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🔸Application of Complex number in Quantum mechanics:
The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg’s matrix mechanics – make use of complex numbers.
The quantum theory provides a quantitative explanation for two types of phenomena that classical mechanics and classical electrodynamics cannot account for:
• Some observable physical quantities, such as the total energy of a black body, take on discrete rather than continuous values. This phenomenon is called quantization, and the smallest possible intervals between the discrete values are called quanta (singular:quantum, from the Latin word for “quantity”, hence the name “quantum mechanics.”) The size of the quanta typically varies from system to system.
• Under certain experimental conditions, microscopic objects like atoms or electrons exhibit wave-like behavior, such as interference. Under other conditions, the same species of objects exhibit particle-like behavior (“particle” meaning an object that can be localized to a particular region ofspace), such as scattering. This phenomenon is known as wave-particle duality.
@EngMath_aut
The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg’s matrix mechanics – make use of complex numbers.
The quantum theory provides a quantitative explanation for two types of phenomena that classical mechanics and classical electrodynamics cannot account for:
• Some observable physical quantities, such as the total energy of a black body, take on discrete rather than continuous values. This phenomenon is called quantization, and the smallest possible intervals between the discrete values are called quanta (singular:quantum, from the Latin word for “quantity”, hence the name “quantum mechanics.”) The size of the quanta typically varies from system to system.
• Under certain experimental conditions, microscopic objects like atoms or electrons exhibit wave-like behavior, such as interference. Under other conditions, the same species of objects exhibit particle-like behavior (“particle” meaning an object that can be localized to a particular region ofspace), such as scattering. This phenomenon is known as wave-particle duality.
@EngMath_aut
🔸Application of complex number in Computer Science:
1️⃣ Arithmetic and logic in computer system
Arithmetic and Logic in Computer Systems provides a useful guide to a fundamental subject of computer science and engineering. Algorithms for performing operations like addition, subtraction, multiplication, and division in digital computer systems are presented, with the goal of explaining the concepts behind the algorithms, rather than addressing any direct applications. Alternative methods are examined, and explanations are supplied of the fundamental materials and reasoning behind theories and examples.
2️⃣ Recticing Software engineering in 21st century
This technological manual explores how software engineering principles can be used in tandem with software development tools to produce economical and reliable software that is faster and more accurate. Tools and techniques provided include the Unified Process for GIS application development, service-based approaches to business and information technology alignment, and an integrated model of application and software security. Current methods and future possibilities for software design are covered
@EngMath_aut
1️⃣ Arithmetic and logic in computer system
Arithmetic and Logic in Computer Systems provides a useful guide to a fundamental subject of computer science and engineering. Algorithms for performing operations like addition, subtraction, multiplication, and division in digital computer systems are presented, with the goal of explaining the concepts behind the algorithms, rather than addressing any direct applications. Alternative methods are examined, and explanations are supplied of the fundamental materials and reasoning behind theories and examples.
2️⃣ Recticing Software engineering in 21st century
This technological manual explores how software engineering principles can be used in tandem with software development tools to produce economical and reliable software that is faster and more accurate. Tools and techniques provided include the Unified Process for GIS application development, service-based approaches to business and information technology alignment, and an integrated model of application and software security. Current methods and future possibilities for software design are covered
@EngMath_aut
🔸Application of Complex number In Electrical Engineering:
The voltage produced by a battery is characterized by one real number (called potential), such as +12 volts or -12 volts😶. But the “AC” voltage in a home requires two parameters. One is a potential, such as 120 volts, and the other is an angle (called phase). The voltage is said to have two dimensions. A 2-dimensional quantity can be represented mathematically as either a vector or as a complex number☺️ (known in the engineering context as phasor). In the vector representation, the rectangular coordinates are typically referred to simply as X and Y. But in the complex number representation, the same components are referred to as real and imaginary. When the complex number is purely imaginary, such as a real part of 0 and an imaginary part of 120, it means the voltage has a potential of 120 volts and a phase of 90°, which is physically very real.
@EngMath_aut
The voltage produced by a battery is characterized by one real number (called potential), such as +12 volts or -12 volts😶. But the “AC” voltage in a home requires two parameters. One is a potential, such as 120 volts, and the other is an angle (called phase). The voltage is said to have two dimensions. A 2-dimensional quantity can be represented mathematically as either a vector or as a complex number☺️ (known in the engineering context as phasor). In the vector representation, the rectangular coordinates are typically referred to simply as X and Y. But in the complex number representation, the same components are referred to as real and imaginary. When the complex number is purely imaginary, such as a real part of 0 and an imaginary part of 120, it means the voltage has a potential of 120 volts and a phase of 90°, which is physically very real.
@EngMath_aut
🔸Application in electronics engineering:
Information that expresses a single dimension, such as linear distance, is called a scalar quantity in mathematics. Scalar numbers are the kind of numbers students use most often🙃. In relation to science, the voltage produced by a battery, the resistance of a piece of wire (ohms), and current through a wire (amps) are scalar quantities.😌
When electrical engineers analyzed alternating current circuits, they found that quantities of voltage, current and resistance (called impedance in AC) were not the familiar one-dimensional scalar quantities that are used when measuring DC circuits. These quantities which now alternate in direction and amplitude possess other dimensions (frequency and phase shift) that must be taken into account.
In order to analyze AC circuits, it became necessary to represent multi-dimensional quantities. In order to accomplish this task, scalar numbers were abandoned andcomplex numberswere used to express the two dimensions of frequency and phase shift at one time.
In mathematics, i is used to represent imaginary numbers🙂. In the study of electricity and electronics, j is used to represent imaginary numbers so that there is no confusion with i, which in electronics represents current. It is also customary for scientists to write the complex number in the form a+jb.
In electrical engineering, the Fourier transform😍 is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, andvideosignals.
Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier. This seems odd at first, as the concept of using a mix of real and ‘imaginary’ numbers to explain things in the real world seem crazy!. To help you get a clear picture of how they’re used and what they mean we can look at a mechanical example…
We can now reverse the above argument when considering a.c. (sine wave) oscillations in electronic circuits. Here we can regard the oscillating voltages and currents as ‘side views’ of something which is actually ‘rotating’ at a steady rate. We can only see the ‘real’ part of this, of course, so we have to ‘imagine’ the changes in the other direction. This leads us to the idea that what the oscillation voltage or current that we see is just the ‘real’ portion’ of a ‘complex’ quantity that also has an ‘imaginary’ part. At any instant what we see is determined by aphase anglewhich varies smoothly with time.
We can now consider oscillating currents and voltages as being complex values that have a real part we can measure and an imaginary part which we can’t. At first it seems pointless to create something we can’t see or measure, but it turns out to be useful in a number of ways.
1️⃣ It helps us understand the behaviour of circuits which contain reactance (produced by capacitors or inductors) when we apply a.c. signals.😊
2️⃣ It gives us a new way to think about oscillations. This is useful when we want to apply concepts like the conservation of energy to understanding the behaviour of systems which range from simple a mechanical pendulums to a quartz-crystal oscillator.
@EngMath_aut
Information that expresses a single dimension, such as linear distance, is called a scalar quantity in mathematics. Scalar numbers are the kind of numbers students use most often🙃. In relation to science, the voltage produced by a battery, the resistance of a piece of wire (ohms), and current through a wire (amps) are scalar quantities.😌
When electrical engineers analyzed alternating current circuits, they found that quantities of voltage, current and resistance (called impedance in AC) were not the familiar one-dimensional scalar quantities that are used when measuring DC circuits. These quantities which now alternate in direction and amplitude possess other dimensions (frequency and phase shift) that must be taken into account.
In order to analyze AC circuits, it became necessary to represent multi-dimensional quantities. In order to accomplish this task, scalar numbers were abandoned andcomplex numberswere used to express the two dimensions of frequency and phase shift at one time.
In mathematics, i is used to represent imaginary numbers🙂. In the study of electricity and electronics, j is used to represent imaginary numbers so that there is no confusion with i, which in electronics represents current. It is also customary for scientists to write the complex number in the form a+jb.
In electrical engineering, the Fourier transform😍 is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, andvideosignals.
Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier. This seems odd at first, as the concept of using a mix of real and ‘imaginary’ numbers to explain things in the real world seem crazy!. To help you get a clear picture of how they’re used and what they mean we can look at a mechanical example…
We can now reverse the above argument when considering a.c. (sine wave) oscillations in electronic circuits. Here we can regard the oscillating voltages and currents as ‘side views’ of something which is actually ‘rotating’ at a steady rate. We can only see the ‘real’ part of this, of course, so we have to ‘imagine’ the changes in the other direction. This leads us to the idea that what the oscillation voltage or current that we see is just the ‘real’ portion’ of a ‘complex’ quantity that also has an ‘imaginary’ part. At any instant what we see is determined by aphase anglewhich varies smoothly with time.
We can now consider oscillating currents and voltages as being complex values that have a real part we can measure and an imaginary part which we can’t. At first it seems pointless to create something we can’t see or measure, but it turns out to be useful in a number of ways.
1️⃣ It helps us understand the behaviour of circuits which contain reactance (produced by capacitors or inductors) when we apply a.c. signals.😊
2️⃣ It gives us a new way to think about oscillations. This is useful when we want to apply concepts like the conservation of energy to understanding the behaviour of systems which range from simple a mechanical pendulums to a quartz-crystal oscillator.
@EngMath_aut
🔸Applications in Fluid Dynamics:
complex functions are used to describe potential flow in two dimensions. Fractals.
Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set
Fluid Dynamics and its sub disciplines aerodynamics, hydrodynamics, and hydraulics have a wide range of applications. For example, they are used in calculating forces and moments onaircraft, the mass flow of petroleum through pipelines, and prediction of weather patterns.
The concept of a fluid is surprisingly general. For example, some of the basic mathematical concepts in traffic engineering are derived from considering traffic as a continuous fluids.
@EngMath_aut
complex functions are used to describe potential flow in two dimensions. Fractals.
Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set
Fluid Dynamics and its sub disciplines aerodynamics, hydrodynamics, and hydraulics have a wide range of applications. For example, they are used in calculating forces and moments onaircraft, the mass flow of petroleum through pipelines, and prediction of weather patterns.
The concept of a fluid is surprisingly general. For example, some of the basic mathematical concepts in traffic engineering are derived from considering traffic as a continuous fluids.
@EngMath_aut
🔸In Civil and Mechanical Engineering:
The concept of complex geometry and Argand plane is very much useful in constructing buildings and cars. This concept is used in 2-D designing of buildings and cars. It is also very useful in cutting of tools. Another possibility to use complex numbers in simple mechanics might be to use them to represent rotations.
@EngMath_aut
The concept of complex geometry and Argand plane is very much useful in constructing buildings and cars. This concept is used in 2-D designing of buildings and cars. It is also very useful in cutting of tools. Another possibility to use complex numbers in simple mechanics might be to use them to represent rotations.
@EngMath_aut
دوستان سلام 😊
اميدوارم حالتون عالي باشه
يكي از دوستان به بنده پيام دادن كه چجوري ميتونيم راحت بسط هاي مكلورن توابع مهم رو حفظ كنيم🙂
دوستان به ويس بنده با دقت گوش بديد🍀
@EngMath_aut
اميدوارم حالتون عالي باشه
يكي از دوستان به بنده پيام دادن كه چجوري ميتونيم راحت بسط هاي مكلورن توابع مهم رو حفظ كنيم🙂
دوستان به ويس بنده با دقت گوش بديد🍀
@EngMath_aut
Forwarded from ریاضیات مهندسی (NaCm)
Forwarded from ریاضیات مهندسی (NaCm)
Forwarded from ریاضیات مهندسی (NaCm)
امتحان مجازی پایانترم .pdf
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❌نمونه سوال امتحان مجازی پایانترم رياضي مهندسي (تستی+تشریحی)
📅 ۱۴ تیرماه سال ۹۹
🏫 دانشگاه صنعتی اميركبير
ممنونم از اقاي محمد ماركي كه توي اين ترم خيلي كمكمون كردند🌸🌸
@EngMath_aut
📅 ۱۴ تیرماه سال ۹۹
🏫 دانشگاه صنعتی اميركبير
ممنونم از اقاي محمد ماركي كه توي اين ترم خيلي كمكمون كردند🌸🌸
@EngMath_aut