http://w.atwiki.jp/tklab/ tklab ja 2008-10-29T14:50:31+09:00 1225259431 https://w.atwiki.jp/tklab/pages/36.html Principle of Virtual Work <math> \sum \mathbf{F}_i = 0 \Longleftrightarrow \delta W = 0 </math> <math>(\Longrightarrow)</math> If <math>\sum \mathbf{F}_i = 0</math>, then it must be true that for any virtual displacement <math>\delta\mathbf{r}</math>, <math>\delta W = \sum \mathbf{F}_i \cdot \delta\mathbf{r} = 0</math>. <math>(\Longleftarrow)</math> If <math>\delta W = 0</math> holds for any virtual displacement <math>\delta\mathbf{r}</math>, then it must be true that <math> \mathbf{F}_i = 0 </math>. Thus, <math> \sum \mathbf{F}_i = 0 </math>. (Otherwise, <math>\delta W = \sum \mathbf{F}_i \cdot \delta\mathbf{r} = 0</math> wouldn't hold for *any* <math>\delta\mathbf{r}</math>.) 2008-10-29T14:50:31+09:00 1225259431 https://w.atwiki.jp/tklab/pages/35.html Properties of Eigenvalues and Eigenvectors http://www.miislita.com/information-retrieval-tutorial/matrix-tutorial-3-eigenvalues-eigenvectors.html - The absolute value of a determinant <math>|\det \mathrm{A} |</math> is the product of the absolute values of the eigenvalues of matrix <math>\mathrm{A}</math> - <math>c=0</math> is an eigenvalue of <math>\mathrm{A}</math> if <math>\mathrm{A}</math> is a singular matrix - If <math>\mathrm{A}</math> is an <math>n \times n</math> triangular matrix or diagonal matrix, the eigenvalues of <math>\mathrm{A}</math> are the diagonal entries of <math>\mathrm{A}</math> - <math>\mathrm{A}</math> and its transpose matrix have the same eigenvalues. 2008-09-27T04:50:10+09:00 1222458610 https://w.atwiki.jp/tklab/pages/34.html Wulff Net (Stereographic Projection[http://en.wikipedia.org/wiki/Stereographic_projection]) [http://www26.atwiki.jp/tklab/pub/wulff.png] 2008-09-13T02:29:16+09:00 1221240556 https://w.atwiki.jp/tklab/pages/33.html Stereographic Projection [http://en.wikipedia.org/wiki/Stereographic_projection] Hyperbolic Transformation [http://www26.atwiki.jp/tklab/pub/hyperbolic.png] 2008-09-13T03:13:52+09:00 1221243232 https://w.atwiki.jp/tklab/pages/32.html =Tensor Notation= ==Gradient== <math> \Phi_{,i} = \frac{\partial \Phi}{\partial x^i} </math> <math> g^{im}\Phi_{,m} </math> ==Divergence== <math> \mathrm{div} A^r = A^r_{,r} </math> <math> A^r_{,k} = \frac{\partial A^r}{\partial x^k} + \left\{ \begin{array}{c} r \\ m \, k \end{array} \right\} A^m </math> ==Curl== <math> C^i = \epsilon^{ijk} A_{k,j} </math> 2008-09-08T13:42:24+09:00 1220848944 https://w.atwiki.jp/tklab/pages/31.html <math> i \hbar \frac{\partial \Psi(\mathbf{r},t)}{\partial t}=\hat{H} \Psi(\mathbf{r},t) </math> 2008-09-05T12:58:06+09:00 1220587086 https://w.atwiki.jp/tklab/pages/30.html <math> w = f(z) = \frac{az + b}{cz + d} </math> 2008-09-05T12:36:22+09:00 1220585782 https://w.atwiki.jp/tklab/pages/29.html Maxwell's Equations <math> \nabla \cdot \mathbf{D} = \rho </math> <math> \displaystyle \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} </math> <math> \nabla \cdot \mathbf{B} = 0 </math> <math> \displaystyle \nabla \times \mathbf{H} = \mathbf{j} + \frac{\partial \mathbf{D}}{\partial t} </math> 2008-09-02T08:07:46+09:00 1220310466 https://w.atwiki.jp/tklab/pages/28.html <math> \[ \phi_n(\kappa) = \frac{1}{4\pi^2\kappa^2} \int_0^\infty \frac{\sin(\kappa R)}{\kappa R} \frac{\partial}{\partial R} \left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR \] </math> 2008-08-29T11:44:52+09:00 1219977892 https://w.atwiki.jp/tklab/pages/27.html #math(130){{{ \rho\left( \frac{\partial \epsilon}{\partial t} + \mathbf{u}\cdot\nabla\epsilon \right) - \nabla\cdot(K_H \nabla T ) + p \nabla \cdot \mathbf{u} = 0 }}} 2008-09-02T11:42:56+09:00 1220323376