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For Real Understanding of Mathematical Concepts for Scientists and Engineers

The Simplest Way to Understand the Concept of Taylor Series Expansion
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Understanding Euler's Formula through its Derivation
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Orthogonality is the Key to Understand Fourier Series Expansion
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The Simplest Way to Understand the Concept of Taylor Series Expansion

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Video Available Hello. I am going to give you the simplest way to understand the concept of Taylor series expansion, or simply Taylor series. With this video and text, I believe you will understand the concept of Taylor series intuitively. In engineering and science, we deal with waveforms. A waveform may represent a characteristic of a phenomenon with respect to a certain variable, or variation of a quantity with the evolution of a certain variable for instance time. In order to apply a mathematical method to a given waveform, the waveform must be represented by a simple mathematical expression. If a given waveform is represented by a complicated equation or by just numerical data, then applying a certain mathematical method would be quite complicated. For representing a given waveform by a simple mathematical expression, note that there are basically three different categories. If you are interested in local behavior of a waveform, you should use Taylor series. Local behavio...
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Orthogonality is the Key to Understand Fourier Series Expansion

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Video Available I would say that "orthogonality" is the key to understand Fourier series expansion. In engineering and science, we deal with waveforms. A waveform may represent a characteristic of a phenomenon with respect to a certain variable, or variation of a quantity with the evolution of a certain variable for instance time. In order to apply a mathematical method to a given waveform, the waveform must be represented by a simple mathematical expression. If a given waveform is represented by a complicated equation or just by numerical data, then applying a certain mathematical method would be quite complicated. For representing a given waveform by a simple mathematical expression, note that there are basically three different categories. If you are interested in local behavior of a waveform, use Taylor series. If a given waveform is periodic and you want to capture the entire behavior of the waveform, then use Fourier series. If a given waveform is single-shot and...
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Understanding Euler's Formula through its Derivation

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Video Available Hello. I am going to tell you how to intuitively understand Euler's formula through its derivation. I presume that you understand the concept of Taylor series expansion. If not, please visit the video or the text entitled "The Simplest Way to Understand the Concept of Taylor Series Expansion" before reading this text. Look at this beautiful equation. This is called Euler's formula. It relates the three different mathematical concepts: exponential function, imaginary number and trigonometric functions. If you substitute , then we get This one connects exponential function, imaginary number and pi. The most important point of Euler's formula is that the term plays an important role in science and engineering as the "oscillation kernel." Ok. Let's derive Euler's formula by applying the Taylor series expansion at to . Recall Taylor series expansion at is where Keep in mind that the derivative of exponential functio...
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