Difference between revisions of "LaTeX/C2/Equations/English"
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| − | | | + | || Initial conditions |
| − | + | || | |
| − | | | + | # Copy equations.tex into numbers.tex |
# Both equations.tex and numbers.tex should be sized | # Both equations.tex and numbers.tex should be sized | ||
| − | # Both should be opened in | + | # Both should be opened in TeXworks |
# Both should be at the first line | # Both should be at the first line | ||
# equations.tex should be at the top, and numbers.tex should be below | # equations.tex should be at the top, and numbers.tex should be below | ||
| Line 15: | Line 15: | ||
# All should be arranged, as recorded in the tutorial | # All should be arranged, as recorded in the tutorial | ||
# equations.tex should be directly below the slides file | # equations.tex should be directly below the slides file | ||
| + | |- | ||
| + | || TeXworks: Lucida | ||
| + | || Grande, 22pt, 20px. Preview: 75%, 200dpi | ||
| + | |- | ||
| + | |} | ||
| + | <br> | ||
| + | <br> | ||
| − | + | {| border = 1 | |
|- | |- | ||
| − | | | + | ||'''Visual Cue''' |
| − | | | + | ||'''Narration''' |
| − | + | ||
|- | |- | ||
| − | | | + | || Opening slide |
| − | + | || Welcome to this tutorial on writing equations using '''LaTeX'''. | |
| − | | | + | |
| − | My name is Kannan Moudgalya | + | My name is Kannan Moudgalya. |
|- | |- | ||
| − | | | + | || Learning objectives slide |
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Down arrow | Down arrow | ||
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Down arrow | Down arrow | ||
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Down arrow | Down arrow | ||
| − | | | + | || In this tutorial, we will learn |
| + | *How to create an equation in '''LaTeX'''. | ||
| − | How to write | + | *How to write multiple equations |
| − | How to | + | *How to align multiple equations and |
| − | Why we should avoid the dollar mode and blank lines while writing equations | + | *Why we should avoid the '''dollar mode''' and blank lines while writing equations |
|- | |- | ||
| − | | | + | || System requirements slide |
| − | + | || One may use any of '''Linux, Windows''' or the '''Mac''' operating system. | |
| − | | | + | |
| − | One needs LaTeX | + | One needs '''LaTeX'''. |
| − | I shall use | + | I shall use '''TexWorks''', although it is not required. |
|- | |- | ||
| − | | | + | || Prerequisites slide |
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Press Down arrow | Press Down arrow | ||
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Down arrow | Down arrow | ||
| − | | | + | || What are the prerequisites? |
| − | One needs to be familiar with spoken tutorials that introduce LaTeX | + | One needs to be familiar with spoken tutorials that introduce '''LaTeX'''. |
| − | Exposure to the recommended method of side-by-side learning | + | Exposure to the recommended method of '''side-by-side learning'''. |
| − | This information is available at the Spoken Tutorial webpage | + | This information is available at the '''Spoken Tutorial''' webpage. |
|- | |- | ||
| − | | | + | || Additional material slide |
| − | | | + | || I shall use the file '''equations.tex''' |
| − | Open in | + | |- |
| − | + | ||Open in TeXworks | |
| − | + | ||I have opened it in '''TeXworks'''. | |
| − | + | ||
| + | |- | ||
| + | ||Back to slides file, press down arrow | ||
Press down arrow | Press down arrow | ||
| − | |||
Press down arrow | Press down arrow | ||
| − | | | + | ||It is available as a code file along with this tutorial. |
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | It is available as a | + | |
| − | You need to have it to practise along with me | + | You need to have it to practise along with me. |
| − | If you don't have it, please download it from the Spoken Tutorial | + | If you don't have it, please download it from the '''Spoken Tutorial webpage'''. |
|- | |- | ||
| − | | | + | || Slide on Leslie Lamport’s book |
| − | + | ||
| + | || I will next call your attention to this standard textbook by '''Leslie Lamport'''. | ||
| + | I learnt my '''LaTeX''' from this book. | ||
| + | It is available for about Rs. 500. | ||
| + | |- | ||
| + | ||Web browser is opened | ||
| + | ||Let me show on the web browser where one can buy it. | ||
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|- | |- | ||
| − | | | + | || |
| − | | | + | || Let us begin this tutorial |
| − | + | ||
|- | |- | ||
| − | | | + | || Open equations.tex |
| − | + | ||
| + | || Let us learn how to write equations using '''LaTeX'''. | ||
| + | I have already opened the file '''equations.tex''' in '''TeXworks'''. | ||
| + | I have saved it as '''numbers.tex''' to use in this tutorial. | ||
| + | I will keep the file '''equations.tex''' intact, and not change it. | ||
| + | |- | ||
| + | ||Open numbers.tex | ||
| + | ||Let me open '''numbers.tex''' | ||
| + | |- | ||
| + | || | ||
| + | ||Please pause the tutorial, make this copy and open it. | ||
| + | Whenever you want to start afresh, you can make a new copy of '''equations.tex''' | ||
| + | I recommend this method whenever a code file is given with ANY spoken tutorial. | ||
| + | |- | ||
| + | || Press the '''Typeset''' key | ||
| + | || Let us '''compile''' this file by pressing the '''Typeset''' arrow at the top left hand corner. | ||
| + | |- | ||
| + | ||Pdf file | ||
| + | ||You can see the '''pdf file''' on the right. | ||
| + | |- | ||
| + | ||Switch to the tex file. | ||
| + | ||Let us look at the '''tex''' file, which begins with its usual '''commands'''. | ||
| + | |- | ||
| + | ||Highlight 8cm | ||
| + | ||I have chosen the width of the '''pdf file''' to be 8cm. | ||
| + | This is the only way I can show the entire file and also use a large font. | ||
| + | |- | ||
| + | || | ||
| − | + | Change 8 to 15 | |
| − | + | ||
| − | + | Put back 8 | |
| − | + | || You may want to increase it to a more reasonable value, such as 15cm. | |
| − | I will | + | But I will continue to use 8cm. |
| − | + | |- | |
| + | ||Click amsmath line | ||
| + | ||I am using the '''package amsmath'''. | ||
| − | + | |- | |
| + | || Click the first align | ||
| + | || I write the first equation within the '''align star environment''', as shown here. | ||
| − | + | |- | |
| + | ||Switch to the pdf and highlight the equation. | ||
| + | ||You can see how it gets '''typeset''' in the '''pdf file'''. | ||
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|- | |- | ||
| − | | | + | ||Click u(t) line in tex |
| − | | | + | ||On the left hand side of this equation, we have '''u of t'''. |
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| − | | | + | ||Click u(t) in the pdf file |
| − | | | + | ||On the right hand side, I have produced big square brackets. |
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Hover on \left and click \right | Hover on \left and click \right | ||
| − | | | + | ||How did I do this? |
| − | + | With '''back slash left square''' and '''back slash right square''' brackets as shown here. | |
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| − | With | + | |
|- | |- | ||
| − | | | + | || Scroll the tex file up |
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Click align* line | Click align* line | ||
| + | || Similarly, I write a second equation using '''align star'''. | ||
| + | |- | ||
| + | ||Click the x_1 line | ||
| + | ||On the left hand side, we have the '''derivative''' of a '''vector'''. | ||
| − | + | |- | |
| − | + | ||Click on the eqn in the pdf file | |
| − | + | ||We can see the '''pdf output''' here. | |
| − | Click on the eqn | + | |
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| + | |- | ||
| + | ||Click on matrix tex | ||
| + | || On the right hand side, we have a '''3 by 3 matrix'''. | ||
| + | |- | ||
| + | ||Click on matrix pdf | ||
| + | ||See the corresponding output. | ||
| + | |- | ||
| + | ||Delete row 1 compile | ||
| + | ||Suppose we delete the first row in this '''matrix''' and '''compile'''. | ||
| + | |- | ||
| + | || | ||
Click the pdf | Click the pdf | ||
| + | ||We get a '''2 by 3 matrix pdf output'''. | ||
| + | The first line is gone. | ||
| + | It is your responsibility to make the equations correct and consistent. | ||
| + | |- | ||
| + | ||Undo the last command in tex | ||
| + | ||Let me put the row back. | ||
| + | |- | ||
| + | ||Compile | ||
| + | ||Let me '''compile'''. | ||
| − | + | |- | |
| − | | | + | || |
| + | ||You can also add more rows, if necessary. | ||
| + | Similarly, you can change the columns, too. | ||
| − | + | |- | |
| + | || | ||
| + | || Recall that we used the '''dollar mode''' to write formulae in another tutorial. | ||
| − | + | |- | |
| + | ||Click on '''begin align''' in tex | ||
| + | ||Here, '''align''' plays the role of the '''dollar mode'''. | ||
| − | + | In fact, one should not put the '''dollar sign''' within the '''align environment'''. | |
| − | + | |- | |
| + | ||Put $ and compile | ||
| − | |||
| + | ||Suppose we put the '''alpha symbol''' within '''$ signs''' and '''compile'''. | ||
| − | We get | + | |- |
| + | ||Error message | ||
| + | ||We get an error message about '''dollar'''. | ||
| − | + | |- | |
| + | ||Press red cross symbol to abort | ||
| + | ||Press the cross symbol on the top left red box and abort the previous '''compilation'''. | ||
| − | + | |- | |
| − | + | ||Remove '''$ signs''' | |
| − | Let | + | ||Let us remove the '''dollar''' sign. |
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| − | + | '''Compile''' again, it works! | |
|- | |- | ||
| − | | | + | || Below the line with alpha, leave a blank line |
| − | + | ||
| + | || Another common mistake is leaving blank lines in the '''align environment'''. | ||
| + | Let me leave an empty line and '''compile'''. | ||
| + | |- | ||
| + | ||Error message | ||
| + | ||It gives an error message. | ||
| + | |- | ||
| + | ||Put % | ||
| + | ||If you need this gap, put '''%''', the '''comment''' character, in the first column. | ||
| + | |- | ||
| + | ||Press red cross symbol to abort | ||
| + | ||Abort '''compiling'''. | ||
| + | |- | ||
| + | ||Press Typeset symbol to compile | ||
| + | ||'''Compile''' again - it '''compiles ok'''. | ||
| + | So it is important to remember that you should not leave a blank line within the '''align statement'''. | ||
| + | |- | ||
| + | ||Remove the blank line | ||
| + | ||Let me remove this line and '''compile'''. | ||
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| − | | | + | ||Next, I want to align the two equations by the '''equal to''' sign. |
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| − | + | For this, we need to put both equations inside one '''align star environment'''. | |
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| − | | | + | || Remove \end{align*} |
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| − | Remove \end{align*} | + | |
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Remove \begin{align*} | Remove \begin{align*} | ||
| + | || Remove the end '''align* command''' of the first equation. | ||
| + | Remove the begin '''align* command''' of the second equation. | ||
| + | |- | ||
| + | ||Compile | ||
| + | || Let us '''compile'''. | ||
| + | |- | ||
| + | ||In the pdf file | ||
| + | ||We see only one equation appearing. | ||
| + | The text in between the two equations appears as a large string. | ||
| + | The second equation does not even show up. | ||
| + | |- | ||
| + | || | ||
| + | || This is not what we wanted. | ||
| + | This problem is solved with the help of the '''intertext command'''. | ||
| + | |- | ||
| + | ||Add braces before and after the text. | ||
| + | ||Let us put the text in between braces. | ||
| + | |- | ||
| + | ||Put the text into \intertext{} | ||
| + | ||Let us also put this '''command, back slash intertext''' as I type now. | ||
| + | Note that '''intertext''' is one word, without a space in between. | ||
| + | |- | ||
| + | ||Point to the '''\''' | ||
| + | ||Do not forget the '''back slash''' before '''intertext'''. | ||
| + | |- | ||
| + | ||Point to the closing brace | ||
| + | ||Also, do not forget the closing brace. | ||
| + | These are all common mistakes. | ||
| − | + | |- | |
| + | ||Compile | ||
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| + | ||On compiling, we get this '''pdf''' file. | ||
| + | You see both equations now. But unfortunately, you see both the equations right aligned. | ||
| + | Not aligned by the '''equal to''' sign. | ||
| + | |- | ||
| + | || | ||
| + | ||So, I am not happy with it. | ||
| + | I want to align by the '''equal to''' sign. | ||
| + | |- | ||
| + | ||Put & sign before = | ||
| + | ||To do this, let us put the '''ampersand''' sign in front of the '''equal to''' sign. | ||
| + | Let's do it for the first equation. | ||
| + | |- | ||
| + | ||Repeat in the 2<sup>nd</sup> eqn. | ||
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| + | ||And then let's do it for the second equation. | ||
| + | If you have more equations to be aligned, do this for all of them. | ||
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| − | | | + | ||Compile |
| − | | | + | ||On '''compiling''', we see the two equations aligned by the '''equal to''' sign. |
| − | | | + | |- |
| + | || | ||
| + | || The equations are aligned even though there is text in between. | ||
| − | Remember to use the | + | Remember to use the '''intertext command''' whether you want the equations aligned or not. |
|- | |- | ||
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Delete text lines | Delete text lines | ||
| + | || What do we do if we don’t have text between equations? | ||
| + | To understand this, let us delete the entire text between the two equations. | ||
| + | |- | ||
| + | ||Put \\ | ||
| + | ||Note that we should not leave an empty line. | ||
| + | |- | ||
| + | ||Compile | ||
| + | ||On '''compiling''', we see both equations coming in the same line. | ||
| + | We didn’t tell '''LaTeX''' to start the second equation in the next line. | ||
| + | |- | ||
| + | ||Add '''\\''' | ||
| + | ||We solve this by putting two '''back slashes''' at the end of the first equation. | ||
| + | |- | ||
| + | ||Compile | ||
| + | ||Let us compile, to get the correct '''pdf''' file. | ||
| + | |- | ||
| + | ||Undo | ||
| + | ||Let us now undo and put the text in between two equations. | ||
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|- | |- | ||
| − | | | + | ||Compile |
| − | | | + | ||Let us '''compile''' once more. |
| − | | | + | |- |
| + | || | ||
| + | || This is where we will stop. | ||
| − | Let me go to the slides | + | Let me go to the slides. |
|- | |- | ||
| − | | | + | || Summary slide |
| − | + | ||
Down arrow | Down arrow | ||
Down arrow | Down arrow | ||
| − | | | + | || This page summarises what we learnt in this tutorial. |
| − | Please recall each of these commands | + | Please recall each of these '''commands'''. |
| − | This is your first assignment | + | This is your first assignment. |
|- | |- | ||
| − | | | + | || More assignments slide |
| − | + | ||
Down arrow | Down arrow | ||
| − | | | + | || I will give some more assignments. |
Can you pause the video and try to produce these equations? | Can you pause the video and try to produce these equations? | ||
| − | The required code is given here | + | The required code is given here. |
|- | |- | ||
| − | | | + | || Assignment 3 |
| − | + | || Can you produce these equations? | |
| − | | | + | |
| − | Notice that these equations are now aligned at the equal to sign | + | Notice that these equations are now aligned at the '''equal to''' sign. |
|- | |- | ||
| − | | | + | || Assignment 4 |
| − | + | || What happens if you forget a required '''ampersand''' sign? | |
| − | | | + | |
| − | Please pause the video and try this input | + | Please pause the video and try this input. |
| − | You should get this output | + | You should get this output. |
| − | Forgetting ampersand signs is a common mistake made by everyone! | + | Forgetting '''ampersand''' signs is a common mistake made by everyone! |
|- | |- | ||
| − | | | + | || Assignment 5 |
| − | + | || Can you pause this video and try to produce this output? | |
| − | | | + | |
| − | Use the hint given here | + | Use the hint given here. |
|- | |- | ||
| − | | | + | || Assignment 6 |
| − | + | || In this assignment, you have to align three equations. | |
| − | | | + | |
| − | Please pause the video here, and try to produce them by yourself | + | Please pause the video here, and try to produce them by yourself. |
| − | I have given the answer here | + | I have given the answer here. |
|- | |- | ||
| − | | | + | || Assignment 7 |
| − | + | || I have the last assignment in this slide. | |
| − | | | + | |
| − | Insert some text between equations | + | Insert some text between equations. |
|- | |- | ||
| − | | | + | || About the ST slide |
| − | + | || I will now give some general information. | |
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| + | This video summarises the '''Spoken Tutorial project'''. | ||
|- | |- | ||
| − | | | + | || Workshops slide |
| − | + | || We conduct workshops using '''Spoken Tutorials'''. | |
| − | | | + | |
|- | |- | ||
| − | | | + | || Answers slide |
| − | + | ||
Press down arrow | Press down arrow | ||
| Line 618: | Line 503: | ||
Down arrow twice | Down arrow twice | ||
| − | | | + | || Do you have questions in <u>THIS</u> '''Spoken Tutorial?''' |
| + | * Please visit this site | ||
* Choose the minute and second where you have the question | * Choose the minute and second where you have the question | ||
* Explain your question briefly | * Explain your question briefly | ||
| Line 624: | Line 510: | ||
| − | + | You will have to register on this website to ask questions. | |
| − | You will have to register on this website to ask questions | + | |
|- | |- | ||
| − | | | + | || Where to post slide |
| − | + | || For topics not covered in spoken tutorials, visit '''stack exchange''' at this address. | |
| − | | | + | |
|- | |- | ||
| − | | | + | || Contact us slide |
| − | + | || For any other questions, please contact us. | |
| − | | | + | |
|- | |- | ||
| − | | | + | || Acknowledgement slide |
| − | + | || '''Spoken Tutorial''' project is funded by MHRD, Government of India. | |
| − | | | + | |
|- | |- | ||
| − | | | + | || |
| − | | | + | || Thanks for joining. |
| − | + | ||
| − | Goodbye | + | Goodbye. |
|- | |- | ||
|} | |} | ||
Latest revision as of 19:24, 22 October 2019
| Initial conditions |
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| TeXworks: Lucida | Grande, 22pt, 20px. Preview: 75%, 200dpi |
| Visual Cue | Narration |
| Opening slide | Welcome to this tutorial on writing equations using LaTeX.
My name is Kannan Moudgalya. |
| Learning objectives slide
Down arrow Down arrow Down arrow |
In this tutorial, we will learn
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| System requirements slide | One may use any of Linux, Windows or the Mac operating system.
One needs LaTeX. I shall use TexWorks, although it is not required. |
| Prerequisites slide
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What are the prerequisites?
One needs to be familiar with spoken tutorials that introduce LaTeX. Exposure to the recommended method of side-by-side learning. This information is available at the Spoken Tutorial webpage. |
| Additional material slide | I shall use the file equations.tex |
| Open in TeXworks | I have opened it in TeXworks. |
| Back to slides file, press down arrow
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It is available as a code file along with this tutorial.
You need to have it to practise along with me. If you don't have it, please download it from the Spoken Tutorial webpage. |
| Slide on Leslie Lamport’s book
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I will next call your attention to this standard textbook by Leslie Lamport.
I learnt my LaTeX from this book. It is available for about Rs. 500. |
| Web browser is opened | Let me show on the web browser where one can buy it. |
| Let us begin this tutorial | |
| Open equations.tex
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Let us learn how to write equations using LaTeX.
I have already opened the file equations.tex in TeXworks. I have saved it as numbers.tex to use in this tutorial. I will keep the file equations.tex intact, and not change it. |
| Open numbers.tex | Let me open numbers.tex |
| Please pause the tutorial, make this copy and open it.
Whenever you want to start afresh, you can make a new copy of equations.tex I recommend this method whenever a code file is given with ANY spoken tutorial. | |
| Press the Typeset key | Let us compile this file by pressing the Typeset arrow at the top left hand corner. |
| Pdf file | You can see the pdf file on the right. |
| Switch to the tex file. | Let us look at the tex file, which begins with its usual commands. |
| Highlight 8cm | I have chosen the width of the pdf file to be 8cm.
This is the only way I can show the entire file and also use a large font. |
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Change 8 to 15 Put back 8 |
You may want to increase it to a more reasonable value, such as 15cm.
But I will continue to use 8cm. |
| Click amsmath line | I am using the package amsmath. |
| Click the first align | I write the first equation within the align star environment, as shown here. |
| Switch to the pdf and highlight the equation. | You can see how it gets typeset in the pdf file. |
| Click u(t) line in tex | On the left hand side of this equation, we have u of t. |
| Click u(t) in the pdf file | On the right hand side, I have produced big square brackets. |
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How did I do this?
With back slash left square and back slash right square brackets as shown here. |
| Scroll the tex file up
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Similarly, I write a second equation using align star. |
| Click the x_1 line | On the left hand side, we have the derivative of a vector. |
| Click on the eqn in the pdf file | We can see the pdf output here. |
| Click on matrix tex | On the right hand side, we have a 3 by 3 matrix. |
| Click on matrix pdf | See the corresponding output. |
| Delete row 1 compile | Suppose we delete the first row in this matrix and compile. |
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Click the pdf
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We get a 2 by 3 matrix pdf output.
The first line is gone. It is your responsibility to make the equations correct and consistent. |
| Undo the last command in tex | Let me put the row back. |
| Compile | Let me compile. |
| You can also add more rows, if necessary.
Similarly, you can change the columns, too. | |
| Recall that we used the dollar mode to write formulae in another tutorial. | |
| Click on begin align in tex | Here, align plays the role of the dollar mode.
In fact, one should not put the dollar sign within the align environment. |
| Put $ and compile
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Suppose we put the alpha symbol within $ signs and compile. |
| Error message | We get an error message about dollar. |
| Press red cross symbol to abort | Press the cross symbol on the top left red box and abort the previous compilation. |
| Remove $ signs | Let us remove the dollar sign.
Compile again, it works! |
| Below the line with alpha, leave a blank line
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Another common mistake is leaving blank lines in the align environment.
Let me leave an empty line and compile. |
| Error message | It gives an error message. |
| Put % | If you need this gap, put %, the comment character, in the first column. |
| Press red cross symbol to abort | Abort compiling. |
| Press Typeset symbol to compile | Compile again - it compiles ok.
So it is important to remember that you should not leave a blank line within the align statement. |
| Remove the blank line | Let me remove this line and compile. |
| Next, I want to align the two equations by the equal to sign.
For this, we need to put both equations inside one align star environment. | |
| Remove \end{align*}
Remove \begin{align*} |
Remove the end align* command of the first equation.
Remove the begin align* command of the second equation. |
| Compile | Let us compile. |
| In the pdf file | We see only one equation appearing.
The text in between the two equations appears as a large string. The second equation does not even show up. |
| This is not what we wanted.
This problem is solved with the help of the intertext command. | |
| Add braces before and after the text. | Let us put the text in between braces. |
| Put the text into \intertext{}
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Let us also put this command, back slash intertext as I type now.
Note that intertext is one word, without a space in between. |
| Point to the \ | Do not forget the back slash before intertext. |
| Point to the closing brace | Also, do not forget the closing brace.
These are all common mistakes. |
| Compile
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On compiling, we get this pdf file.
You see both equations now. But unfortunately, you see both the equations right aligned. Not aligned by the equal to sign. |
| So, I am not happy with it.
I want to align by the equal to sign. | |
| Put & sign before = | To do this, let us put the ampersand sign in front of the equal to sign.
Let's do it for the first equation. |
| Repeat in the 2nd eqn.
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And then let's do it for the second equation.
If you have more equations to be aligned, do this for all of them. |
| Compile | On compiling, we see the two equations aligned by the equal to sign. |
| The equations are aligned even though there is text in between.
Remember to use the intertext command whether you want the equations aligned or not. | |
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Delete text lines |
What do we do if we don’t have text between equations?
To understand this, let us delete the entire text between the two equations. |
| Put \\ | Note that we should not leave an empty line. |
| Compile | On compiling, we see both equations coming in the same line.
We didn’t tell LaTeX to start the second equation in the next line. |
| Add \\ | We solve this by putting two back slashes at the end of the first equation. |
| Compile | Let us compile, to get the correct pdf file. |
| Undo | Let us now undo and put the text in between two equations. |
| Compile | Let us compile once more. |
| This is where we will stop.
Let me go to the slides. | |
| Summary slide
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This page summarises what we learnt in this tutorial.
Please recall each of these commands. This is your first assignment. |
| More assignments slide
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I will give some more assignments.
Can you pause the video and try to produce these equations? The required code is given here. |
| Assignment 3 | Can you produce these equations?
Notice that these equations are now aligned at the equal to sign. |
| Assignment 4 | What happens if you forget a required ampersand sign?
Please pause the video and try this input. You should get this output. Forgetting ampersand signs is a common mistake made by everyone! |
| Assignment 5 | Can you pause this video and try to produce this output?
Use the hint given here. |
| Assignment 6 | In this assignment, you have to align three equations.
Please pause the video here, and try to produce them by yourself. I have given the answer here. |
| Assignment 7 | I have the last assignment in this slide.
Insert some text between equations. |
| About the ST slide | I will now give some general information.
This video summarises the Spoken Tutorial project. |
| Workshops slide | We conduct workshops using Spoken Tutorials. |
| Answers slide
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Do you have questions in THIS Spoken Tutorial?
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| Where to post slide | For topics not covered in spoken tutorials, visit stack exchange at this address. |
| Contact us slide | For any other questions, please contact us. |
| Acknowledgement slide | Spoken Tutorial project is funded by MHRD, Government of India. |
| Thanks for joining.
Goodbye. |
