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25. ### Double Convolution

Stephen, you are right in pointing out that such notation is confusing. Let me then rephrase my question clearly: Let us denote a convolution $$\int_0^{t} A(t-\tau) x(\tau) \mathrm{d}\tau$$ With the notation $$A \star x$$ I would like to write down the expression for the double convolution...
26. ### Double Convolution

Well, there is no mention in Wikipedia of Stjeltes convolutions (although they are very strongly related to the measure theory approach described in the Section "Measure"), would it help if I edited my question using Riemann integrals notation?
27. ### Expanding delta in Field Theory Derivation of Euler-Lagrange Equations

Let me preface I know very little about Field Theory, so please take my reply with caution. On the other hand, the step in question seems to me stemming from basic calculus of variations (but care needs to be exterted as there are different definitions of variations around). If you search...
28. ### Double Convolution

Stephen, many thanks for your input. I am not using an engineering notation, whatever yo mean by that, I am simply writing the convolution down as a Stijeltes convolution (using Stjeltes instead of Riemann integration), as the notation is more compact. Under certain technical conditions...

31. ### Can not get Integral Right

Thank you ever so much. That is massive progress. The third term, $$-2ln(a)$$ is easily explained, all there is to do now is to understand the origin of the fourth and fifth terms... They should originate from the integral $$\int_{-a}^{a} ln (\vert r - \zeta \vert) \mathrm{d}r$$ which I...
32. ### Can not get Integral Right

Hi All, I met the following function to evaluate, $$v(x)=\int_{-\infty}^{-a} G(t) ln(\vert t - x \vert) \mathrm{d}t + \int_{-a}^{a} -N ln(\vert t - x \vert) \mathrm{d}t + \int_{a}^{\infty} G(t) ln(\vert t - x \vert) \mathrm{d}t$$, where G is an unknown even function, N is a constant. After...
33. ### Lagrangian Mechanics - Non Commutativity rule

Haeel, your comment is interesting. It would be certianly the most welcome shouldyou expnad upon it: for example, how would a better notation shed light on the topic of non-commutativity? thanks a lot
34. ### Lagrangian Mechanics - Non Commutativity rule

I am reading Cornelius Lanczos' "The Variational principles of Mechanics", and Vujanovic, Atackanovic "Introduction to Modern Variational tecniques in Mechanics and Engineering". I understand that in some derivations q and its time derivative are treated as independent, that they are to be...
35. ### Lagrangian Mechanics - Non Commutativity rule

Hi there, I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are the same is presented in the book I am reading as a rule, commutativity, and...
36. ### Cannot work out change of variables in Integral

Hi there, in a paper the author obtains the integral $$\int_{a}^{\infty} \frac {g(\lambda(r))}{r}\mathrm{d}r$$ which is claimed to be equivalent to $$\int_{a/A}^{1} \frac {g(\lambda(r))}{\lambda (\lambda^3-1)}\mathrm{d}\lambda$$ making use of the relationship (previously physically...
37. ### Integral Computation

Many thanks to all of you for help, very useful and appreciated. Simon Bridge, yes my solution for the question posed by Arildno was wrong, not a typo but considering g as the constant, again my error. Thanks again
38. ### Integral Computation

Arildno, thanks for your hint. I suggest functions $$f = \sqrt{log x}$$ and $$g = \sqrt{log (x/e^{a^2})}$$. Simon, My best attempt so far: $$\int \frac{1}{(x^2 + (y-s)^2)^2} \mathrm{d}s$$, change variable y-s = z, $$\int \frac{-1}{(x^2 + z^2)*(x^2+z^2)} \mathrm{d}s =$$ and now I...
39. ### Deformation gradient f(3,3) vs Coordinates

I am more and more confused... So you know Young's modulus! (actually you know the constant of the Neo-Hookean model, is not a Young's modulus strictly speaking, also the formula you reported, Sigma=Youngsmodulus * Epsilon, is not valid at all), you know the stress (you are applying it and it...
40. ### Deformation gradient f(3,3) vs Coordinates

No worries for the quick replies, boring day in the office... Ok maybe I get it know, i was confused by the fact you claimed in the first post wanted to read the value of the Elasticity Modulus in the interphase. I interpreted this as, I do not know such value in advance. Well then, how can...
41. ### Deformation gradient f(3,3) vs Coordinates

Both F and strain tensor are a function only of gradients of displacement, as is clear from their definition. Physically this is an essential requirement, the principle of objectivity: moving a body by a rigid motion can not alter its energetic content (which is supposed to be a function of its...
42. ### Deformation gradient f(3,3) vs Coordinates

AccessTUD, finding F is the problem of elasticity itself! Analytically, you have to solve the Navier or Cauchy PDE, not always easy. Numerically, it is the deliverable of a FE computation, for example. Now back to your point: your example confuses me. In the beam under tension, the strain is...
43. ### Deformation gradient f(3,3) vs Coordinates

AccessTUD, I am unsure I understand the problem. So you know already the deformation gradient (as a tensor, not only the F(3,3) component I assume) on your domain: your elastic problem is solved already! You wonder then about how to compute the stress tensor: I do not understand why would...
44. ### Integral Computation

Hi All, I am having some troble with the following integral $$\int_{-c}^{c} \frac{x^3}{(x^2+(y-s)^2)^2}\mathrm{d}s$$ Many thanks as usual
45. ### Variational calculus Euler lagrange Equation

Hi All, I found it easier to master the functional derivative concept by performing the calculations out, leaving aside for a moment the symbol $$\delta$$ to denote variations, a useful notation that might though hide the mechanics of what is going on. Then your example becomes transparent...
46. ### Variational Principle and Vectorial Identities

Let me rephrase the question, to make it clearer. How to compute the Euler Lagrange equation of the functional $$\int_{\Omega} \nabla^{(s)} u D (\nabla^{(s)}u)$$ where u is a vectorial function, $$\nabla^{(s)}u = \frac{1}{2} (u_{i,j}+u_{j,i})$$ and D is a (symmetric) constant tensor $$D_{ijkl}$$?
47. ### Variational Principle and Vectorial Identities

Hello there, I am struggling in proving the following. The principle of Minimum energy for an elastic body (no body forces, no applied tractions) says that the equilibrium state minimizes $$\int_{\Omega} \nabla^{(s)} u D (\nabla^{(s)}u)$$ among all vectorial functions u satisfying the...
48. ### Double Integrals: Computing P with Constant Limits

Hi all, I have a question regarding certain double integrals. Assume the function $$l(t)$$ is given as well as the function $$K(t)$$, defined only for positive argument. Also the definition $$n(t) = \int_{-\infty}^{t} K(t-\tau) l(\tau) \mathrm{d}\tau$$ is given. if I wish to compute P =...
49. ### Dirac Delta substitution

PS: I think I might be closer now...I just realized what was in front of me clearly, that the fraction is the Green's function for the operator nabla^2, am I right? Thanks a lot to all
50. ### Dirac Delta substitution

Office_shredder, it looks like I have understood less than what I thought... Your explanation made perfect sense until I tried a direct computation. For simplicity in 1D, I fixed y = 1. Then, if I understood, the indicated function should behave as a delta-distributuion under the integral...