Transform, Z-Transform, Power Series, and Binomial Series are all core tools in applied mathematics. The course's emphasis on mathematical proofs, equation derivations, and problem solutions directly aligns with the work of an Applied Mathematician, who often develops new mathematical models and analytical methods. Entry into this field typically requires an advanced degree, such as a master's or PhD, underscoring the need for a strong theoretical background provided by such a course.

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Signal Processing Engineer

A Signal Processing Engineer designs and implements algorithms to analyze, interpret, and manipulate various types of signals, such as audio, video, or sensor data. This rigorous course in Advanced Calculus of Higher Mathematics is directly applicable to this field. The extensive coverage of Fourier Series, Fourier Transform, and Z-Transform are the bedrock of digital signal processing. Understanding Parseval's Identity, Harmonic Analysis, and Convolution Theorem is crucial for filter design, spectral analysis, and understanding system behavior in both time and frequency domains. This course can help learners develop the deep mathematical intuition and problem-solving skills necessary to excel in developing sophisticated signal processing solutions. It provides the essential mathematical framework for designing and analyzing algorithms that extract meaningful information from complex data.

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Telecommunications Engineer

A Telecommunications Engineer designs, develops, and manages systems for transmitting information across various mediums, including wireless and optical networks. The Advanced Calculus of Higher Mathematics course is exceptionally relevant for this career path. The Fourier Transform, in all its forms, and Fourier Series are indispensable for understanding signal modulation, demodulation, bandwidth, and spectral efficiency. Z-Transform is crucial for analyzing and designing digital communication systems and filters. Concepts like Parseval's Identity and Harmonic Analysis are vital for power distribution and understanding signal integrity. This course helps learners develop a deep understanding of the mathematical principles that govern signal propagation and processing, which is fundamental to building reliable and high-performance communication networks as a Telecommunications Engineer.

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Computational Scientist

A Computational Scientist develops and uses computational methods and software to solve complex scientific and engineering problems. This Advanced Calculus of Higher Mathematics course offers highly relevant mathematical foundations for this career. The course's rigorous treatment of Fourier Transforms, Z-Transforms, and Power Series is crucial for developing numerical algorithms for simulations, data analysis, and modeling physical phenomena. Understanding convergence, mathematical proofs, and problem solutions directly feeds into designing efficient and accurate computational techniques. This role often requires an advanced degree, such as a master's or PhD, as it involves both deep scientific knowledge and advanced computational skills. The course helps build the analytical and mathematical framework essential for translating complex problems into solvable computational models.

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Control Systems Engineer

A Control Systems Engineer designs, develops, and maintains systems that regulate the behavior of other devices or processes, from robotics to industrial automation. The course in Advanced Calculus of Higher Mathematics is highly relevant, particularly through its in-depth exploration of the Z-Transform. The Z-Transform is indispensable for analyzing and designing discrete-time control systems, which are prevalent in modern engineering. Understanding its Region of Convergence, properties, and inverse transformations is critical for modeling system dynamics, ensuring stability, and predicting performance. Furthermore, Fourier analysis may be useful for frequency domain analysis and understanding system response to various inputs. This course helps build a strong analytical foundation crucial for designing robust and efficient control algorithms.

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Acoustical Engineer

An Acoustical Engineer studies and manipulates sound and vibration in various environments, from architectural design to product development. This Advanced Calculus of Higher Mathematics course offers highly relevant mathematical tools for this profession. The in-depth coverage of Fourier Series and Harmonic Analysis is particularly important, as sound waves are inherently periodic and can be decomposed into their constituent frequencies. Understanding Parseval's Identity, which relates energy in time and frequency domains, and the Fourier Transform for transient signals, is key for analyzing sound phenomena, designing noise control solutions, and digital audio processing. The course helps build the analytical framework for understanding the physics of sound and vibration, making it invaluable for tackling complex acoustical challenges.

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Research Scientist (Physics)

A Research Scientist in Physics conducts theoretical and experimental investigations to expand scientific knowledge and solve complex physical problems. This Advanced Calculus of Higher Mathematics course provides fundamental mathematical tools essential for many areas of physics. Fourier Series and Fourier Transforms are critical for understanding wave phenomena, signal analysis, and solving differential equations in physics. Power Series are used extensively in perturbation theory and series solutions to various physical problems. The rigorous approach to mathematical proofs and problem-solving cultivated in this course is invaluable for theoretical modeling and data analysis in physical research. This role typically requires an advanced degree, often a PhD, making a strong foundation in higher mathematics indispensable for success.

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Image Processing Engineer

An Image Processing Engineer designs and implements algorithms for enhancing, transforming, and analyzing digital images for applications ranging from medical imaging to computer vision. The Advanced Calculus of Higher Mathematics course is highly relevant to this field. The Fourier Transform is a fundamental concept in image processing, used extensively for filtering, compression, and feature extraction by analyzing spatial frequencies within an image. Understanding concepts like convolution theorem and properties of transforms helps in designing efficient image manipulation techniques. This course provides the rigorous mathematical foundation needed to understand and innovate in areas like image reconstruction, noise reduction, and pattern recognition, which are critical tasks for an Image Processing Engineer. It helps develop the theoretical background for advanced image analysis.

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Quantitative Analyst

A Quantitative Analyst, often referred to as a quant, applies advanced mathematical and statistical methods to financial markets to develop models for pricing, risk management, and trading strategies. This course in Advanced Calculus of Higher Mathematics provides foundational analytical skills essential for this role. The study of Fourier Series, Fourier Transform, and Z-Transform may be useful for time series analysis, spectral analysis of market data, and understanding stochastic processes. Power Series and Binomial Series also contribute to developing robust financial models and understanding approximations. While typically requiring an an advanced degree like a master's or PhD in a quantitative field, the rigorous problem-solving approach and advanced mathematical tools covered help build the theoretical underpinnings for complex financial modeling challenges.

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Algorithm Developer

An Algorithm Developer designs, analyzes, and optimizes computational procedures for solving specific problems, often in software engineering, data processing, or scientific computing. This Advanced Calculus of Higher Mathematics course can be highly beneficial for learners aiming to become an Algorithm Developer. The mathematical rigor in Fourier Transforms and Z-Transforms helps with designing efficient algorithms for signal processing, data compression, and pattern recognition. Understanding Power Series and convergence properties is essential for analyzing algorithm complexity and ensuring numerical stability. The course's emphasis on mathematical proofs and deriving equations cultivates a precise, logical thinking approach critical for constructing robust and performant algorithms. It helps build a strong analytical foundation for innovative algorithm design across various domains.

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Biomedical Engineer

A Biomedical Engineer applies engineering principles to biology and medicine, designing medical devices, diagnostic equipment, and therapeutic solutions. The Advanced Calculus of Higher Mathematics course offers mathematical tools that may be useful for a Biomedical Engineer, particularly in signal and image processing within medical applications. Fourier Transforms are fundamental for analyzing biological signals like ECG or EEG data, understanding medical imaging modalities (e.g., MRI, CT), and filter design. Z-Transforms are relevant for digital signal processing of discrete biological data. This course helps learners develop the analytical foundation to interpret complex physiological signals, develop algorithms for medical diagnostics, and contribute to the innovation of healthcare technologies.

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Data Scientist

A Data Scientist analyzes and interprets complex data, developing predictive models and extracting actionable insights. The Advanced Calculus of Higher Mathematics course may be useful for learners pursuing a career as a Data Scientist. The concepts of Fourier Transform and Harmonic Analysis, for instance, are highly applicable in time series analysis for identifying periodic patterns, noise reduction, and feature engineering from temporal data. Understanding Power Series and convergence helps in grasping numerical methods and optimization algorithms commonly used in machine learning. While this role often involves statistical modeling and programming, the deep mathematical foundation from this course helps in understanding the underlying principles of various data analytical techniques and developing more sophisticated algorithms for data interpretation.

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Robotics Engineer

A Robotics Engineer designs, builds, and programs robotic systems for various applications, from manufacturing to exploration. The Advanced Calculus of Higher Mathematics course may be useful for a Robotics Engineer, particularly in areas concerning control systems, sensor data processing, and motion planning. The Z-Transform is vital for analyzing and designing discrete-time control algorithms that dictate robot movements and stability. Fourier analysis may be helpful for processing sensor data (e.g., from lidar or accelerometers) to filter noise or detect patterns, and for understanding system resonances. This course helps to build a foundational understanding of the advanced mathematical concepts required for modeling kinematic and dynamic systems, developing sophisticated control strategies, and enhancing the intelligence of robotic platforms.

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Aerospace Engineer

An Aerospace Engineer designs, develops, and tests aircraft, spacecraft, satellites, and missiles. The Advanced Calculus of Higher Mathematics course may be useful for this profession due to its application in areas like control systems and vibration analysis. Concepts such as the Z-Transform are critical for designing discrete-time control systems for flight control and navigation. Fourier analysis may be helpful for analyzing vibrations in aircraft structures, understanding aerodynamic noise, and processing sensor data. While the field encompasses many disciplines, a strong grasp of these advanced mathematical tools helps in modeling dynamic systems, ensuring stability, and optimizing performance. This course contributes to building the analytical rigor required for complex engineering challenges in aerospace.

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Machine Learning Engineer

A Machine Learning Engineer designs, builds, and deploys scalable machine learning models. The Advanced Calculus of Higher Mathematics course may be helpful for a Machine Learning Engineer in developing a deeper understanding of underlying algorithms. Concepts like Fourier Transforms and Power Series may be useful for feature engineering, especially in domains involving time series or spectral data, and for understanding the convergence properties of iterative optimization algorithms. While many machine learning tasks utilize libraries, a solid mathematical foundation helps in comprehending algorithm mechanics, debugging complex models, and contributing to research and development of novel techniques. This course helps to build the analytical rigor necessary for high-level problem-solving and model optimization in machine learning.

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