hardware & software
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The document discusses the basics of computer hardware and software. It defines hardware as the physical parts of a computer like the keyboard, monitor, chips, and data. Software is defined as programs and data that allow a computer to function through a series of instructions. The document outlines that computers require both hardware and software to work, as each is useless without the other. It provides examples of common computer hardware components like the CPU, memory, hard drive, and input/output devices. It also distinguishes system software, application software, and utility software.
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hardware & software
This document discusses the basics of computer hardware and software. It defines hardware as the physical parts of a computer like the keyboard, monitor, chips, and data. Software is defined as programs and data that allow hardware to function through a series of instructions. The document outlines that computers require both hardware and software to work, as each is useless without the other. It provides examples of common computer hardware components like the CPU, memory, hard drive, and input/output devices. It also defines the different types of software like operating systems, applications, and utilities.
hardware & software
The document discusses the basics of computer hardware and software. It defines hardware as the physical parts of a computer like the keyboard, monitor, chips, and data. Software is defined as programs and data that allow a computer to function through a series of instructions. The document outlines that computers require both hardware and software to work, as each is useless without the other. It provides examples of common computer hardware components like the CPU, memory, hard drive, and input/output devices. It also distinguishes system software like the operating system from application software and utility software.
hardware & software
This document discusses the basics of computer hardware and software. It defines hardware as the physical parts of a computer like the keyboard, monitor, chips, and data. Software is defined as programs and data that allow hardware to function through a series of instructions. The document outlines that computers require both hardware and software to work, as each is useless without the other. It provides examples of common internal hardware components like the CPU, memory, and hard drive, and external components like keyboards, monitors, and printers. Input and output devices are described as how information flows between the user and computer. The document also defines operating systems, application software, and utility software and their different functions for running and maintaining a computer.
hardware & software
The document discusses the basics of computer hardware and software. It defines hardware as the physical parts of a computer like the keyboard, monitor, chips, and data. Software is defined as programs and data that allow a computer to function through a series of instructions. The document outlines that computers require both hardware and software to work, as each is useless without the other. It provides examples of common computer hardware components like the CPU, memory, hard drive, and input/output devices. It also distinguishes system software like the operating system from application software and utility software.
Hardware and software
This document defines and describes computer hardware and software. It discusses the different types of hardware, including internal hardware like the CPU, motherboard, RAM and ROM. It also covers input hardware like keyboards and mice, and output hardware such as monitors and printers. The document then defines software as a set of instructions that allows hardware components to work together. It outlines the main types of software, including system software like operating systems, programming software used by programmers, and application software that users interact with directly.
Software And Hardware
Software refers to computer programs, procedures and documentation that make up a computer system. It includes application software, firmware, middleware and system software. Software testing ensures software is functional and meets requirements before release through tools like QTP and Loadrunner. Testware refers to all utilities and software used for testing but not for operational purposes. Examples of software include websites, programs and video games coded in languages like C and C++.
Hardware vs. Software Presentations
Hardware refers to the physical components of a computer like the monitor, keyboard, and storage devices. Software refers to programs and instructions that are not physical objects. A computer needs both hardware and software to function, with the hardware providing the basic components and structure and software providing instructions to operate the hardware and perform tasks. Common types of software include operating systems, which facilitate communication between the user and hardware, and applications, which allow users to perform specific tasks.
Parts of a computer
Software includes programs and instructions that control hardware but cannot be touched. There are two main types: application programs that perform tasks like word processing, and operating systems that manage hardware and allow application programs to function. Hardware are the physical devices of a computer like the CPU that processes data, input devices like keyboards and mice to enter data, and output devices like monitors and printers that display or print results.
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hardware & software
This document discusses the basics of computer hardware and software. It defines hardware as the physical parts of a computer like the keyboard, monitor, chips, and data. Software is defined as programs and data that allow hardware to function through a series of instructions. The document outlines that computers require both hardware and software to work, as each is useless without the other. It provides examples of common computer hardware components like the CPU, memory, hard drive, and input/output devices. It also defines the different types of software like operating systems, applications, and utilities.
hardware & software
The document discusses the basics of computer hardware and software. It defines hardware as the physical parts of a computer like the keyboard, monitor, chips, and data. Software is defined as programs and data that allow a computer to function through a series of instructions. The document outlines that computers require both hardware and software to work, as each is useless without the other. It provides examples of common computer hardware components like the CPU, memory, hard drive, and input/output devices. It also distinguishes system software like the operating system from application software and utility software.
hardware & software
This document discusses the basics of computer hardware and software. It defines hardware as the physical parts of a computer like the keyboard, monitor, chips, and data. Software is defined as programs and data that allow hardware to function through a series of instructions. The document outlines that computers require both hardware and software to work, as each is useless without the other. It provides examples of common internal hardware components like the CPU, memory, and hard drive, and external components like keyboards, monitors, and printers. Input and output devices are described as how information flows between the user and computer. The document also defines operating systems, application software, and utility software and their different functions for running and maintaining a computer.
hardware & software
The document discusses the basics of computer hardware and software. It defines hardware as the physical parts of a computer like the keyboard, monitor, chips, and data. Software is defined as programs and data that allow a computer to function through a series of instructions. The document outlines that computers require both hardware and software to work, as each is useless without the other. It provides examples of common computer hardware components like the CPU, memory, hard drive, and input/output devices. It also distinguishes system software like the operating system from application software and utility software.
Hardware and software
This document defines and describes computer hardware and software. It discusses the different types of hardware, including internal hardware like the CPU, motherboard, RAM and ROM. It also covers input hardware like keyboards and mice, and output hardware such as monitors and printers. The document then defines software as a set of instructions that allows hardware components to work together. It outlines the main types of software, including system software like operating systems, programming software used by programmers, and application software that users interact with directly.
Software And Hardware
Software refers to computer programs, procedures and documentation that make up a computer system. It includes application software, firmware, middleware and system software. Software testing ensures software is functional and meets requirements before release through tools like QTP and Loadrunner. Testware refers to all utilities and software used for testing but not for operational purposes. Examples of software include websites, programs and video games coded in languages like C and C++.
Hardware vs. Software Presentations
Hardware refers to the physical components of a computer like the monitor, keyboard, and storage devices. Software refers to programs and instructions that are not physical objects. A computer needs both hardware and software to function, with the hardware providing the basic components and structure and software providing instructions to operate the hardware and perform tasks. Common types of software include operating systems, which facilitate communication between the user and hardware, and applications, which allow users to perform specific tasks.
Parts of a computer
Software includes programs and instructions that control hardware but cannot be touched. There are two main types: application programs that perform tasks like word processing, and operating systems that manage hardware and allow application programs to function. Hardware are the physical devices of a computer like the CPU that processes data, input devices like keyboards and mice to enter data, and output devices like monitors and printers that display or print results.
What is Peripheral devies
This document discusses computer peripheral devices and their functions. It defines peripheral devices as external objects that provide input and output for computers. It describes several types of input devices like keyboards, mice, scanners, and microphones that feed data into computers. It also discusses various output devices like monitors, printers, and projectors that display or print the computer's output. The document explains how interface chips and decoders help synchronize data transfer between the CPU and input/output devices.
Computer
Peripheral devices are external hardware devices attached to a computer that are either input or output devices. Input devices receive signals or data into the system and include keyboards, mice, webcams, barcode readers, image scanners, and touch screens. Output devices send signals or data from the computer and include display devices like computer monitors, LCD projectors, and printers.
2do. class, Peripheral Device
This document defines and categorizes peripheral devices that can be connected to a computer. Peripheral devices are described as auxiliary equipment that are connected to the CPU independently of the computer and help the computer communicate with the outside world. The document outlines four main categories of peripheral devices: input devices that enter data for processing, output devices that display or project information, storage devices that store data and information, and communication devices that connect computers to other machines to send and receive information.
Advance peripheral devices
Peripheral devices expand the functionality of computers and include input and output devices. Some examples are:
- A light pen allows a user to point at and draw on a display screen like a touchscreen but with greater accuracy.
- A cooling pad helps reduce the operating temperature of laptops to protect them from overheating.
- An image scanner optically scans images, text, and objects and converts them to digital files.
- Memory cards store digital data and are commonly used in portable electronic devices like cameras, phones, and laptops.
Peripherals
Peripheral devices are external hardware attached to a computer. There are three types: input devices which produce input signals; output devices which present processed data externally through printing or displaying; and storage devices which store information and documents so they can be retrieved when needed. Examples of storage devices include hard disks, diskettes, CD ROMs, and thumb drives.
Hardware and software
The document discusses the hardware and software components of a computer. It describes the central processing unit, arithmetic/logic unit, control unit, primary and secondary memory as hardware components. Input devices include keyboards, mice, digital cameras, touch screens and bar code scanners. Output devices are displays, printers, headphones and speakers. Software components include systems software like operating systems and utility programs that control hardware operations, and application software that helps users solve problems. The operating system manages tasks, multitasking, time-sharing and networking capabilities.
Intro Ch 01B.ppt
This document discusses the four main parts of a computer system: hardware, software, data, and users. It defines each part and provides examples. Hardware includes processing devices like the CPU, memory devices like RAM and ROM, and input/output and storage devices. Software tells the hardware what to do and includes operating systems and application programs. Data are pieces of information stored and processed by the computer. Users are people who operate the computer and provide instructions on what tasks to complete.
Peripheral devices
Keyboards are external input devices used to type data into computers and include alphabetic, numerical, and symbol keys. They connect to devices via serial or USB ports. Mice are also input devices that send click information to computers and offer an alternative to keyboard interaction. Typical mice have two buttons and a scroll wheel. The motherboard connects all computer parts like the CPU, memory, drives, and ports and acts as the computer's backbone. Processors are microchips that process instructions from software and come from manufacturers like Intel and AMD, with higher gigahertz capacities providing faster processing. Printers are external devices that take computer data and print hard copies of text, images, and photos. Speakers receive audio input from sound cards
BASIC COMPUTER PERIPHERALS/DEVICES/SYSTEMS
Computer malfunctions can range from a minor setting that is incorrect, to spyware, viruses, and as far as replacing hardware and an entire operating system.
While computer hardware configurations vary widely, a "Computer OEM & Repair" technician will work with five general categories of hardware; desktop computers, laptops, servers, computer clusters and smartphones / mobile computing. Technicians also work with and occasionally repair a range of peripherals, including input devices (like keyboards, mice, and scanners), output devices (like displays, printers, and speakers), and data storage devices such as internal and external hard drives and disk arrays. Technicians involved in system administration might also work with networking hardware, including routers, switches, fiber optics, and wireless networks. OEM= Original Equipment Manufacturer.
What is computer?
A computer accepts data as input, processes it using programs, and produces output. It has four basic functions: input, processing, output, and storage. Data becomes information after being processed. A computer system has two main components - hardware which are the physical devices, and software which are the programmed instructions. Common hardware includes input devices like keyboards, processing devices like the CPU, output devices like monitors, storage devices like hard drives, and communication devices like modems. System software manages the hardware and allows other programs to run, while application software allows users to perform tasks.
hardware and software
The document discusses the key components of a computer system including the central processing unit, memory, hardware, and software. It defines hardware as the physical parts of a computer like the motherboard, processor, hard drive, and memory. Software is defined as the instructions and data used by computer systems like operating systems and applications. The CPU processes input using the fetch-execute cycle and works with memory, both primary storage like RAM and secondary storage like hard disks, to store and retrieve data and instructions. Input and output devices allow the computer to receive and display information.
Software and hardware
Software consists of programs that run on a computer to perform functions, while hardware refers to the physical components. There are two main types of software: system software that manages computer resources, and application software for specific tasks. Hardware includes basic components needed for operation like RAM and ROMs, as well as complementary components like webcams. Common hardware components are networks, hard disks, USB drives, processors, keyboards, and monitors. Together, software and hardware allow computers to function and be used by people.
Hardware and Software
Computer Science / ICT - Hardware and Software
This presentation explains the brief of Hardware and Software with example that you will need to know
Computer system, devices and peripherals
1. Gather the necessary tools including a screwdriver set, pliers, and allen wrenches.
2. Assemble the toolbox by screwing in handles and hinges using the screwdrivers.
3. Organize the tools inside the toolbox using the pliers and allen wrenches to securely fasten compartments or holders.
ITFT_Peripheral devices
This document provides information about various computer peripheral and network devices. It describes input devices like keyboards, mice, and scanners. It outlines output devices such as monitors, printers, and speakers. Storage devices covered include hard drives, flash drives, CDs/DVDs, and tape. The document also discusses network interface cards, hubs, switches, routers, and servers. It provides brief explanations of modems, data transmission rates, and common network types.
ISBB_Chapter2 (1).pptx
This chapter discusses computer hardware. It describes the main components of a computer, including the central processing unit (CPU), memory, storage, input/output devices, and motherboard. It explains how Moore's Law has led to exponential growth in computing power over time. It also discusses how the personal computer market has become commoditized, with low profit margins forcing manufacturers to focus on lowering costs.
Peripheral devices
Peripheral devices expand the capabilities of a computer but are not part of the core architecture. There are three types: input devices which send data to the computer, output devices which provide output to the user, and storage devices which store processed data. Examples of input devices include keyboards and mice, while monitors and printers are common output devices. Peripherals connect to computers through various connection methods and cabling technologies such as USB, serial, and fiber optic cables.
Peripheral devices by Tahir Yasin
This document defines and describes various common peripheral devices used with computers. It discusses CD-ROMs, the CPU, joysticks, keyboards, LCD displays, mice, printers, speakers, USB ports, and webcams. For each device it provides 1-2 sentences on what the device is and its basic function.
PPT on INPUT & OUTPUT DEVICES
Input And Output Devices
Prepared By Harsh ,Mehar , Astha and Kalpana
Students Of Birla Institute Of Technology,Patna Campus
Department: BBA Session 2016-2019
Guided By : Ritesh Ravi Sir (Our Computer Teacher)
Grade9_Unit 1.pdf
The document summarizes the basic components of a computer system. It discusses that a computer system consists of hardware, software, and liveware. It describes the major hardware components including input devices like keyboards and mice, processing devices like the CPU, output devices like monitors and printers, and storage devices like hard disks. It also discusses the different types of software including operating systems, utility software, and application software. Finally, it covers liveware which refers to the users of the computer system.
HARDWARE AND SOFTWARE.pptx
The hardware components are everything you can physically touch and see in a computer, including all the input and output devices from keyboards, mics, and mice to screens and speakers
Computer components
The document discusses various components of a computer system. It describes both hardware and software components. The key hardware components include the monitor, motherboard, CPU, RAM, expansion cards, power supply, drives, and peripherals like keyboards and mice. Software is divided into system software, which controls the computer, and application software, which users interact with directly. Different types of hardware and software are described in detail.
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What is Peripheral devies
This document discusses computer peripheral devices and their functions. It defines peripheral devices as external objects that provide input and output for computers. It describes several types of input devices like keyboards, mice, scanners, and microphones that feed data into computers. It also discusses various output devices like monitors, printers, and projectors that display or print the computer's output. The document explains how interface chips and decoders help synchronize data transfer between the CPU and input/output devices.
Computer
Peripheral devices are external hardware devices attached to a computer that are either input or output devices. Input devices receive signals or data into the system and include keyboards, mice, webcams, barcode readers, image scanners, and touch screens. Output devices send signals or data from the computer and include display devices like computer monitors, LCD projectors, and printers.
2do. class, Peripheral Device
This document defines and categorizes peripheral devices that can be connected to a computer. Peripheral devices are described as auxiliary equipment that are connected to the CPU independently of the computer and help the computer communicate with the outside world. The document outlines four main categories of peripheral devices: input devices that enter data for processing, output devices that display or project information, storage devices that store data and information, and communication devices that connect computers to other machines to send and receive information.
Advance peripheral devices
Peripheral devices expand the functionality of computers and include input and output devices. Some examples are:
- A light pen allows a user to point at and draw on a display screen like a touchscreen but with greater accuracy.
- A cooling pad helps reduce the operating temperature of laptops to protect them from overheating.
- An image scanner optically scans images, text, and objects and converts them to digital files.
- Memory cards store digital data and are commonly used in portable electronic devices like cameras, phones, and laptops.
Peripherals
Peripheral devices are external hardware attached to a computer. There are three types: input devices which produce input signals; output devices which present processed data externally through printing or displaying; and storage devices which store information and documents so they can be retrieved when needed. Examples of storage devices include hard disks, diskettes, CD ROMs, and thumb drives.
Hardware and software
The document discusses the hardware and software components of a computer. It describes the central processing unit, arithmetic/logic unit, control unit, primary and secondary memory as hardware components. Input devices include keyboards, mice, digital cameras, touch screens and bar code scanners. Output devices are displays, printers, headphones and speakers. Software components include systems software like operating systems and utility programs that control hardware operations, and application software that helps users solve problems. The operating system manages tasks, multitasking, time-sharing and networking capabilities.
Intro Ch 01B.ppt
This document discusses the four main parts of a computer system: hardware, software, data, and users. It defines each part and provides examples. Hardware includes processing devices like the CPU, memory devices like RAM and ROM, and input/output and storage devices. Software tells the hardware what to do and includes operating systems and application programs. Data are pieces of information stored and processed by the computer. Users are people who operate the computer and provide instructions on what tasks to complete.
Peripheral devices
Keyboards are external input devices used to type data into computers and include alphabetic, numerical, and symbol keys. They connect to devices via serial or USB ports. Mice are also input devices that send click information to computers and offer an alternative to keyboard interaction. Typical mice have two buttons and a scroll wheel. The motherboard connects all computer parts like the CPU, memory, drives, and ports and acts as the computer's backbone. Processors are microchips that process instructions from software and come from manufacturers like Intel and AMD, with higher gigahertz capacities providing faster processing. Printers are external devices that take computer data and print hard copies of text, images, and photos. Speakers receive audio input from sound cards
BASIC COMPUTER PERIPHERALS/DEVICES/SYSTEMS
Computer malfunctions can range from a minor setting that is incorrect, to spyware, viruses, and as far as replacing hardware and an entire operating system.
While computer hardware configurations vary widely, a "Computer OEM & Repair" technician will work with five general categories of hardware; desktop computers, laptops, servers, computer clusters and smartphones / mobile computing. Technicians also work with and occasionally repair a range of peripherals, including input devices (like keyboards, mice, and scanners), output devices (like displays, printers, and speakers), and data storage devices such as internal and external hard drives and disk arrays. Technicians involved in system administration might also work with networking hardware, including routers, switches, fiber optics, and wireless networks. OEM= Original Equipment Manufacturer.
What is computer?
A computer accepts data as input, processes it using programs, and produces output. It has four basic functions: input, processing, output, and storage. Data becomes information after being processed. A computer system has two main components - hardware which are the physical devices, and software which are the programmed instructions. Common hardware includes input devices like keyboards, processing devices like the CPU, output devices like monitors, storage devices like hard drives, and communication devices like modems. System software manages the hardware and allows other programs to run, while application software allows users to perform tasks.
hardware and software
The document discusses the key components of a computer system including the central processing unit, memory, hardware, and software. It defines hardware as the physical parts of a computer like the motherboard, processor, hard drive, and memory. Software is defined as the instructions and data used by computer systems like operating systems and applications. The CPU processes input using the fetch-execute cycle and works with memory, both primary storage like RAM and secondary storage like hard disks, to store and retrieve data and instructions. Input and output devices allow the computer to receive and display information.
Software and hardware
Software consists of programs that run on a computer to perform functions, while hardware refers to the physical components. There are two main types of software: system software that manages computer resources, and application software for specific tasks. Hardware includes basic components needed for operation like RAM and ROMs, as well as complementary components like webcams. Common hardware components are networks, hard disks, USB drives, processors, keyboards, and monitors. Together, software and hardware allow computers to function and be used by people.
Hardware and Software
Computer Science / ICT - Hardware and Software
This presentation explains the brief of Hardware and Software with example that you will need to know
Computer system, devices and peripherals
1. Gather the necessary tools including a screwdriver set, pliers, and allen wrenches.
2. Assemble the toolbox by screwing in handles and hinges using the screwdrivers.
3. Organize the tools inside the toolbox using the pliers and allen wrenches to securely fasten compartments or holders.
ITFT_Peripheral devices
This document provides information about various computer peripheral and network devices. It describes input devices like keyboards, mice, and scanners. It outlines output devices such as monitors, printers, and speakers. Storage devices covered include hard drives, flash drives, CDs/DVDs, and tape. The document also discusses network interface cards, hubs, switches, routers, and servers. It provides brief explanations of modems, data transmission rates, and common network types.
ISBB_Chapter2 (1).pptx
This chapter discusses computer hardware. It describes the main components of a computer, including the central processing unit (CPU), memory, storage, input/output devices, and motherboard. It explains how Moore's Law has led to exponential growth in computing power over time. It also discusses how the personal computer market has become commoditized, with low profit margins forcing manufacturers to focus on lowering costs.
Peripheral devices
Peripheral devices expand the capabilities of a computer but are not part of the core architecture. There are three types: input devices which send data to the computer, output devices which provide output to the user, and storage devices which store processed data. Examples of input devices include keyboards and mice, while monitors and printers are common output devices. Peripherals connect to computers through various connection methods and cabling technologies such as USB, serial, and fiber optic cables.
Peripheral devices by Tahir Yasin
This document defines and describes various common peripheral devices used with computers. It discusses CD-ROMs, the CPU, joysticks, keyboards, LCD displays, mice, printers, speakers, USB ports, and webcams. For each device it provides 1-2 sentences on what the device is and its basic function.
PPT on INPUT & OUTPUT DEVICES
Input And Output Devices
Prepared By Harsh ,Mehar , Astha and Kalpana
Students Of Birla Institute Of Technology,Patna Campus
Department: BBA Session 2016-2019
Guided By : Ritesh Ravi Sir (Our Computer Teacher)
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The document summarizes the basic components of a computer system. It discusses that a computer system consists of hardware, software, and liveware. It describes the major hardware components including input devices like keyboards and mice, processing devices like the CPU, output devices like monitors and printers, and storage devices like hard disks. It also discusses the different types of software including operating systems, utility software, and application software. Finally, it covers liveware which refers to the users of the computer system.
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Computer is a device that accepts digital data as input, manipulates it based on a program's instructions, and provides processed results as output. It has two main parts: hardware and software. Hardware are the physical and tangible parts like the keyboard, monitor, mouse, and central processing unit. Hardware is divided into input devices to feed data, storage devices to store data, and output devices to display or print data. Software includes the operating system and programs that tell the computer what to do. Common system software includes Windows and Linux, while application software includes programs like Word and Excel.
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- Components of a computer system including hardware, software, and users. Hardware includes input/output devices and the central processing unit.
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- The main components of a computer system including the CPU, memory, secondary storage, and different types of software like system software, application software, and utility software.
Introduction to computer
This document provides an introduction to computers. It describes the basic components of a computer system including the central processing unit, memory, storage, motherboard, and peripheral devices. It also discusses input and output devices, ports, software types, and common uses of computers in daily life such as communication, documentation, and entertainment. The advantages and disadvantages of computer usage are outlined.
ITFT - Hardware & software concept
The document discusses hardware and software components of a computer system. It describes the four main hardware components as the CPU, memory, input devices, and output devices. It then explains the two main types of software: application software which performs specific tasks, and system software like operating systems and compilers that support the execution of other programs. Finally, it provides some common examples of application software like Microsoft Word, PowerPoint, web browsers, and multimedia programs.
Lecture 01 concept of computer system
This document provides an overview of computer components and software. It defines a computer as an electronic device that can accept data as input, manipulate it according to programmed rules, produce output from that processing, and store results. The central processing unit (CPU) contains a control unit that interprets instructions and an arithmetic logic unit that performs logical and arithmetic processes. Input and output devices allow entering and presenting data. Primary storage like RAM stores active data and software, while secondary storage like hard disks store inactive data. The document outlines types of computer systems and peripherals, input and output technologies, and categories of system and application software.
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04. features hardware and software - ipo of computer
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lec34.ppt
1. The document discusses analytic functions of complex variables through examples. It defines analytic functions as those whose derivatives of all orders exist in the region of analyticity.
2. The Cauchy-Riemann equations are derived and their implications are explored, including that they imply the Laplace equation and orthogonality of level curves.
3. Several examples are worked through to determine if functions are analytic by checking if they satisfy the Cauchy-Riemann equations. The Cauchy-Riemann equations are also derived in polar coordinates.
lec33.ppt
The Cauchy Riemann (CR) conditions provide a necessary and sufficient condition for a function f(z) = u(x, y) + iv(x, y) to be analytic in a region. The CR conditions require that the partial derivatives of u and v satisfy ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. If a function satisfies these conditions at all points in a region, then it is analytic in that region. The document proves this using cases where ∆y = 0 and ∆x = 0, showing the derivatives must be equal. Examples are provided to demonstrate checking functions for analytic
lec31.ppt
Complex numbers allow solutions to equations like x2 + 1 = 0 by extending real numbers to include imaginary numbers. A complex number z is defined as z = x + iy, where x and y are real numbers and i is the imaginary unit equal to √-1. Complex numbers can be added and multiplied following specific rules, such as z1 + z2 = (x1 + x2) + i(y1 + y2) for addition and z1z2 = (x1x2 - y1y2) + i(y1x2 + x1y2) for multiplication. The inverse of a complex number z is calculated as z-1 = (x/(x2+y
lec32.ppt
The document discusses properties of complex numbers including:
- Commutativity and associativity of addition and multiplication
- Additive and multiplicative identities and inverses
- Conjugates, modulus, and triangle inequality
- Polar form representation using modulus and argument
- Exponential form for products, quotients, and powers
- Roots of complex numbers and finding nth roots
- Representing functions of a complex variable using modulus and argument
lec42.ppt
1) The document discusses examples of calculating the Jacobian of transformations. It defines the Jacobian as the determinant of the partial derivatives of the transformed coordinates.
2) It then discusses Möbius transformations, which are fractional linear transformations of the form (az+b)/(cz+d). The Jacobian of a Möbius transformation depends only on z.
3) Several examples are given of using Möbius transformations to map one geometric region to another, such as mapping a circle to a line.
lec41.ppt
This document discusses conformal mapping, which maps curves and regions in such a way that preserves angles and their directions. It provides examples of conformal mappings:
1) The mapping w = ez maps a vertical line in the z-plane to a circle in the w-plane, with the phase angle increasing along the circle.
2) The mapping ω = eiθ0(z-z0)/(z-z0) maps an area in the upper half z-plane to the interior of a unit circle in the ω-plane. Points on the x-axis in z are mapped to the boundary of the circle.
lec39.ppt
The document discusses Taylor and Laurent series expansions. It provides examples of using these expansions to represent functions around points.
Taylor series provides a power series representation of an analytic function around a point. Laurent series allows representing functions in annular regions, including points where the function is not analytic, using both positive and negative powers of (z - z0). Examples show deducing Laurent series expansions for simple functions like z4 and 1/z4 around various points, and evaluating coefficients via contour integrals and the residue theorem. The document also gives an example of using a contour integral to compute a Greens function in many-particle physics.
lec38.ppt
1) Jordan's lemma is used to convert real integrals over the infinite real axis into complex integrals over a contour enclosing the real axis in the complex plane.
2) Several examples are provided of using residues and Jordan's lemma to evaluate definite integrals over the real line or infinite intervals that involve functions with poles, including integrals of x^2, sin(x)/x, 1/(x^2+a^2)^2, and sin(x)/(x(x^2+a^2)).
3) The technique involves closing the contour with a semicircle at infinity where the integral over the semicircle goes to zero by Jordan's lemma, leaving the original integral equal to the residue theorem applied to the
lec37.ppt
1) The document discusses evaluating contour integrals using the residue theorem. It provides examples of calculating residues and evaluating integrals where the contour encloses poles.
2) The residue of a function f(z) at a pole z=a is the coefficient of the (z-a)^-1 term in the Laurent series expansion of f(z) about z=a.
3) According to the residue theorem, the value of a contour integral of a function along a closed loop is equal to 2πi times the sum of the residues of the function enclosed by the contour.
lec23.ppt
1) The document discusses representation of the Dirac delta function in cylindrical and spherical coordinate systems. It shows that δ(r - r') = δ(ρ - ρ')δ(φ - φ')δ(z - z')/ρ in cylindrical coordinates and δ(r - r') = δ(r - r')δ(θ - θ')δ(φ - φ')/r^2 in spherical coordinates.
2) It also derives the important relation ∇^2(1/r) = -4πδ(r) and shows its application to the Laplace equation for electrostatic potential.
3) The completeness of eigenfunctions of harmonic oscillators and Legend
lec21.ppt
1. The Dirac delta function is an important concept in quantum mechanics and electrodynamics that describes an impulse or large force acting over a very short time interval.
2. The key properties of the Dirac delta function are that it is equal to infinity at a single point and zero everywhere else, and that the integral of the function over its entire range is equal to one.
3. The Dirac delta function can be used to find the value of an arbitrary function f(x) at a specific point a, as the integral of f(x) multiplied by the Dirac delta function over all x is equal to f(a).
lec20.ppt
This document contains a series of tutorial problems related to matrices and linear algebra. Problem 1 asks to invert a 3x3 matrix. Problem 2 asks to write a vector as a linear combination of two other vectors. Problem 3 involves finding the inverse, trace, and determinant of related matrices. Problem 4 proves a property about powers of similar matrices. Problem 5 diagonalizes a 2x2 matrix and finds its eigenvalues and eigenvectors.
lec19.ppt
This document discusses finding the eigenvalues and eigenfunctions of a spin-1/2 particle pointing along an arbitrary direction. It shows that the eigenvalue equation reduces to a set of two linear, homogeneous equations. The eigenvalues are found to be ±1/2, and the corresponding eigenvectors are written in terms of the direction angles θ and Φ. As an example, it shows that for a spin oriented along the z-axis, the eigenvectors reduce to simple forms as expected for a spin-1/2 particle. It also introduces the Gauss elimination method for numerically solving systems of linear equations that arise in eigenvalue problems.
lec18.ppt
This document discusses solving a mass-spring system as an eigenvalue problem. It:
1) Sets up differential equations to model the displacements of two masses connected by springs.
2) Transforms the coupled differential equations into a matrix eigenvalue equation.
3) Solves the eigenvalue equation to obtain the frequencies of oscillation for the two masses.
4) Combines the eigenvectors with complex exponential functions to obtain general solutions for the displacements of each mass over time.
lec17.ppt
This document discusses linear transformations and matrices. It introduces how linear transformations on physical quantities are usually described by matrices, where a column vector u representing a physical quantity is transformed into another column vector Au by a transformation matrix A. As an example, it discusses orthogonal transformations, where the transformation matrix A is orthogonal. It proves that for an orthogonal transformation, the inner product of two vectors remains invariant. It also discusses properties of other types of matrices like Hermitian, skew-Hermitian and unitary matrices.
lec16.ppt
This document discusses properties of symmetric, skew-symmetric, and orthogonal matrices. It defines each type of matrix and provides examples. Key points include:
- Symmetric matrices have Aij = Aji for all i and j. Skew-symmetric matrices have Aij = -Aji. Orthogonal matrices satisfy AT = A-1.
- The eigenvalues of symmetric matrices are always real. The eigenvalues of skew-symmetric matrices are either zero or purely imaginary.
- Any real square matrix can be written as the sum of a symmetric matrix and skew-symmetric matrix.
lec30.ppt
1) The document discusses calculating the moment of inertia tensor for a cylinder with radius R and height H. It is shown that the only non-zero components of the inertia tensor are Ixx = (3MH + 4MR2)/12, Iyy = Ixx, and Izz = MR2/2.
2) Equations for velocity, acceleration, and the Christoffel symbols in an arbitrary coordinate system are presented. Expressions for calculating acceleration in cylindrical coordinates using the metric tensor and Christoffel symbols are given.
lec28.ppt
Tensors obey algebraic properties including addition, multiplication, contraction, and symmetrization. Addition of tensors combines their components. Multiplication of tensors combines their indices and ranks to form a new tensor. Contraction sets equal a covariant and contravariant index, reducing the tensor's rank. Symmetric tensors do not change sign under index interchange, while antisymmetric tensors change sign.
lec27.ppt
1) The document discusses tensors with multiple indices and the cross product of two vectors A and B. The components of the cross product vector C are given by Ai Bj - Aj Bi.
2) It describes how tensor components transform between coordinate systems using transformations of partial derivatives. The transformation property for cross products is derived.
3) Tensors are defined by their rank, with the number of covariant and contravariant indices specifying a tensor's rank. Vectors have a rank of 1. Examples calculate tensor components in different coordinate systems.
lec26.ppt
1) There are two types of vectors - contravariant vectors whose components transform according to Equation 1, and covariant vectors whose components transform according to Equation 2.
2) The dot product of two contravariant or two covariant vectors is not independent of the coordinate system.
3) The dot product of a contravariant and a covariant vector is independent of the coordinate system.
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hardware & software
- 2. SCH.NO.5677
- 4. 4 Hardware and Software • Hardware – the physical, tangible parts of a computer – keyboard, monitor, wires, chips, data • Software – programs and data – a program is a series of instructions • A computer requires both hardware and software • Each is essentially useless without the other
- 5. Technology in Your Life Introducing Your Computer Your computer may use hardware like the components shown here. Hardware Basics
- 6. Technology in Your Life Hardware can be inside or outside the computer: The CPU and hard drive are inside the computer. The keyboard, monitor, and printer are outside the computer and must be connected by cables. Hardware Basics
- 7. • System Unit: Motherboard, CPU (Central Processing Unit), Memory • Hard disk • Floppy disk drive • CD drive/DVD drive • Power supply unit
- 8. Technology in Your Life Introducing Your Computer What Are Input and Output Devices? Information flows between you and your computer through input and output devices. Examples of input devices are: keyboard mouse scanner digital camera joystick Examples of output devices are: monitor printer speakers input Information that is entered into a computer. output Information that a computer produces and delivers back to the user. Hardware Basics
- 10. Technology in Your Life Introducing Your Computer Software is the set of instructions that lets you “talk” to your computer. Software translates commands into bits and bytes. Different types of software have different functions: Operating system (OS) software - makes your computer work Application software - lets you do different tasks on your computer Utility software - helps you maintain your computer and keep it in good running condition SOFTWARE A set of instructions, also called a program or application, that tells a computer how to perform tasks. OPERATING SYSTEM (OS) Software that controls all the other software programs and allows a computer to perform basic tasks. Software Basics
- 12. Copyright © 2000 by Brooks/Cole Publishing Company A division of International Thomson Publishing Inc.
- 13. Computer Software Relationships User Interface Basic Input and Output Services (BIOS) • needed for a computer to boot up User Interface Operating System User Interface Application Programs Computer Hardware
- 14. Summary