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# Gradients and Related Operators Civil Engineering (CE) Notes | EduRev

## Civil Engineering (CE) : Gradients and Related Operators Civil Engineering (CE) Notes | EduRev

The document Gradients and Related Operators Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Advanced Solid Mechanics - Notes, Videos, MCQs & PPTs.
All you need of Civil Engineering (CE) at this link: Civil Engineering (CE)

In this section, we consider scalar and vector-valued functions that assign a scalar and vector to each element in the subset of the set of position vectors for the points in 3D space. If x denotes the position vector of points in the 3D space, then   is a function that assigns a scalar Φ to each point in the 3D space of interest and Φ is called the scalar field. A few examples of scalar fields are temperature, density, energy. Similarly, the vector field û(x) assigns a vector to each point in the 3D space of interest. Displacement, velocity, acceleration are a few examples of vector fields.

Let  denote the region of the 3D space that is of interest, that is the set of position vectors of points that is of interest in the 3D space. Then a scalar field Φ, defined on a domain     is said to be continuous if

where  denote the set of all position vectors of points in the 3D space  a is a constant vector. The scalar field φ is said to be differential if there exist a vector field w such that

It can be shown that there is at most one vector field, w satisfying the above equation (proof omitted). This unique vector field w is called the gradient of Φ and denoted by grad(Φ).
The properties of continuity and differentiability are attributed to a vector field u and a tensor field T defined on a domain D, if they apply to the scalar field u·a and a · Tba and b Given that the vector field u, is differentiable, the gradient of u denoted by

The divergence and the curl of a vector u denoted as div(u) and curl(u), are respectively scalar valued and vector valued and are defined by

When the tensor field T is differentiable, its divergence denoted by div(T), is the vector field defined by

If grad(Φ) exist and is continuous, Φ is said to be continuously differentiable and this property extends to u and T if grad(u · a) and grad(a · Tb) exist and are continuous for all a, b ∈

An important identity: For a continuously differentiable vector field, u

(Proof omitted)
Let Φ and u be some scalar and vector field respectively. Then, the Laplacian operator, denoted by ∆ (or by ∇2 ), is defined as

The Hessian operator, denoted by ∇∇ is defined as

If a vector field u is divergence free (i.e. div(u) = 0) then it is called solenoidal. It is called irrotational if curl(u) = o. It can be established that

curl(curl(u)) = grad(div(u)) − ∆(u).                                       (2.214)

Consequently, if a vector field, u is both solenoidal and irrotational, then ∆(u) = o (follows from the above equation) and such a vector field is said to be harmonic. If a scalar field Φ satisfies ∆(Φ) = 0, then Φ is said to be harmonic.

Potential theorem: Let u be a smooth point field on an open or closed simply connected region R and let curl(u) = o. Then there is a continuously differentiable scalar field on R such that

From the definition of grad (2.207), it can be seen that it is a linear operator. That is,

Consequently, all other operators div, ∆ are also linear operators.

Before concluding this section we collect a few identities that are useful subsequently. Here Φ, Ψ are scalar fields, u, v are vector fields and A is second order tensor field.

Till now, in this section we defined all the quantities in closed form but we require them in component form to perform the actual calculations. This would be the focus in the reminder of this section. In the following, let ei denote the Cartesian basis vectors and xi i = 1, 2, 3, the Cartesian coordinates.

Let Φ be differentiable scalar field, then it follows from equation (2.206), on replacing a by the base vectors ei in turn, that the partial derivatives   exist in D and that, moreover, w=  . Hence

Let u be differentiable vector field in D and a some constant vector in E. Then, using (2.228) we compute

where ap and up are the Cartesian components of a and u respectively. Consequently, appealing to definition (2.207) we obtain

Then, according to the definition for divergence, (2.208)

Next, we compute div(u ∧ a) to be

using (2.231). Comparing the above result with the definition of curl in (2.209) we infer

Finally, let T be a differentiable tensor field in  , then we compute

using equation (2.231). Then, we infer

from the definition (2.210).

The component form for the other operators, namely Laplacian and Hessian can be obtained as respectively.

Above we obtained the components of the different operators in Cartesian coordinate system, we shall now proceed to obtain the same with respect to other coordinate systems. While using the method illustrated here we could obtain the components in any orthogonal curvilinear system, we choose the cylindrical polar coordinate system for illustration.

First, we outline a general procedure to find the basis vectors for a given orthogonal curvilinear coordinates. A differential vector dp in the 3D vector space is written as dp = dxiei relative to Cartesian coordinates. The same coordinate independent vector in orthogonal curvilinear coordinates is written as

where zdenotes curvilinear coordinates. But

where gi is the basis vectors in the orthogonal curvilinear coordinates. Thus,

Comparing equations (2.239) and (2.240) we obtain the desired transformation relation between the bases to be:

The basis gso obtained will vary from point to point in the Euclidean space and will be orthogonal (by the choice of curvilinear coordinates) at each point but not orthonormal. Hence we define

and use these as the basis vectors for the orthogonal curvilinear coordinates.

Let (x, y, z) denote the coordinates of a typical point in Cartesian coordinate system and (r, θ, Z) the coordinates of a typical point in cylindrical polar coordinate system. Then, the coordinate transformation from cylindrical polar to Cartesian and vice versa is given by:

respectively. From these we compute

Consequently, the orthonormal cylindrical polar basis vectors (er, eθ, eZ) and Cartesian basis vectors (e1, e2, e3) are related through the equations

obtained using equation (2.242) along with a change in notation: ((ec)1, (ec )2,(ec )3) = (er, eθ, eZ).
Now, we can compute the components of grad(Φ) in cylindrical polar coordinates. Towards this, we obtain

using the chain rule for differentiation.

To obtain the components of grad(u) one can follow the same procedure outlined above or can compute the same using the above result and the definition of grad (2.207) as detailed below:

where we have used the following identities:

obtained by recognizing that only the Cartesian components of a are to be treated as constants and not the cylindrical polar components, ar = acos(θ) + ay sin(θ), aθ = −ax sin(θ) + ay cos(θ), aZ = az. From (2.255) and (2.207) the matrix components of grad(u) in cylindrical polar coordinates are written as

Using the techniques illustrated above the following identities can be established:

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## Advanced Solid Mechanics - Notes, Videos, MCQs & PPTs

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MathSciNet bibliographic data MR2282011 (2007k:53019) 53C10 (53C29) Bryant, Robert L. Some remarks on \$G\sb 2\$$G\sb 2$-structures. Proceedings of Gökova Geometry-Topology Conference 2005, 75–109, Gökova Geometry/Topology Conference (GGT), Gökova, 2006. Links to the journal or article are not yet available

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.

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# <namedCaches> Element (Cache Settings)

Specifies a collection of configuration settings for the named MemoryCache instances. The NamedCaches property references the collection of configuration settings from one or more namedCaches elements of the configuration file.

<configuration>
<system.runtime.caching>
<memoryCache>
<namedCaches>

## Syntax

<namedCaches>
</namedCaches>

## Type

None

## Attributes and Elements

The following sections describe attributes, child elements, and parent elements.

### Attributes

Attribute Description
cacheMemoryLimitMegabytes An integer value that specifies the maximum allowable size, in megabytes, that an instance of a MemoryCache can grow to. The default value is 0, which means that the autosizing heuristics of the MemoryCache class are used by default.
name The name of the cache.
physicalMemoryLimitPercentage An integer value between 0 and 100 that specifies the maximum percentage of physically installed computer memory that can be consumed by the cache. The default value is 0, which means that the autosizing heuristics of the MemoryCache class are used by default.
pollingInterval A value that indicates the time interval after which the cache implementation compares the current memory load against the absolute and percentage-based memory limits that are set for the cache instance. This value is entered in "HH:MM:SS" format.

### Child Elements

Element Description
<add> Adds a named cache to the namedCaches collection for a memory cache.
<clear> Clears the namedCaches collection for a memory cache.
<remove> Removes a named cache entry from the namedCaches collection for a memory cache.

### Parent Elements

Element Description
<configuration> Specifies the root element in every configuration file that is used by the common language runtime and .NET Framework applications.
<memoryCache> Defines an element that is used to configure a cache that is based on the MemoryCache class.
<system.runtime.caching> Contains types that let you implement output caching in applications that are built into the .NET Framework.

## Remarks

The memory cache configuration section of the Web.config file can contain add, remove, and clear attributes for the namedCaches collection. Each namedCaches entry is uniquely identified by the name attribute.

You can retrieve instances of memory cache entries by referencing the information in the application configuration files. By default, only the default cache instance has an entry in the configuration file. The default cache instance is the instance that is returned from the Default property.

If you set the name attribute to "default", the element uses the default memory cache instance.

## Example

The following example shows how to set the name of the cache to the default cache entry name by setting the name attribute to "default".

The cacheMemoryLimitMegabytes attribute and the physicalMemoryPercentage attribute are set to zero. Setting these attributes to zero means that the autosizing heuristics of the MemoryCache class are used. The cache implementation compares the current memory load against the absolute and percentage-based memory limits every two minutes.

<configuration>

<system.runtime.caching>
<memoryCache>
<namedCaches>


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It was mid 2012, first time when I installed TeXstudio version 2.3, on my PC. ‘Peaceful‘ was the word, I used to describe it then after. TeXstudio has now gradually inched to version 2.7 and by time being even more peaceful and wonderful. Wonder why did I call it ‘peaceful’?

Writing LaTeX is a real pain, when you are willing to write a long document or article, which in my case are mathematics blogs and books. It was a great headache to write math commands points by point until I installed TeXstudio.  After I played with TeXstudio for Windows a couple of days, I understood how it functions. and that was the start of great time for me. No more remembering all the LaTeX codes, no more external compilers and no more annoying extra plugins to be installed.  TeXstudio sports such great features which literally reduces your effort to half or even less than that. It offers beautiful designs (yep, a Mac OS feeling on Windows), easy template wizards and quick compilers. Some other noticeable features of this app are auto-fill suggestions, syntax-highlighting, real-time viewing, spell and reference check, bibliography support and profile management. I’ll be discussing more about these features later, but before we follow-up , let’s recall the background of the app.

TeXstudio is an integrated writing environment for creating LaTeX documents. It is an Open Source application, available for major desktop platforms, Windows, Linux, BSD and Mac OS. TeXStudio is a team project of Benito van der Zander, Jan Sundermeyer, Daniel Braun and Tim Hoffmann, forked from the TeXMaker application, a non-open source application, which open-source development stalled in 2009. Actually, TeXstudio was aimed to be a set of extensions for TeXMaker and was then called TeXMakerX.  In 2009,TeXMakerX was renamed as TeXstudio and made completely open-source application. TeXstudio completely differs to TeXMaker in philosophy and slightly in features. TeXMaker still exists, is free of charge and get updates regularly but the development is strictly non-open.

# Features

## Easy Writing and Editing

TeXstudio provides simpler writing experience, with bunch of premade TeX templates and quick wizards. As first time user, you can just open the application, open File –> New from template and you are ready to go. If you want to save the edited .tex file to somewhere else and not in the TeXstudio’s default folder, you may also select your preferred destination folder.

Document Wizard in TeXstudio

Remember that, TeXstudio builds five or more files at once, therefore you should better use a blank folder for a single document. This will place all files to that blank folder and there will not arise any hindrance in compiling the final document.

five files, when you start creating a document from an article template

Experienced users can start with a blank new file from File –> New. The regular Microsoft Office shortcuts also work with TeXstudio.

Some useful features used while writing are:

• auto completion

Auto Completion: As you write any Math/LaTeX command, editor suggests you what should come the next and auto-completes your command.

• Easy content navigation:  TeXstudio supports easy content and Word by Word navigation.
• Syntax Highlighting: In all the screen-shots above, you can see how Texstudio interprets the TeX commands and syntax.
• Spell Check: TeXstudio automatically checks for spelling errors and provides suggestions for them.
• Others: You can place bookmarks within a document at ease, customize the shortcuts, menus and toolbars, drag and drop images to the document directly and can add tables too.

## Instant Previews

Write your document, go to Tools –> Build & View or simply press F1 key and you’ll see instant pdf/ps previews on the right side of your editor.  There is no need of any  external apps/plugins. The preview supports zooming and dragging, and is very convenient to use.

Write a bit

Build it and Preview!

## Compiling

You can compile and publish a document by Tools –> Compile or by F6 key. The compiled document can be found in the destination folder you chosen earlier, while creating the document. After compiling is complete, you folder will have at least seven files instead of five. You may read them using suitable readers.

compiled files

## Profile Management

You may customize and save TeXstudio’s behavior in profiles. Each time you need a different behavior from the app, chose a different profile.

## Bibliography Support

In the reference section of your document, you can easily add foot-notes, credits and other bibliographic items easily using Bibliography menu.

PDF preview of compiled document

## Other features

Some other remarkable features are:

• SVN Support
• Structure view
• Code folding
• Convert to HTML
• Automatic detection of MikTeX, TeX Live, Ghostscript and Standardlatex

##### TeXStudio

TeXStudio is the most complete LaTeX Editor.

Editor's Rating:
4.7
1. Faiz Alam says:

this is lovable post. Gaurav i really like your writing skills…

2. Proserpine says:

Hello,

You write: “TeXMaker application, a non-open source application” and “the development is strictly non-open”. These are two different things. Texmaker is released under the GNU General Public License => http://www.xm1math.net/texmaker/index.html.

“TeXstudio completely differs to TeXMaker in (…) features”: False. I use both. TeXstudio and TeXmaker are very close. Most features are identical. TeXstudio is better in some areas but TeXmaker scores on other. Both are very good LaTeX editors.

1. On the first note, the development… uh.. Open Development of TexMaker is no further possible. I tried to obtain the source codes of both the applications but TexMaker was nowhere available.
On the second, yes you were correct. I have corrected the post with your inputs.
Thank you Proserplne.

1. Proserpine says:

Also, to claim that “the [TeXmaker] development is strictly non-open” seems excessive. On the author’s site, you can read “To propose some code, just make a patch from the latest source tarball and click on the “Contact” link.”

3. shekhar says:

hello, i have problem while compiling latex document on texstudio it is showing file utf8x.def is not found. i used ” \usepackage[utf8x]{inputenc}” this package on windows 7.

1. Hello Shekhar! Thanks for stopping by.
TeXstudio doesn’t support extended encoding for utf8 (utf8x) natively. To enable this, you need to install miktex.

Bye.

1. shekhar says:

thnx got it.

4. Juan says:

Hello.

What is the difference between compiling and building?

1. PSP says:

what is the difference between ‘clicking’ and ‘tapping the mouse button’ ?

5. Daryl says:

1. The first and foremost advantage is the continued development. Others are already mentioned in the post, like:
* An all in one package for all LaTeX editing needs
* Available for all desktop OS
* Inbuilt compiler and viewer
If you are really looking for a $\LaTeX$ based solution – I’ll surely suggest TexStudio.

How to customize wizards? I need to customize the Table wizard

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