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Add eSign PDF Computer. Investigate probably the most user-pleasant experience with airSlate SignNow. Handle your complete document digesting and discussing method digitally. Change from handheld, papers-dependent and erroneous workflows to computerized, computerized and flawless. You can easily produce, deliver and sign any documents on any gadget just about anywhere. Ensure that your important organization instances don't slip over the top.
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FAQs
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I'm looking for a solution to collect someone's signature online
Many businesses actively collect e-signature online while working with partners and customers in order to add one or more signatures to an already typed document by using some specialized tool that easily integrates into a document workflow.For many use cases, it is needed to sign an already typed document and avoid signing digital documents by hand - which is tedious otherwise cause you need to print, sign and scan all that stuff - that’s why such electronic signature tools are used - and, in this case, it is better to use a specialized tool, like Draw Your Signature Online and Sign PDF - ...
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Information Security: How can I get a Digital Signature?
Digital signatures are being widely used across the globe. There is a specific process to acquiring the signature. The way of acquisition is standard, no matter what country you’re trying to get the signature in. Digital signatures are created and issued by qualified individuals. For anyone to get a valid digital certificate, they must get it from a signNowing authority (CA). The signNowing Authority (CA) is a kind of Trust Service Provider - a third party provider designated and trusted by the country. It has the power of issuing citizens digital signatures. These CAs have rules and regulations they abide by. While in the USA, you can use the following CAs signNow US Globalsign Hello Sign When in the UK, you can use the following CAs signNow E-sign.co.uk signNow UK When you are in India, you can use the following CAs to get your digital signature certificate. eMudra Digital Signature India Government Approved signNowing Authorities These are some of the trusted sites that you can use to get your digital signature certificate in India, the UK, and the USA. They comply with every rule that governs electronic signatures, and you will get the best experience with them. Meanwhile, if you’re looking for e-signature software for your work, I recommend checking out signNow - with a high level of security, plenty of advanced features and overall ease of use, this application is a good fit for both small and medium-sized companies, startups, law-firms, and individual use as well. With signNow, you can: MANAGE SIGNATURE TASKS ● Visual progress bar - Monitor signature tasks by intuitively checking all signers’ status ● Timeline of Personal Activities - Display and record activities of all your personal tasks ● Void signature requests - Cancel signature tasks with one tap ●Search tool - Find your documents easily by searching with names of people or documents ASSIGN SIGNATURE TASKS TO MULTIPLE SIGNERS ●Invite multiple signers by adding them straight from your contact list or entering their email accounts ● Assign various fields to signers in a designated order, including signatures, texts, and dates ● Send documents to multiple signers at one time ● Show your signers where to fill in at a glance IMPORT DOCUMENTS TO START SIGNING ●Get documents from camera, photos, or the iOS file app ●Obtain documents from various cloud services, including Dropbox, Google Drive, and more ●Open-in documents from email attachments and the web PERSONALIZE YOUR SIGNATURES ● Create signatures with free-hand drawing ● Make stamps by using your camera or photos ● Pre-fill your personal information and quickly drag and drop it to the document ● Add signatures, initials, texts, and dates to documents All these features keep your documents well-organized, while the ability to track the entire signing process eases the overall task. With top-notch security, legally-binding audit trails and 2-factor authentication, this application will improve your workflow and save plenty of both time and money. Plus, the multi-platform option gives you the freedom to work across various devices. Disclaimer: I am part of Kdan’s team, and my answers might be a bit biased.
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How do I register a trademark license? How do I create a digital signature?
If you want to register your trademark internationally, you will have to carefully consider in which countries you do. For example, it is usually unnecessary to register in 3 classes internationally, and one class is enough. The costs to register trademarks internationally are dependent on the country.As far as online signatures:It’s rather easy! All you need to do is go to Sign PDF Online with DigsignNower. Follow the instructions below and when you’re down, download the final image with the signature to your device.(Works on mobile devices as well!)Here’s a step-by-step guide, it only takes a few seconds to create a digital signature.1. Upload a fileStart out by simply clicking the choose file link to upload the PDF, Word, TXT, IMG, TXT or XLS document that you want to sign off. Alternatively, you can also grab it right from your desktop and drop it as shown in the image below:In just a few seconds your file will be fully available once the upload is completed.2. Apply your editsOnce you’re in, you will be able to select the type of edits you want to apply to your document.This tool is fully equipped with everything you need not only to create an electronic signature but also to fill out your forms online by adding text, selecting checkboxes, inserting the current date or even initialing where necessary.2.1 Create a free electronic signatureHover the mouse over the areas where you want to apply the edits.Feel free to sign documents the way you like. This signature maker tool offers three different options to create electronic signatures:A ) Draw an e-signature using a mouse or touchpad.B) Type your name, or scan an image of your signature.C) And last but not least, upload it to the document.2.2 Fill out the documentAs stated before, you can also add check marks, dates, text or initials in any page of your documents. Simply select the type of function you want to use, select the area and type in the information. Once you’re done click add.3. Download your document for free!When ready, clickto proceed. Our free tool will create a new file with the same format that can be downloaded completely for free without any type of registration!Click download and save the file on your computer.Here’s also a quick video demonstration on how to use this tool!
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How do I add a digital signature in a WhatsApp image?
All you need to do is go to Sign PDF Online with DigsignNower. Follow the instructions below and when you’re down, download the final image with the signature to your device.(Works on mobile devices as well!)1. Upload a fileStart out by simply clicking the choose file link to upload the PDF, Word, TXT, IMG, TXT or XLS document that you want to sign off. Alternatively, you can also grab it right from your desktop and drop it as shown in the image below:In just a few seconds your file will be fully available once the upload is completed.2. Apply your editsOnce you’re in, you will be able to select the type of edits you want to apply to your document.This tool is fully equipped with everything you need not only to create an electronic signature but also to fill out your forms online by adding text, selecting checkboxes, inserting the current date or even initialing where necessary.2.1 Create a free electronic signatureHover the mouse over the areas where you want to apply the edits.Feel free to sign documents the way you like. This signature maker tool offers three different options to create electronic signatures:A ) Draw an e-signature using a mouse or touchpad.B) Type your name, or scan an image of your signature.C) And last but not least, upload it to the document.2.2 Fill out the documentAs stated before, you can also add check marks, dates, text or initials in any page of your documents. Simply select the type of function you want to use, select the area and type in the information. Once you’re done click add.3. Download your document for free!When ready, clickto proceed. Our free tool will create a new file with the same format that can be downloaded completely for free without any type of registration!Click download and save the file on your computer.Here’s also a quick video demonstration on how to use this tool!
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What are some useful tech hacks for everyday use?
Here are some I find useful and fun:Instead of "Ctrl, Alt & Del", press "Ctrl, Shift and Escape" to get straight to the task-manager in Windows.If you want to download a Youtube video, just add "ss" to the URL between www. and Youtube.The program DeTune will transfer all of the songs from an iPod/iPhone and put them on you computer.If you search "do the harlem shake" on YouTube the page itself will do the harlem shake.(This one's just for fun)The "Hola Unblocker" extension on Google Chrome will allow you to access to UK version of Netflix; thus unlocking many more shows and movies.Need to focus on studying? Screen Time is an app that lets you limit the time you use on your iphone or ipad. Set the time, press start and when it expires it closes whatever you were doing.If you play YouTube videos through Safari you can still listen to them with your phone's screen turned off.Replace the "en" in a Wikipedia link with "simple" to strip away the complex and mostly irrelevant information on the page.Accidentally erase something you just typed on your iPhone? To undo that, just shake it!1. Go to Google and type in "50 most popular women" 2. Click on the first link 3. Check out #7 xDTo move frame by frame on Youtube, pause the video and then use J or L to go backward or forward respectively.Just in case something caught your attention then I am Rohan Bhatia.You could have gone anyway.
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What is the best website very few people know about? What's great about it?
1 Couchsurfing:If you travel a lot over to new places, this is the website to make memories. It’s a social network to meet locals in new places and experience the place from their angle. You can invite people to stay with you at your place as a host.2. How Stuff Works: Well, the name says it all. How Stuff works gives you the insights into different processes. Be it something as easy as how does an IP protocol works or how does Alcohol affects your body, I binge read the website for hours. I love how they have linked the website to each other.3. Spreeder: Spreeder teaches you to read faster and understand things quickly. It will for sure make you productive with time.4. Highbrow: Highbrow is an website that provides you daily lessons on different topics ranging from Productivity to Business to Technology. You can find all sorts of lessons delivered to your mailbox every morning. Every lesson is worth 5 minutes of your time and you cannot subscribe to more than one course at a time as it will play with your attention.5. StumbleUpon: StumbleUpon is a place where you literally stumble upon things. Choose your interests from the list and click on the stumble upon icon which looks like ohm. Every time you will be presented with an article, image, infographic or video from all over the web. You will be presented with cool stuff all the time.6. Nerd Fitness: There is a lot of bullshit regarding fitness out there. These guys are helping you out with all that. Videos, Blogs and Training regimes for nerds, jockeys and everyone who wants to level up their game in fitness. This site has taught me how small lessons can boost up your life with exercise. There is a lot of free stuff on the website that you can get help from.7. Cooking for Engineers : Cooking for Engineers is again a cooking site for nerds. There’s a technical twist to cooking on this website. All the recipes and ingredients are analysed for best results. If you like cooking with everything to be perfect with accurate facts, go visit this website now.8. Information is beautiful: If you’re bored of information in just text format, visit this website. This website provides information on various topics in beautiful pictures, charts and infographics. You can roam around on website for hours and never get bored.9. Duolingo: Duolingo is a great website to learn new languages. You will have to be persistent with the website in order to learn a new language here and over time you will see the results. It’s fun to learn new language on this website(or App).10. Lumosity: Learning new things via games has been fun since Kindergarten and this website is just the same. You can learn calculations, improve your vocabulary and much more on this website.11. Khanacdemy: What started off as a teaching program for cousins ended up into one of the biggest knowledge store for users all around the world. Khanacademy has a collection of usefull video lessons on various topics ranging from Maths to Economics and the best part is: It’s totally free since it’s non-profit.12. Codeacademy: Have you ever tried to learn a new programming language but ended up only cramming text from books and never got to the practical part. This is where Codeacademy comes in. Choose your language and start practicing on it. There are lessons first, then assignments and if you are struck, help is always around the corner. If you’re just started programming in college or thinking of taking programming as a career, try this website. You will love it.13. Grammarly: Grammarly is an online website which looks out for grammatical errors in your writing. It has a pretty cool browser extension that sorts out mistakes in your emails or wherever you’re writing. If you’r serious about writing and cannot afford a writer, their pro version will do the job.
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How can I protect my PDF book from being pirated?
Simple - don’t put your ebook (in this dreadful ebook format) on the web. I suspect that’s not the answer you want though so here’s another: Don’t bother worrying about this. The format you have chosen is designed for printing, not for reading on an ebook reader, most serious readers won’t bother with it because doing all that scrolling is irritating and PDF files can often contain malware. Those who pirate books will thank you though since you have made their task easier. A better format would be EPUB or for Amazon MOBI. Both of these can have DRM applied which prevents an average computer user sharing the file. However I suggest you DON’T add DRM since it won’t prevent those who pirate books from removing it. It will prevent most legitimate users from transferring the files between devices though. If your book is worth copying, there is nothing you can do to prevent it being pirated once it is available for sale on the web. Those who download your pirated books though probably wouldn’t have bought it anyway. If they like it, there’s a chance they may buy another of your books, post a review or talk about it to friends. Look on those pirate copies as free advertising.
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What must be my strategy to score full marks in the GATE mathematics (also suggest best book for practicing prev. year’s questio
Mathematics is a very scoring subject. And it is very important for those who just want to pass or to get below 2000 or top 100. Who want to miss easy 10+ marks??? Some of the problems students face related to mathematics are: * Students have studied it in the first year and have completely forgotten * Mathematics is widely regarded as one of the toughest subjects * Students think they can’t learn that subject since it is beyond them So overcome your fears and see that it is very easy provided, you study well. Did I tell you, my most favorite subject is mathematics. I learnt mathematics by practice only! When I see a new topic, I too feel blank inside my head. Then I realize that all I need to do is to understand the concepts by practicing then I feel comfortable in doing those concepts. I will try to explain important topics here and give some tips to prepare mathematics. Finally I will give some timetable to study it so that you finish that subject. Before we start, see the GATE syllabus here: * A [ http://www.gate.iitg.ac.in/Syllabi/AE_Aerospace-Engineering.pdf ]E [ http://www.gate.iitg.ac.in/Syllabi/AE_Aerospace-Engineering.pdf ], Civil [ http://www.gate.iitg.ac.in/Syllabi/CE_Civil-Engineering.pdf ], Chemical [ http://www.gate.iitg.ac.in/Syllabi/CH_Chemical-Engineering.pdf ], CS [ http://www.gate.iitg.ac.in/Syllabi/CS_Computer-Science-and-Information-Technology.pdf ], ECE [ http://www.gate.iitg.ac.in/Syllabi/EC_Electronics-and-Communications_Engineering.pdf ], EE [ http://www.gate.iitg.ac.in/Syllabi/EE_Electrical-Engineering.pdf ], ME [ http://www.gate.iitg.ac.in/Syllabi/ME_Mechanical-Engineering.pdf ], Petroleum [ http://www.gate.iitg.ac.in/Syllabi/PE_Petroleum_Engineering.pdf ], Instrumentation [ http://www.gate.iitg.ac.in/Syllabi/IN_Instrumentation-Engineering.pdf ] All branches except CS have almost same syllabus. So I will focus on these common topics. You can study from either ACE material or Made Easy material or B. S. Gravel or any good textbook along with previous year papers. Try to spend daily an hour on Mathematics till you become confident in it. * Linear Algebra: Generally the topics asked are related to finding rank, determinant, Eigenvalues and Eigenvectors, Solution of simultaneous equations, Finding inverse and power of matrix. * * Rank: It is the minimum of ( independent rows or columns ) in the given matrix. To find the rank. try to write the given equations in to matrix form and reduce it into row echelon form [ http://stattrek.com/matrix-algebra/echelon-transform.aspx ]. The number of non-zero rows left in the matrix is the rank of the matrix. Ex: Rank of matrix [ https://math.stackexchange.com/questions/2080554/how-to-calculate-matrix-rank ] * Determinant: You can solve these problems in two methods: * * Expand the matrix by finding co-factors and then simplifying it gives the determinant and straightforward. Refer here for complete procedure: The Determinant of a Square Matrix [ https://people.richland.edu/james/lecture/m116/matrices/determinant.html ] * Next approach is very simple and easy. Try to convert the given matrix into row echelon form. Then the determinant is just the product of diagonal terms. Only point you need to keep in mind is, when you interchange two adjacent rows, rank doesn’t change but determinant changes to opposite sign. So if you interchange two rows say [math]i^{th}[/math] and [math]j^{th}[/math]. Then, determinant changes the sign to [math](-1)^{j-i}[/math]. Also another point to remember is not to multiply any row by a constant. Rank doesn’t change but determinant changes. * Solution of simultaneous equations: To solve there questions, try to write the Augmented matrix [math][A|b][/math] from [math]Ax=b[/math]. Find the rank of [math][A|b][/math] and also for A. * * If [math]R(A) = R([A|b]) = n[/math] (no. of rows or columns in the given matrix), then we have unique solution. * If [math]R(A) = R([A|b]) %3C n[/math] (no. of rows or columns in the given matrix), then we have multiple solutions. * If [math]R(A) \neq R([A|b])[/math], then we have no solution. * Eigenvalues and Eigenvectors: Eigenvalues and Eigenvectors are the solutions the equation [math]Ax=\lambda x[/math], [math]\lambda[/math] is the Eigenvalue and [math]x[/math] is the Eigenvector. To solve these types of problems, since we have one equation and two unknowns ([math]\lambda[/math] and [math]x[/math]) so there is no unique solution to given equation. Refer here for a solved example: Eigenvalues and Eigenvectors example [ https://www.scss.tcd.ie/~dahyotr/CS1BA1/SolutionEigen.pdf ] * * To find Eigenvalues, write given equation as [math](A-\lambda I)x=0[/math]. Find the determinant of [math](A-\lambda I)[/math]. [math]\lambda[/math] is Eigenvalues obtained. * Next for each Eigenvalue [math]\lambda[/math] obtained, find the Eigenvector corresponding to it. Since the equation becomes to some [math]Bx=0[/math], we don’t have a unique solution. Assume one of the values in vector as 1. Find the rest of the terms. * Finding inverse and Power of matrix: use Cayley Hamilton theorem to solve these problems. Every matrix satisfies its own characteristic equation [math]|A-\lambda I|=0[/math]. From this find the equation in terms of [math]\lambda[/math]. Replace [math]\lambda[/math] with [math]A[/math] and then we get characteristic equation. From this, we can find any power or inverse easily. Refer here for a solved problem: Cayley Hamilton Theorem Examples [ https://www.math.upenn.edu/~rimmer/math240/8_9powers.pdf ] * Calculus: Some topics to focus are finding maximum and minimum values, Finding Limits, Continuity and Differentiability of functions, Taylor and Maclaurin series, Vector calculus, Finding area and Volumes, * * Maximum and Minimum values: Differentiate the given equation and find [math]f'(x)=0[/math]. Then, find [math]f"(x)[/math] at the same values obtained. Refer this for further clarification: Minimum and Maximum Values [ http://tutorial.math.lamar.edu/Classes/CalcI/MinMaxValues.aspx ] * * [math]f"(x)%3C0[/math], then value is local maxima * [math]f"(x)%3E0[/math], then value is local minima * [math]f"(x)=0[/math], then nothing can be said * Also check with the boundaries too. They could be maximum or minimum in the given interval. * Limit, Continuity and Differentiability: Refer this for some solved examples: Continuity and Differentiability [ http://tekoclasses.com/ENGLISH%20PDF%20PACKAGE/28%20CONTINUITY%20&%20DIFFRENTIABILITY%20PART%201%20of%201.pdf ] * * Limits: To find the limit of form, [math]0/0[/math] or [math]\infty/\infty[/math] we need to use L-Hospital rule. [math]0/\infty 0[/math] and [math]\infty/0 \to \pm \infty[/math] and so limit does not exist, if limit is checked from left and right of zero. For [math]0\times \infty[/math] form, write it as [math]0/(1/\infty) = 0/0[/math] form and use L-Hospital rule. A simple approach to solve majority of the trigonometric and logarithmic problems is by expanding the terms in x. And then simplifying gives directly the limit. * Ex. of above limit application: [math]\lim_{x \to 0} \dfrac{\sin{x}-x}{x^3} =\lim_{x \to 0} \dfrac{(x-x^3/6+x^5/120-...)-x}{x^3}=\frac{-1}{6}[/math] * Continuity: To find whether function is continuous or not, simply find the [math]\lim_{x \to 0^+}[/math] and [math]\lim_{x \to 0^-}[/math] and [math]f(0)[/math] and see that all three values are equal. * Differentiability: Along with continuity, the function need to have a unique limit of differentiated value. [math]f'(x)|_{x=0} = lim_{h \to 0^+} \dfrac{f(x+h)-f(x)}{h}=lim_{h \to 0^-} \dfrac{f(x+h)-f(x)}{h}[/math] * For two variable problems, limit has to be same in any order we apply for continuity and differentaiability. Also if limit doesn’t exist, just solve the problems with one the given order. * Taylor series and Maclaurin Series: To solve problems related to this, find the function first, second and third etc. derivatives at x=a (for Taylor) and x=0 (for Maclaurin. Then get the expansion of the function. Taylor series is the expansion of function along x=a (from: Taylor Series [ http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx ]) Maclaurin series is the expansion of function along x=0 * * Definite and Indefinite integrals: Indefinite integrals are very tricky. Try to solve solutions and check if we get given question. Definite integrals are the toughest. Learn formulas of [math]\int \sqrt{x}[/math], [math]\int_{ 0}^{ \pi/2} \sin^n {x}, \int_{0}^{\pi/2} \cos^n {x}, [/math] * * Solve some problems of definite integrals from here: Computing Definite Integrals [ http://tutorial.math.lamar.edu/Problems/CalcI/ComputingDefiniteIntegrals.aspx ] * Most of the definite problems can be solved by method of substitutions like [math]x=t^2[/math] for [math]\sqrt{x}[/math], [math]x=a\tan{x}[/math] for [math]x^2+a^2[/math] in denominator, [math]x=a\sin{x}[/math] for [math]\sqrt{a^2-x^2}[/math] and so on. Refer here for examples of all models: Substitution Rule for Indefinite Integrals [ http://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinite.aspx ], More Substitution Rule [ http://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinitePtII.aspx ] * Area between two curves is [math]\int_{a}^b {f(x)-g(x)} dx[/math] is the area between [math]f(x)[/math] and [math]g(x)[/math] where a and b are the two intersection points of the curve. To find area of the curve and x axis, take [math]g(x)[/math] as zero. Refer here for some solved examples of Area Between Curves [ http://tutorial.math.lamar.edu/Classes/CalcI/AreaBetweenCurves.aspx ] * To calculate volumes, use [math]\int Ady[/math] or [math]\int Adx[/math]. Where, [math]A[/math] represents the area of the typical disc. Refer here for some solved example: Volume of Revolution [ http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-volumes-2009-1.pdf ] * Vector Calculus: The questions asked from this topic are very easy if we know the formulas. Very important operator is Del: [math]\nabla = \dfrac{\partial}{\partial x} \overrightarrow{i}+\dfrac{\partial}{\partial y} \overrightarrow{j}+\dfrac{\partial}{\partial z} \overrightarrow{k}[/math] * * Curl [math]\overrightarrow{F}[/math]is defined as the cross product between [math]\nabla[/math] and [math]\overrightarrow{F}[/math]= [math]\nabla \times \overrightarrow{F}[/math] * * * Divergence or div [math]\overrightarrow{F}[/math] is defined as [math]\nabla. \overrightarrow{F}[/math]. Refer here to solve some problems: Curl and Divergence [ http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx ] * * * Gradient is defined for a scalar only where as we defined Curl and Divergence to a vector. Gradient is defined as [math]\overrightarrow{\nabla} F=\dfrac{\partial \overrightarrow{F}}{\partial x}\overrightarrow{i} +\dfrac{\partial \overrightarrow{F}}{\partial y}\overrightarrow{j} +\dfrac{\partial \overrightarrow{F}}{\partial z}\overrightarrow{k}[/math]. The rate of change of function is defined by gradient. * To get directional derivative of the function, simply calculate the gradient and calculate [math]\overrightarrow{\nabla} f.\overrightarrow{u}[/math] (where . is dot product and [math]\overrightarrow{u}[/math] is unit vector in the direction of given vector. Refer here for some solved problems: Directional Derivatives [ http://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx#Gradient_Defn ]. To calculate the maximum rate of change given function [math]f(x,y,z)[/math], simply compute [math]\overrightarrow{\nabla} f.\dfrac{\overrightarrow{\nabla} f}{|\overrightarrow{\nabla} f|}[/math] * To compute tangent and normal of a function, compute gradient first. Refer here for solved problems: Gradient Vector, Tangent Planes and Normal Lines [ http://tutorial.math.lamar.edu/Classes/CalcIII/GradientVectorTangentPlane.aspx ] Tangent of a function [math]f(x,y,z)[/math] at [math](x_0,y_0,z_0)[/math] is represented by: Similarly normal is represented by * * * If the field is conservative [ http://tutorial.math.lamar.edu/Classes/CalcIII/ConservativeVectorField.aspx ], we can calculate potential which is nothing but the gradient of function. * For line integrals, just take the equation of the path given and integrate along it. Refer here: Line, Surface and Volume Integrals [ https://www.robots.ox.ac.uk/~dwm/Courses/2VA/2VA-N4.pdf (https://www.robots.ox.ac.uk/~dwm/Courses/2VA/2VA-N4.pdf ] * Greens Theorem: Refer here for example: Green's Theorem [ http://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx ]. It converts line integral to area integral. It is defined only for 2D as shown: * * * Stokes Theorem: It is the higher version of Greens Theorem. It converts surface integral to volume integral. Refer here for example: Stokes' Theorem [ http://tutorial.math.lamar.edu/Classes/CalcIII/StokesTheorem.aspx ]. It is defined as: * * * Gauss Divergence Theorem: It converts volume integral to surface integral (or reverse of Stokes theorem). Refer here for example: Gauss Divergence Theorem [ http://tutorial.math.lamar.edu/Classes/CalcIII/DivergenceTheorem.aspx (http://tutorial.math.lamar.edu/Classes/CalcIII/DivergenceTheorem.aspx ] * Transformations: Key points to focus would be finding of transform for given function specially half sine, cosine, unit step function, and Dirac Delta function, and its inverse or convolution and its application to compute the differential equations. * * Fourier Series: * * Fourier Series Tutorial I [ http://www.cse.salford.ac.uk/physics/gsmcdonald/H-Tutorials/Fourier-series-tutorial.pdf (http://www.cse.salford.ac.uk/physics/gsmcdonald/H-Tutorials/Fourier-series-tutorial.pdf ] * Fourier Series Tutorial II [ http://nptel.ac.in/courses/111103021/15.pdf ] * Fourier Series Examples [ https://www.math.psu.edu/tseng/class/Math251/Notes-PDE%20pt2.pdf ] * Laplace Transforms: * * Laplace Transforms Overview I [ http://www.vyssotski.ch/BasicsOfInstrumentation/LaplaceTransform.pdf ] * Laplace Transforms Overview II [ http://www.math.psu.edu/shen_w/250/NotesLaplace.pdf ] * Step Functions [ https://www.math.psu.edu/tseng/class/Math251/Notes-LT2.pdf ] * Initial Value Problems (IVP) [ https://www.math.psu.edu/tseng/class/Math251/Notes-LT1.pdf ] * Dirac Delta Function [ https://www.math.psu.edu/tseng/class/Math251/Notes-LT3.pdf ] * Formulas sheet: Laplace Transforms Formulas [ http://personal.maths.surrey.ac.uk/st/Mark.Holland/Old_ms200/formula_sheet.pdf (http://personal.maths.surrey.ac.uk/st/Mark.Holland/Old_ms200/formula_sheet.pdf ] * Inverse Laplace Transforms [ https://sites.ualberta.ca/~csproat/Homework/MATH%20334/Chapter%20Solutions/Chapter%20Part%202.pdf ] * Dirac Delta function: Function is infinite at a single value and is zero at rest of the values. Applying Fourier and Laplace transforms to it very important. Refer here for its applications [ http://materia.dfa.unipd.it/salasnich/dfl/dfl.pdf ] * z‐Transform: Key points to note are applying z-transforms, inverse z-transforms, finding poles and zeros, and application to compute power series. * * Zeros are locations where numerator of function is zero and Pole is a location where the denominator is zero so function is infinite there. For ex. consider [math]H(z) = \dfrac{z-1}{z+1}[/math]. Zeros are [math]z=1[/math] and poles are [math]z=-1[/math]. * Z-Transforms Complete [ http://www.uobabylon.edu.iq/uobcoleges/ad_downloads/4_7629_305.doc ] * Z-Transforms [ https://wolfweb.unr.edu/~fadali/ee472/Ztransform.pdf ] * Z-Transforms Overview [ http://nptel.ac.in/courses/106106097/pdf/Lecture12_ExampleZTransforms.pdf ] * Properties of Z-Transforms [ http://nptel.ac.in/courses/106106097/11 ] * Power series expansion exists only if region of convergence [math]|R| %3C 1[/math]Power Series and Functions [ http://tutorial.math.lamar.edu/Classes/CalcII/PowerSeriesandFunctions.aspx ] * Ordinary Differential Equations: Some key points to remember is learning different first order types, for second order with particular integral as [math]\sin {x},\cos{x},e^x,x^n,[/math] combination of any of these, application of method of variation of parameters to any general function like [math]\tan {x}, \sec{x}, \log{x}[/math] etc. and application of Euler-Cauchy method for variable coefficients. Solving IVP, BVP. * * Order [ http://www3.ul.ie/cemtl/pdf%20files/bm2/DegreeOrder.pdf ]is the highest order differential equation. Degree [ http://www3.ul.ie/cemtl/pdf%20files/bm2/DegreeOrder.pdf ]is the power of highest order in given differential equation after differential equation is radical free (meaning free from [math]1/3[/math] power etc.) * First Order Linear equations: If only [math]y'[/math] term is present it is first order. Linear if [math]y'[/math] is free linear. Some of the types of standard models available are: * * Linear * Homogeneous * Exact * Variable Separable * Bernoulli's equation * Above types can be found here: Recognizing types of ODE [ http://www.math.hawaii.edu/~lee/calculus/DE.pdf ] and First Order DE's [ http://tutorial.math.lamar.edu/Classes/DE/IntroFirstOrder.aspx ] and Problems I [ https://rutherglen.science.mq.edu.au/wchen/lnfycfolder/fyc15.pdf ] * Second Order Equations with constant coefficients: Standard form is [math]ay"+by'+cy=f(x) = (aD^2+bD+c)y=f(x)[/math]. Where, [math]D=d/dx[/math]. To solve these equations, * * First form an characteristic equation i,e, [math]aD^2+bD+c=0[/math]. Solve for D. Let [math]D=a[/math] be a root, * If roots are unique, solution is [math]y=c_1e^{ax}[/math]. By using superposition combine all the roots. Real & Distinct Roots [ http://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx ] * If two roots are equal say [math]D=a[/math], then solution is [math]y=(C_0+xC_1)e^{ax}[/math].Differential Equations - Repeated Roots [ http://tutorial.math.lamar.edu/Classes/DE/RepeatedRoots.aspx ] * If roots are complex say [math]D=a\pm ib[/math], then the solution is [math]y=(c_0\sin{bx}+c_1\cos{bx})e^{ax}[/math]. Differential Equations - Complex Roots [ http://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx ] * After finding the roots, we need to particular integral (PI). i,e, solution which is just satisfying the given function. We have [math]F(D)y=f(x)[/math] * If PI is of form [math]e^{ax}[/math], then solution is [math]\dfrac{1}{F(a)} e^{ax}[/math] * If PI is of form [math]\sin {ax}[/math], [math]\cos {ax}[/math] then replace [math]D^2[/math] with [math]-a^2[/math]. * If PI is of [math]x^{ax}[/math], then simply find partial fractions of [math]\dfrac{1}{F(D)}[/math] and then integrate. * If we have some combination of [math]e^{ax},\sin{bx}[/math], then first write [math]\sin{bx}[/math] or [math]\cos{bx}[/math] into exponential form using Euler identity [math]e^{ibx}=\cos{bx}+i\sin{bx}[/math]. Then combine both of the exponential terms and solve. * There is an alternative way to find solution. If PI is of form [math]e^{ax}[/math], the solution is of form [math]Ae^{ax}[/math]. If PI is of form [math]\sin {ax}[/math], or [math]\cos {ax}[/math] the solution is of form [math]A\cos {ax}+B\sin{ax}[/math]. If PI is of form [math]c_0x^2+c_1x+c_2[/math], then solution is of form [math]ax^2+bx+c[/math]. Once we assume the solution, then substitute in the given differential equation and find the coefficients. * Refer here for solved problems: Problems I [ https://rutherglen.science.mq.edu.au/wchen/lnfycfolder/fyc16.pdf ] and Problems II [ http://epsassets.manchester.ac.uk/medialand/maths/helm/19_3.pdf ] * Application of Differential Equations [ http://epsassets.manchester.ac.uk/medialand/maths/helm/19_4.pdf ] * Method of variation of parameters: It is used for any particular integral. Refer here how to solve: Variation of Parameters I [ https://math.berkeley.edu/~ehallman/math1B/variation-sols.pdf ], Solving tan (x) [ http://home.iitk.ac.in/~sghorai/TEACHING/MTH203/ode10.pdf ] * Variable coefficients: If the differential equation is of form [math]c_0x^nD^n+c_1x^{n-1}D^{n-1}+...+c_n)y=f(x)[/math], we need to assume [math]y=x^r[/math] as solution and substitute in the differential equation and solve for [math]r[/math]. Refer here for solved examples: Differential Equations - Euler Equations [ http://tutorial.math.lamar.edu/Classes/DE/EulerEquations.aspx ] * Partial Differential Equations: variable separable method is widely used in these problems. Some key points are: * * Partial derivatives determination [ https://www.math.psu.edu/tseng/class/Math251/Notes-Partial%20Differentiation.pdf ]is important to solve these problems. * If First order PDE is of form, [math]a\dfrac{\partial z}{\partial x}+b\dfrac{\partial z}{\partial y}=c[/math], The solution is solved by using [math]\dfrac{dx}{a}=\dfrac{dy}{b}=\dfrac{dz}{c}[/math]. Refer this to solve some problems Linear PDE [ http://nptel.ac.in/courses/111103021/7 ] * Refer this to solve non linear first order PDE: Problems I [ http://nptel.ac.in/courses/111103021/8 ] and Problems II [ http://nptel.ac.in/courses/111103021/10 ] * General second order equation is [math]au_{xx}+bu{xy}+cu_{yy}+du_x+eu_y+fu=g(x,y)[/math]. * * If [math]b^2-4ac%3C0[/math], PDE is elliptic. Ex: Laplace equation * If [math]b^2-4ac=0[/math], PDE is parabolic. Ex: Heat equation * If [math]b^2-4ac%3C0[/math], PDE is hyperbolic. Ex: Wave equation * One-Dimensional diffusion equation: [math]\dfrac{\partial u}{\partial t} =\kappa\dfrac{\partial^2 u}{\partial^2 x}[/math]. Where [math]\kappa = \dfrac{K_0}{c\rho}[/math] is called thermal diffusion. Refer here for solved example: 1D Diffusion equation [ https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/lecture-notes/heateqni.pdf ] * Heat equation: [math]\alpha^2 u_{xx}=u_t[/math]. * * We get the solution in variable separable form by assuming of [math]y=X(x)T(t)[/math]. * Substituting the above yields to [math]\alpha^2X"(x)T(t)=X(x)T'(t)[/math] * [math]\Rightarrow \dfrac{X"(x)}{X(x)} =\dfrac{T'(t)}{\alpha^2 T(t) }=C[/math](Constant. Since LHS is a function of x and RHS is a function of t). * Solve the two equations and substitute boundary conditions. * Solved example of Heat equation [ https://www.math.psu.edu/tseng/class/Math251/Notes-PDE%20pt1.pdf ] * Wave equation: [math]\alpha^2 u_{xx}=u_{tt}[/math]. Solve as above and solve by variable separable method. Refer here for solved example: Wave Equation [ https://www.math.psu.edu/tseng/class/Math251/Notes-PDE%20pt4.pdf ] * Numerical Methods: Key topics to focus are solution of algebraic equation by Newton’s method, single step numerical differentiation and Simpson and Trapezoidal methods. * * Simpson’s 1/3 rule: Used to determine area approximately. * * Given by: [math]\int\limits^{b}_{a}f(x)dx = \frac{h}{3}\left [y_{0}+y_n+4\left(\sum_{i = odd}y_{i}\right) + 2\left (\sum_{i=even}y_{i}\right)\right][/math] * Error [math]E= -\frac{b-a}{180}\overline{\Delta^4 y}[/math], where [math]\overline{\Delta^4 y}[/math] is fourth forward difference. * Solved example here: Simpson’s 1/3 Rule Example [ http://nptel.ac.in/courses/122104018/node121.html ] * Trapezoidal rule: * * Given by [math]\int\limits^{b}_{a}f(x)dx = h\left[\frac{y_0+y_n}{2}+\sum_{i=1}^{n-1} y_i\right][/math] * Error [math]E=-\frac{b-a}{12}\overline{\Delta^2 y}[/math] * Solved example here: Trapezoidal Rule Example [ http://nptel.ac.in/courses/122104018/node120.html ] * Simpson’s 3/8 rule: * * Given by: [math]\int\limits^{b}_{a}f(x)dx = \frac{3h}{8} \left[y_{0}+y_n+3\left(\sum_{i = 1,2,4,5,\cdots} y_{i} \right)+2\left(\sum_{i=3,6,\cdots} y_{i}\right)\right][/math] * Error [math]E= -\frac{3(b-a)}{80}h^4\overline{\Delta^4 y}[/math] * Solved example here: Simpson’s 3/8 Rule Example [ http://mathforcollege.com/nm/mws/gen/07int/mws_gen_int_txt_simpson3by8.pdf ] * Euler’s forward method: Refer this for solved example [ http://nptel.ac.in/courses/122104018/node124.html ] * * Given by: [math]y_{i+1} = y_i + h f(x_i, y_i) + \frac{h^2}{2!} f(x_i, y_i) + \cdots[/math] * It is a single step forward method. Error order [math]O(h^2)[/math] * For Euler’s backward method, use [math]y_{i-1}[/math] instead of [math]y_{i+1}[/math] and [math]h[/math] with [math]-h[/math] * Range-Kutta 2nd Order: It has error order 2. Given by [math]y_{k+1} = y_k + \frac{h}{2} \left( f(x_k, y_k) + f(x_k + h, y_k + h f(x_k)) \right)[/math] * Range-Kutta 4th Order: It has error order 4 i.e [math]O(h^5)[/math]. * * Given by [math]y_{k+1} = y_k + \frac{k_1 + 2 k_2 + 2 k_3 + k_4}{6}[/math], where, * [math]k_1 = h f(x_k, y_k)[/math] * [math]k_2 = h f(x_k + \frac{h}{2}, y_k + \frac{k_1}{2})[/math] * [math]k_3 = h f(x_k + \frac{h}{2}, y_k + \frac{k_2}{2})[/math] * [math]k_4 = h f(x_k + h, y_k + k_3)[/math] * Refer here for Newton Interpolating Polynomial [ http://nptel.ac.in/courses/122104019/numerical-analysis/Rathish-kumar/rathish-oct31/fratnode5.html ] and Lagrangian Interpolating Polynomial [ http://nptel.ac.in/courses/122104019/numerical-analysis/Rathish-kumar/rathish-oct31/fratnode4.html ] * Newton-Raphson method: Used for solving roots of algebraic and transcendental equations. Given by [math]x_{i+1}=x_i+\frac{f(x_i)}{f'(x_i)}[/math]. Refer here for solved examples [ https://www.math.ubc.ca/~anstee/math104/104newton-solution.pdf ] * Bisection Method and False Position Method - Linear convergence; Secant Method - 1.618 and Newton’s Raphsons Method - 2 * Probability and Statistics: Key topics to focus are Bayes Theorem, Conditional and Joint probability, Binomial, Poisson, normal and exponential distribution — there mean, variance and standard deviation, Correlation and regression. * * Properties of Probability [ https://onlinecourses.science.psu.edu/stat414/node/7 ]: * * [math]0\leq P(A)\leq 1[/math] * [math]P(A)=1-P(A')[/math] * [math]P(\varnothing)=0 [/math] * [math]P(\cup)=1[/math] * Conditional Probability [ https://onlinecourses.science.psu.edu/stat414/node/10 ]: Given by * * [math]P(A \cap B) = P(A | B) × P(B)[/math] * [math]P(A \cap B) = P(B | A) × P(A)[/math] * Bayes Theorem: [math]P(E_i|A)=\frac{P(E_i)P(E_i|A)}{\sum\limits_{k=0}^{n}P(E_k)P(A|E_k)}[/math], where [math]P(E_i|A)=\frac{P(E_i \cap A)}{P(A)}[/math]. Refer here for solved examples [ https://onlinecourses.science.psu.edu/stat414/node/12 ] * Discrete Distribution [ https://onlinecourses.science.psu.edu/stat414/node/48 ]: * * Binomial Distribution [ https://onlinecourses.science.psu.edu/stat414/node/53 ]: [math]f(x)=\binom{n}{x} p^x (1-p)^{n-x}[/math], denoted by [math]X \sim b(n, p)[/math]. Mean [math]\mu=np[/math], Variance [math]\sigma^2 = np(1-p)[/math] * Geometric Distribution [ https://onlinecourses.science.psu.edu/stat414/node/55 ]: [math]f(x)=P(X=x)=(1-p)^{x−1}p[/math]. Cumulative distribution function is [math]F(x)=P(X\leq x)=1-(1-p)^x[/math]. Mean [math]\mu=E(X)=\frac{1}{p}[/math], Variance [math]\sigma^2=Var(X)=\frac{1-p}{p^2}[/math] * Negative Binomial Distribution [ https://onlinecourses.science.psu.edu/stat414/node/78 ]: [math]f(x)=P(X=x)=\binom{x-1}{r-1}(1-p)^{x-r}p^r[/math], Mean [math]\mu = E(X) = \frac{r}{p}[/math] and Variance [math]\sigma^2=\frac{r(1-p)}{p^2}[/math] * Poisson Distribution [ https://onlinecourses.science.psu.edu/stat414/node/54 ]: [math]f(x)=\frac{e^{-\lambda}\lambda^x}{x!}[/math], Mean [math]\mu=\lambda[/math], Variance [math]\sigma^2=\lambda[/math] * Continuous Distribution [ https://onlinecourses.science.psu.edu/stat414/node/86 ]: * * Exponential Distribution [ https://onlinecourses.science.psu.edu/stat414/node/138 ]: [math]f(x)=\frac{1}{\theta}e^{-x/\theta}[/math], Mean [math]\mu = \theta[/math], Variance [math]\sigma^2=\theta^2[/math] * Normal Distribution [ https://onlinecourses.science.psu.edu/stat414/node/139 ]: [math]f(x)=\frac{1}{\sigma \sqrt{2\pi}}-e^{\frac{-1}{2} (\frac{x-\mu}{\sigma})^2}[/math], Mean [math]\mu[/math] and the variance of X is [math]\sigma^2[/math] * Substituting [math]Z=\dfrac{X-\mu}{\sigma}[/math] converts normal distribution into standard normal distribution. Solve example problems here [ https://onlinecourses.science.psu.edu/stat414/node/150 ] * Hypothesis testing can be studied from here [ https://onlinecourses.science.psu.edu/stat414/node/290 ] and Correlation and Regression from here [ https://academic.macewan.ca/burok/Stat141/notes/regression.pdf ] For specific doubts, comment below. It would be grateful if someone could assist me in creating the PDF (with proper links and formulas)of this file and add a link here so that it would be very helpful to all.
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