iMechanica - linear algebra
https://www.imechanica.org/taxonomy/term/1275
enPhD position in X-Ray Microscopy on Microstructure Evolution in Metals
https://www.imechanica.org/node/25300
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/73">job</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/13195">DTU</a></div><div class="field-item odd"><a href="/taxonomy/term/539">phd</a></div><div class="field-item even"><a href="/taxonomy/term/5318">Denmark</a></div><div class="field-item odd"><a href="/taxonomy/term/6615">diffraction</a></div><div class="field-item even"><a href="/taxonomy/term/6300">Solid State Physics</a></div><div class="field-item odd"><a href="/taxonomy/term/1275">linear algebra</a></div><div class="field-item even"><a href="/taxonomy/term/4201">Python</a></div><div class="field-item odd"><a href="/taxonomy/term/920">physics</a></div><div class="field-item even"><a href="/taxonomy/term/288">materials science</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p> </p>
<p> </p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">In our center: “The physics of metal plasticity” we conduct basic science to reveal the physical mechanisms governing the mechanical properties of metals, such as strength and formability. When metals are deformed plastically, dislocations (defects in the crystal lattice) multiply and self-organize in 3D patterns. We have developed a hard X-ray microscope at the European Synchrotron Radiation Source (ESRF) that for the first time allows visualization of how these patterns form. We are looking for an outstanding and motivated candidate to join our international group, where we combine X-ray physics, simulations of experiments and scientific computing with dislocation dynamics simulations to understand metal plasticity. </span></p>
<p>This work is also part of a new Danish Center-of-Excellence on hard materials in 3D, SOLID.<span> </span>Here you will be interacting with 15 other PhDs, all exploiting the latest 3D methods based on large scale x-ray and neutron facilities, within in a broad range of fields spanning ferroelectrics, martensitic transformation, biomineralization and archeology.</p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">Responsibilities and qualifications</span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Your research will be based on studies at the hard X-ray microscope at the European Synchrotron Radiation Source, ESRF, in France. </span></p>
<p><span lang="EN-GB" xml:lang="EN-GB">Your responsibility will be planning and operation of experiments at ESRF, data analysis and interpretation of results.</span></p>
<p>You are excited about fundamental science, and we expect that you enjoy being part of a team, that you have sense of humor, is a good problem solver and that you can work efficiently and independently.</p>
<p>For this position we favor candidates with a degree in physics or materials science. You should</p>
<ul type="disc"><li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Have a solid grasp of diffraction, solid state physics and linear algebra</span></li>
<li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Be skilled in data analysis with proven experience with programming, e.g. Python</span></li>
<li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Be skilled in experimental work</span></li>
</ul><p> </p>
<p class="normal1"><span lang="EN-GB" xml:lang="EN-GB">You must have a two-year master's degree (120 ECTS points) or a similar degree with an academic level equivalent to a two-year master's degree.<br /></span><span lang="EN-GB" xml:lang="EN-GB"><br /></span><strong><span lang="EN-GB" xml:lang="EN-GB">Approval and Enrolment</span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">The scholarship for the PhD degree is subject to academic approval, and the candidate will be enrolled in one of the general degree programmes at DTU. For information about our enrolment requirements and the general planning of the PhD study programme, please see the </span><span lang="DA" xml:lang="DA"><a href="http://www.dtu.dk/english/Education/PhD/Rules/PhDguide"><span lang="EN-GB" xml:lang="EN-GB">DTU PhD Guide</span></a></span><span lang="EN-GB" xml:lang="EN-GB">.</span></p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">Assessment</span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">The assessment of the applications will be made by Professor Henning Friis Poulsen, DTU Physics and Professor Grethe Winther, DTU Mechanical Engineering.</span></p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">We offer<br /></span></strong><span lang="EN-GB" xml:lang="EN-GB">DTU is a leading technical university globally recognized for the excellence of its research, education, innovation and scientific advice. We offer a rewarding and challenging job in an international environment. We strive for academic excellence in an environment characterized by collegial respect and academic freedom tempered by responsibility.</span><span lang="EN-GB" xml:lang="EN-GB"></span></p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">Salary and appointment terms </span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">The appointment will be based on the collective agreement with the Danish Confederation of Professional Associations. The allowance will be agreed upon with the relevant union. The period of employment is 3 years.</span></p>
<p>You can read more about <span lang="DA" xml:lang="DA"><a href="http://www.dtu.dk/english/about/job-and-career/working-at-dtu/career-paths"><span lang="EN-GB" xml:lang="EN-GB">career paths at DTU here</span></a></span><strong><span lang="EN-GB" xml:lang="EN-GB">.</span></strong><span lang="EN-GB" xml:lang="EN-GB"></span></p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">Further information </span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Should you have any queries regarding the positions, please contact Henning Friis Poulsen (</span><span lang="DA" xml:lang="DA"><a href="mailto:hfpo@fysik.dtu.dk"><span lang="EN-GB" xml:lang="EN-GB">hfpo@fysik.dtu.dk</span></a></span><span lang="EN-GB" xml:lang="EN-GB">) or Grethe Winther (</span><span lang="DA" xml:lang="DA"><a href="mailto:grwi@mek.dtu.dk"><span lang="EN-GB" xml:lang="EN-GB">grwi@mek.dtu.dk</span></a></span><span lang="EN-GB" xml:lang="EN-GB">).</span></p>
<p>You can read more about the SOLID center on <span lang="DA" xml:lang="DA"><a href="http://www.solid.dtu.dk"><span lang="EN-GB" xml:lang="EN-GB">www.solid.dtu.dk</span></a></span><span lang="EN-GB" xml:lang="EN-GB">.</span></p>
<p>If you are applying from abroad, you may find useful information on working in Denmark and at DTU at <span lang="DA" xml:lang="DA"><a href="https://www.dtu.dk/english/about/job-and-career/moving-to-denmark"><span lang="EN-GB" xml:lang="EN-GB">DTU – Moving to Denmark</span></a></span><span lang="EN-GB" xml:lang="EN-GB">.</span><span lang="EN-GB" xml:lang="EN-GB"></span></p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">Application procedure </span></strong></p>
<p><span lang="EN-GB" xml:lang="EN-GB">Your complete online application must be submitted no later than<strong> 15 <span>August 2021 (Danish time)</span></strong><span>.</span> </span></p>
<p>Apply online at <span lang="DA" xml:lang="DA"><a href="http://www.career.dtu.dk"><span lang="EN-GB" xml:lang="EN-GB">www.career.dtu.dk</span></a></span><span lang="EN-GB" xml:lang="EN-GB">. </span></p>
<p><span lang="EN-GB" xml:lang="EN-GB">Applications must be submitted as <strong>one PDF file</strong> containing all materials to be given consideration. To apply, please open the link "Apply online", fill out the online application form, and attach <strong>all your materials in English in one PDF file</strong>. The file must include:</span></p>
<ul type="disc"><li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">A letter motivating the application (cover letter)</span></li>
<li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Curriculum vitae </span></li>
<li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Grade transcripts and BSc/MSc diploma</span></li>
<li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Excel sheet with translation of grades to the Danish grading system </span><span lang="EN-GB" xml:lang="EN-GB">(see guidelines and </span><span lang="DA" xml:lang="DA"><a href="http://www.dtu.dk/english/Education/phd/Applicant/Pre_acceptance-1-"><span lang="EN-GB" xml:lang="EN-GB">Excel spreadsheet here</span></a></span><span lang="EN-GB" xml:lang="EN-GB">) </span></li>
</ul><p> </p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">You may apply prior to obtaining your master's degree but cannot begin before having received it.</span> </p>
<p><span lang="EN-GB" xml:lang="EN-GB">A<span>ll interested candidates irrespective of age, gender, race, disability, religion or ethnic background are encouraged to apply.</span></span></p>
<p> </p>
<p class="MsoNormal"><strong><em><span lang="EN-GB" xml:lang="EN-GB"></span></em></strong></p>
<p>Technology for people<br /><em><span lang="EN-GB" xml:lang="EN-GB">DTU develops technology for people. With our international elite research and study programmes, we are helping to create a better world and to solve the global challenges formulated in the UN’s 17 Sustainable Development Goals. Hans Christian Ørsted founded DTU in 1829 with a clear vision to develop and create value using science and engineering to benefit society. That vision lives on today. DTU has 12,900 students and 6,000 employees. We work in an international atmosphere and have an inclusive, evolving, and informal working environment. DTU has campuses in all parts of Denmark and in Greenland, and we collaborate with the best universities around the world.</span></em></p>
<p> </p>
<p> </p>
</div></div></div>Thu, 01 Jul 2021 12:27:57 +0000info@signatur.dk25300 at https://www.imechanica.orghttps://www.imechanica.org/node/25300#commentshttps://www.imechanica.org/crss/node/25300PhD position in Simulations of Dark field X-Ray Microscopy
https://www.imechanica.org/node/25299
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/73">job</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/539">phd</a></div><div class="field-item odd"><a href="/taxonomy/term/5318">Denmark</a></div><div class="field-item even"><a href="/taxonomy/term/13195">DTU</a></div><div class="field-item odd"><a href="/taxonomy/term/13196">X-Ray Microscopy</a></div><div class="field-item even"><a href="/taxonomy/term/13197">dark field</a></div><div class="field-item odd"><a href="/taxonomy/term/6615">diffraction</a></div><div class="field-item even"><a href="/taxonomy/term/6300">Solid State Physics</a></div><div class="field-item odd"><a href="/taxonomy/term/4201">Python</a></div><div class="field-item even"><a href="/taxonomy/term/1275">linear algebra</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p> </p>
<p> </p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">In our center we have built a dark field x-ray microscope, installed at the European Synchrotron Radiation Facility, ESRF.<span> </span>Similar to transmission electron microscopy, this allows visualization of the microstructure (grains, domains, dislocations), but now in 3D and nondestructively. At the same time, we can map the local stress. With this microscope we can for the first time directly see how the microstructure evolves inside real life samples. We are looking for an outstanding and motivated candidate to push the boundaries of the new technique by simulations and experiments.</span></p>
<p>In the center: “The Physics of Metal Plasticity” we combine X-ray simulations and experiments with <span> </span>dislocation dynamics simulations to see how dislocations (defects) self-organize during deformation of metals. The aim is to provide a first principles description of mechanical properties such as strength.</p>
<p>This work is also part of a new Danish Center-of-Excellence on hard materials in 3D, SOLID.<span> </span>Here you will be interacting with 15 other PhDs, all exploiting the latest 3D methods based on large scale x-ray and neutron facilities, within in a broad range of fields spanning ferroelectrics, martensitic transformation, biomineralization and archeology.</p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">Responsibilities and qualifications<span> </span></span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Your research will be key for advancing the instrument at ESRF into a real microscope, with modalities similar to a TEM.</span></p>
<p>Your responsibility will be full scale optical simulation of the microscope coupled with interfacing to dislocation dynamics simulations, design and optimisation of modalities for visualization of the dislocations, and demonstration experiments and associated data analysis.</p>
<p>You are excited about fundamental science, and we expect that you enjoy being part of a team, that you have sense of humour, is a good problem solver and that you can work efficiently and independently.</p>
<p>For this position we favour candidates with a degree in physics or materials science. You should</p>
<ul type="disc"><li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Have a solid grasp of diffraction, solid state physics and linear algebra</span></li>
<li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Understand Fourier optics</span></li>
<li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Have hands on experience with simulations with proven experience with programming, e.g. Python</span></li>
</ul><p> </p>
<p class="normal1"><span lang="EN-GB" xml:lang="EN-GB">You must have a two-year master's degree (120 ECTS points) or a similar degree with an academic level equivalent to a two-year master's degree.</span></p>
<p> <strong><span lang="EN-GB" xml:lang="EN-GB">Approval and Enrolment</span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">The scholarship for the PhD degree is subject to academic approval, and the candidate will be enrolled in one of the general degree programmes at DTU. For information about our enrolment requirements and the general planning of the PhD study programme, please see the </span><span lang="DA" xml:lang="DA"><a href="http://www.dtu.dk/english/Education/PhD/Rules/PhDguide"><span lang="EN-GB" xml:lang="EN-GB">DTU PhD Guide</span></a></span><span lang="EN-GB" xml:lang="EN-GB">.</span></p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">Assessment</span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">The assessment of the applications will be made by Professor Henning Friis Poulsen, DTU Physics and Profesor Grethe Winther, DTU Mechanical Engineering.</span></p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">We offer<br /></span></strong><span lang="EN-GB" xml:lang="EN-GB">DTU is a leading technical university globally recognized for the excellence of its research, education, innovation and scientific advice. We offer a rewarding and challenging job in an international environment. We strive for academic excellence in an environment characterized by collegial respect and academic freedom tempered by responsibility.</span><span lang="EN-GB" xml:lang="EN-GB"><br /></span><strong><span lang="EN-GB" xml:lang="EN-GB"><br />Salary and appointment terms </span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">The appointment will be based on the collective agreement with the Danish Confederation of Professional Associations. The allowance will be agreed upon with the relevant union. The period of employment is 3 years.</span></p>
<p>You can read more about <span lang="DA" xml:lang="DA"><a href="http://www.dtu.dk/english/about/job-and-career/working-at-dtu/career-paths"><span lang="EN-GB" xml:lang="EN-GB">career paths at DTU here</span></a></span><strong><span lang="EN-GB" xml:lang="EN-GB">.</span></strong><span lang="EN-GB" xml:lang="EN-GB"></span></p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">Further information </span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Should you have any queries regarding the positions, please contact Henning Friis Poulsen (</span><span lang="DA" xml:lang="DA"><a href="mailto:hfpo@fysik.dtu.dk"><span lang="EN-GB" xml:lang="EN-GB">hfpo@fysik.dtu.dk</span></a></span><span lang="EN-GB" xml:lang="EN-GB">) or Grethe Winther (</span><span lang="DA" xml:lang="DA"><a href="mailto:grwi@mek.dtu.dk"><span lang="EN-GB" xml:lang="EN-GB">grwi@mek.dtu.dk</span></a></span><span lang="EN-GB" xml:lang="EN-GB">).</span></p>
<p>You can read more about the SOLID center on <span lang="DA" xml:lang="DA"><a href="http://www.solid.dtu.dk"><span lang="EN-GB" xml:lang="EN-GB">www.solid.dtu.dk</span></a></span><span lang="EN-GB" xml:lang="EN-GB">.</span></p>
<p>If you are applying from abroad, you may find useful information on working in Denmark and at DTU at <span lang="DA" xml:lang="DA"><a href="https://www.dtu.dk/english/about/job-and-career/moving-to-denmark"><span lang="EN-GB" xml:lang="EN-GB">DTU – Moving to Denmark</span></a></span><span lang="EN-GB" xml:lang="EN-GB">.</span></p>
<p><strong><span lang="EN-GB" xml:lang="EN-GB">Application procedure </span></strong></p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Your complete online application must be submitted no later than<strong> 15 <span>August 2021 (Danish time)</span></strong><span>. </span></span></p>
<p>Apply online at <span lang="DA" xml:lang="DA"><a href="http://www.career.dtu.dk"><span lang="EN-GB" xml:lang="EN-GB">www.career.dtu.dk</span></a></span><span lang="EN-GB" xml:lang="EN-GB">.</span><strong><span lang="EN-GB" xml:lang="EN-GB"></span></strong></p>
<p><span lang="EN-GB" xml:lang="EN-GB">Applications must be submitted as <strong>one PDF file</strong> containing all materials to be given consideration. To apply, please open the link "Apply online", fill out the online application form, and attach <strong>all your materials in English in one PDF file</strong>. The file must include:</span></p>
<ul type="disc"><li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">A letter motivating the application (cover letter)</span></li>
<li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Curriculum vitae </span></li>
<li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Grade transcripts and BSc/MSc diploma</span></li>
<li class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB">Excel sheet with translation of grades to the Danish grading system </span><span lang="EN-GB" xml:lang="EN-GB">(see guidelines and </span><span lang="DA" xml:lang="DA"><a href="http://www.dtu.dk/english/Education/phd/Applicant/Pre_acceptance-1-"><span lang="EN-GB" xml:lang="EN-GB">Excel spreadsheet here</span></a></span><span lang="EN-GB" xml:lang="EN-GB">) </span></li>
</ul><p class="MsoNormal"> <span lang="EN-GB" xml:lang="EN-GB"> <br />You may apply prior to obtaining your master's degree but cannot begin before having received it.</span> </p>
<p><span lang="EN-GB" xml:lang="EN-GB">A<span>ll interested candidates irrespective of age, gender, race, disability, religion or ethnic background are encouraged to apply.</span></span></p>
<p> </p>
<p class="MsoNormal"><span lang="EN-GB" xml:lang="EN-GB"> </span></p>
<p> </p>
<p class="MsoNormal"><strong><em><span lang="EN-GB" xml:lang="EN-GB">Technology for people<br /></span></em></strong><em><span lang="EN-GB" xml:lang="EN-GB">DTU develops technology for people. With our international elite research and study programmes, we are helping to create a better world and to solve the global challenges formulated in the UN’s 17 Sustainable Development Goals. Hans Christian Ørsted founded DTU in 1829 with a clear vision to develop and create value using science and engineering to benefit society. That vision lives on today. DTU has 12,900 students and 6,000 employees. We work in an international atmosphere and have an inclusive, evolving, and informal working environment. DTU has campuses in all parts of Denmark and in Greenland, and we collaborate with the best universities around the world.</span></em></p>
<p> </p>
<p> </p>
</div></div></div>Thu, 01 Jul 2021 12:22:18 +0000info@signatur.dk25299 at https://www.imechanica.orghttps://www.imechanica.org/node/25299#commentshttps://www.imechanica.org/crss/node/25299Massvolume vs. Spacetime
https://www.imechanica.org/node/15955
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1275">linear algebra</a></div><div class="field-item odd"><a href="/taxonomy/term/9477">spacetime</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p class="MsoNormal">
<span><strong>Apples and oranges</strong>.</span> <span>Each element in a set is a pile containing some number of apples and some number of oranges.<span> </span> Adding two piles means putting them together, resulting in a pile in the set. Multiplying a pile and a real number <em>r</em> means finding in the set a pile <em>r</em> times the amount.<span> </span> We model each pile as a vector, and the set as a two-dimensional vector space over the field of real numbers.</span>
</p>
<p class="MsoNormal">
<span>A vector represents different objects as a single entity. A pile containing some number of apples and some number of oranges is a vector. The addition of two vectors does not require us to add apples and oranges.<span> </span> Rather, in adding two piles, we add apples to apples, and oranges to oranges.<span> </span> The addition of vectors generalizes the addition of numbers:<span> </span> adding two vectors corresponds to adding two lists of numbers in parallel.</span>
</p>
<p class="MsoNormal">
<span><strong>Mass and</strong> <strong>volume</strong>.<span> </span> We can also list different physical quantities together as a single object.<span> </span> Consider a set, each element of which is a piece of some mass and some volume.<span> </span> Adding two pieces means putting them together, resulting in a piece in the set. Multiplying a piece and a real number <em>r</em> means finding in the set a piece <em>r</em> times the amount.<span> </span> This set is a two-dimensional vector space over the field of real numbers. </span>We do not have any familiar name for this vector space, and will call it massvolume.<span> </span>
</p>
<p class="MsoNormal">
<span><strong>Spacetime</strong>.<span> </span> When we list apples and oranges together, or volume and mass together, the results do not surprise us.<span> </span> But when Einstein and Minkowski listed directed segments in space and directed intervals of time together, the result was shocking.<span> </span> Minkowski said, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”<span> </span> What makes spacetime, but not massvolume, so shocking and so enduring?</span>
</p>
<p>
I'm curious how you think about it.
</p>
</div></div></div>Wed, 22 Jan 2014 19:48:54 +0000Zhigang Suo15955 at https://www.imechanica.orghttps://www.imechanica.org/node/15955#commentshttps://www.imechanica.org/crss/node/15955Scalar done wrong
https://www.imechanica.org/node/15857
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1097">tensor</a></div><div class="field-item odd"><a href="/taxonomy/term/1234">scalar</a></div><div class="field-item even"><a href="/taxonomy/term/1235">vector</a></div><div class="field-item odd"><a href="/taxonomy/term/1275">linear algebra</a></div><div class="field-item even"><a href="/taxonomy/term/9412">linear map</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong>Update on 9 April 2016</strong>. At the bottom of this post, I attach a pdf file of my notes on scalar.</p>
<p>Notes on scalars now forms part of <a href="http://imechanica.org/node/19709">my notes on linear algebra</a>.</p>
<p>When I was updating my very brief <a href="14164">notes on tensors</a>, it occurred to me to post on iMechanica a request for recommendation of <a href="15843">textbooks on linear algebra</a>. I was delighted to see Arash respond. I then asked for his opinion about the definition of tensor. He responded again, and we seemed to agree. Then Amit joined the discussion, and then others. That thread has become very interesting and very long.</p>
<p>But I have another issue with the way we use linear algebra. I wish to get your opinion. The issue is about scalars.</p>
<p class="MsoNormal"><span><strong>Vector space</strong>. First some background. A vector space involves two sets and four operations. One set is the number field F, along with two operations: addition of two elements in F gives another element in F, and multiplication of two elements in F gives another element in F. The two operations follow the usual arithmetic rules. See a formal definition of <a href="http://en.wikipedia.org/wiki/Field_(mathematics)">number field</a>. For our purpose, F can be the field of real numbers. There is nothing more to it. The other set V is a set of vectors, along with two more operations: addition of two elements in V gives another element in V, and multiplication of an element in F and an element in V gives an element in V. The two operations follow the usual rules for vectors. See a formal definition of <a href="http://en.wikipedia.org/wiki/Vector_space">vector space</a>.</span></p>
<p class="MsoNormal"><span>A space V is said to be n-dimensional if there exist n linearly independent elements in V, but every n + 1 elements of the space are linearly dependent.</span></p>
<p class="MsoNormal"><span>In particular, for a one-dimensional vector space S on a number field F, any two elements in S are linearly dependent.<span> </span> Let u be an arbitrary, but none-zero, element in S.<span> </span> All other elements in S takes the form au where a<span> </span>is an element in F.</span></p>
<p class="MsoNormal"><span><strong>Is scalar a synonym of number?</strong> In many textbooks on linear algebra, the word <em>scalar</em> is a synonym to the word <em>number</em>, an element in the field F.<span> </span> The word scalar is also commonly used in physics to indicate quantities like mass, energy and entropy.<span> </span> The two usages of the word scalar are incompatible in several ways.<span> </span> First, a physical property like mass is more than just a number; it has a unit.<span> </span> Second, the multiplication defined on a field makes no sense to mass:<span> </span> the multiplication of two elements in F gives yet another element in F, but the multiplication of two masses does not give another mass.<span> </span> Third, if we regard both mass and entropy as elements in the field F, then we need to assign a meaning to the addition of mass and entropy.<span> </span> What does that even mean?</span></p>
<p class="MsoNormal"><span>Thus, we will call an element in the field F a number, and will reserve the word scalar for physical quantities.</span></p>
<p><strong><span>The set of pieces of a substance of all sizes.</span></strong><span><span> </span> Given a substance (e.g., gold), pieces of all different amounts of the substance form a set.<span> </span> We can define the addition of the pieces, but we do not have a sensible definition for the multiplication of the pieces.<span> </span> Thus, this set is not a number field. This set, however, is a one-dimensional vector space. We stipulate the two operations in a natural way. The addition of two pieces of the substance is another piece of the substance, and multiplication of a real number and a piece of the substance is another piece of the substance.</span></p>
<p class="MsoNormal"><strong><span>Extensive property of a substance.</span></strong><span><span> </span> A piece of a substance has many physical properties, such as volume, shape, color, temperature, mass, energy, entropy.<span> </span> A physical property is extensive if it is proportional to the amount of the substance.<span> </span> Volume, mass, energy, and entropy are extensive properties.<span> </span> Shape, color, temperature are not extensive properties.</span></p>
<p class="MsoNormal"><span><strong>Extensive scalar</strong>. We can use a one-dimensional vector space S to model a physical property such as mass.<span> </span> In this model, F is the field of real numbers.<span> </span> We call this one-dimensional vector space a scalar set. We use a particular element in S as the unit for this quantity.<span> </span> For example, for the scalar set S of all masses, the unit mass, kg, is </span><a href="http://en.wikipedia.org/wiki/Kilogram">a block of metal located in Sevres, France</a><span>.<span> </span> All other masses equal this unit times a real number.</span></p>
<p><span>By contrast, temperature cannot be represented as a one-dimensional vector space.<span> </span> The addition of two temperatures does not give another temperature.</span></p>
<p><span><span><strong>Linear form</strong>. In textbooks of linear algebra, a linear form is commonly defined as follows. Let V be a vector space on a number field F. A linear form is a linear map that maps an element in V to an element in F.</span></span></p>
<p><span><span>I believe that this definition is inconsistent with how mechanicians use linear form. </span></span></p>
<p><span><span>Here is my modified definition. Let V be a vector space and S be a scalar set, both on the same number field F. A linear form is a linear map that maps an element in V to an element in S.</span></span></p>
<p><strong>Dual space</strong>. The set of all linear forms from V to S is also a vector space. We denote this space by V', and call it the dual space of V with respect to the scalar set S. The vector space V and its dual space V' have the same dimensions. </p>
<p><span><span><strong>Work, displacement, and force. </strong> Here is an example how linear form arises in mechanics. Work cannot form a number field: the multiplication of two amounts of work does not give another work. Work, however, is a scalar. If the displacement is small, we know the work is linear in the displacement. That is, there exists a linear map that maps the displacement to work. The displacement is an element of a three-dimensional vector space U, and the work is an element of a scalar set. The linear map fits the definition of the linear form, and we call the linear map the force. The force is an element of the dual space U' with respect to the scalar set of work.</span></span></p>
<p><span><span><strong>Tensor</strong>. We have had a <a href="15843">long thread on the definition of tensor</a>. Here is a definition that I like. Let V1, V2..., Vp and V be vector spaces on the same field F. These vector spaces may represent objects of different kinds, and may even have different dimensions. A tensor is a <a href="http://en.wikipedia.org/wiki/Multilinear_map">multilinear map</a> that maps an element in V1, an element in V2,..., and an element in Vp to an element in V. </span></span></p>
<p><span><span><strong>Remark 1</strong>. Tensor is a generalization of linear form. </span></span></p>
<p><span><span><strong>Remark 2</strong>. The set all multilinear maps from V1, V2..., Vp to V is a vector space. This vector space generalizes dual space.</span></span></p>
<p><span><span><strong>Remark 3</strong>. We can use this new vector space to generate other tensors. To be consistent, a vector is a special case of tensor, so is a scalar. They live in different vector spaces.</span></span></p>
<p><span><span>This is the main theme of linear algebra: vector spaces, and linear maps between them. Linear maps themseves form new vector spaces. We map the maps. The never-ending strory of linear algebra.</span></span> </p>
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<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://www.imechanica.org/files/scalar%202017%2002%2002.pdf" type="application/pdf; length=2220534">scalar 2017 02 02.pdf</a></span></td><td>2.12 MB</td> </tr>
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</div></div></div>Tue, 31 Dec 2013 07:17:51 +0000Zhigang Suo15857 at https://www.imechanica.orghttps://www.imechanica.org/node/15857#commentshttps://www.imechanica.org/crss/node/15857Textbook on linear algebra
https://www.imechanica.org/node/15843
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/549">continuum mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/1097">tensor</a></div><div class="field-item even"><a href="/taxonomy/term/1275">linear algebra</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Linear algebra is significant to many aspects of mechanics. For some years I have been using the <a href="http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X/ref=sr_1_1?s=books&ie=UTF8&qid=1388169733&sr=1-1&keywords=shilov+linear+algebra">book by Shilov</a>. But this book may or may not be a good one to recommend to a student, depending on his or her prior experience. On StackExchange Mathematics, there are several excellent threads discussing textbooks of linear algebra. A particular recommendation was made for <a href="http://math.stackexchange.com/questions/433858/high-level-linear-algebra-book">textbooks of linear algebra at three levels</a>. Do you have any recommendations?</p>
</div></div></div>Fri, 27 Dec 2013 23:39:31 +0000Zhigang Suo15843 at https://www.imechanica.orghttps://www.imechanica.org/node/15843#commentshttps://www.imechanica.org/crss/node/15843How do find Inverse to Rectangular Matrix?
https://www.imechanica.org/node/7078
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1275">linear algebra</a></div><div class="field-item odd"><a href="/taxonomy/term/4534">matrix inverse</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear All,
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How do find the inverse of the rectangular matrix.
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</div></div></div>Sat, 14 Nov 2009 05:50:38 +0000Muthukumar M7078 at https://www.imechanica.orghttps://www.imechanica.org/node/7078#commentshttps://www.imechanica.org/crss/node/7078