\documentclass[12pt,addpoints]{exam} \usepackage{enumitem} \usepackage[top=1in, bottom=1in, left=1in, right=1in]{geometry} \usepackage{amsmath} \usepackage{amssymb} \usepackage{bm} \usepackage{graphicx} \usepackage{hyperref} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\ddiv}{div} \DeclareMathOperator{\curl}{curl} \begin{document} \Large \noindent \textbf{Problem Set 1} \\ \normalsize ME 444: Continuum Mechanics \\ D.~H.\ Kelley \\ \numpoints~points \\ \begin{questions} \question[2] Show that if the vectors $\bm{a}$, $\bm{b}$, and $\bm{c}$ are linearly dependent, then $\bm{a} \cdot ( \bm{b} \times \bm{c} ) = 0$. \question Given \[ \left[ S_{ij} \right] = \left[ \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 2 \\ 3 & 0 & 3 \end{array} \right], \] evaluate the following: \begin{parts} \part[1] $S_{ii}$ \part[1] $S_{ij} S_{ij}$ \part[1] $S_{jk} S_{jk}$ \part[1] $S_{mn} S_{nm}$ \end{parts} \question Given \[ \left[ a_i \right] = \left[ \begin{array}{c} 1 \\ 0 \\ 2 \end{array} \right], \, \left[ B_{ij} \right] = \left[ \begin{array}{ccc} 2 & 3 & 0 \\ 0 & 5 & 1 \\ 0 & 2 & 1 \end{array} \right], \, \left[ C_{ij} \right] = \left[ \begin{array}{ccc} 0 & 3 & 1 \\ 1 & 0 & 2 \\ 2 & 4 & 3 \end{array} \right], \] demonstrate the equivalence of the following subscripted equations and the corresponding matrix equations. In each, indices $i$ and $j$ retain their meanings as defined above (indicating the row and column, respectively, in the \emph{original} quantities $a_i$, $B_{ij}$, and $C_{ij}$); no indices are renamed. \begin{parts} \part[1] $ D_{ji} = B_{ij} \Leftrightarrow \bm{D} = \bm{B}^T $ \part[1] $ b_i = B_{ij} a_j \Leftrightarrow \bm{b} = \bm{B} \bm{a} $ \part[1] $ c_j = B_{ji} a_i \Leftrightarrow \bm{c} = \bm{B}^T \bm{a} $ \part[1] $ s = B_{ij} a_i a_j \Leftrightarrow s = \bm{a}^T \bm{B} \bm{a} $ \end{parts} \question Given \[ [ a_i ] = \left[ \begin{array}{c} 1 \\ 2 \\ 0 \end{array} \right], \, [ b_i ] = \left[ \begin{array}{c} 0 \\ 2 \\ 3 \end{array} \right], \, [ S_{ij} ] = \left[ \begin{array}{ccc} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 4 & 0 & 1 \end{array} \right], \] calculate \begin{parts} \part[2] $ \epsilon_{ijk} a_k $ \part[2] $ \epsilon_{ijk} S_{jk} $ \part[2] $ \epsilon_{ijk} a_i b_j $ \end{parts} \question[2] \label{prob:epsdelta} Verify that $ \epsilon_{ijm} \epsilon_{klm} = \delta_{ik} \delta_{jl} - \delta_{il} \delta_{jk}$. \question Calculate and simplify \begin{parts} \part[1] $ \epsilon_{ilm} \epsilon_{jlm}$ \part[1] $ \epsilon_{ijk} \epsilon_{ijk}$ \end{parts} \question[3] Using the result of the problem~\ref{prob:epsdelta}, show that $ \bm{a} \times ( \bm{b} \times \bm{c} ) = ( \bm{a} \cdot \bm{c} ) \bm{b} - ( \bm{a} \cdot \bm{b} ) \bm{c} $. \question[3] Find the components of the tensor $ \bm{T} $ that transforms any vector $ \bm{a} $ into a vector $ \bm{b} = ( \bm{a} \cdot \hat{\bm{n}} ) \hat{\bm{m}} $, where $ \hat{\bm{m}} = (\hat{\bm{e}}_1 + \hat{\bm{e}}_2)/\sqrt{2} $ and $ \hat{\bm{n}} = (\hat{\bm{e}}_1 - \hat{\bm{e}}_3)/\sqrt(2) $. Write the tensor $ \bm{T} $ as a dyadic product. \question[3] Determine the components of the tensor $ \bm{D} $ that transforms any vector $ \bm{a} $ into a vector $ \bm{b} = \bm{a} + \bm{a} \times \bm{g} $, where $ \bm{g} = ( \hat{\bm{e}}_1 + \hat{\bm{e}}_2 - \hat{\bm{e}}_3 )/\sqrt{3} $. \question[4] With respect to the Cartesian basis $ \{ \hat{\bm{e}}_1, \, \hat{\bm{e}}_2, \, \hat{\bm{e}}_3 \} $, the tensor $ \bm{T} $ has components \[ [ T_{ij} ] = \left[ \begin{array}{ccc} 1 & 2 & -2 \\ 2 & 0 & 0 \\ -3 & 0 & 1 \end{array} \right], \] and the vector $ \bm{v} $ has components $ \bm{v} = \hat{\bm{e}}_1 + \hat{\bm{e}}_2 -3 \hat{\bm{e}}_3 $. Find the components of $ \bm{T} $ and $ \bm{v} $ with respect to a Cartesian basis $ \{ \hat{\bm{e}}_{1^\prime}, \, \hat{\bm{e}}_{2^\prime}, \, \hat{\bm{e}}_{3^\prime} \} $, where $ \hat{\bm{e}}_{1^\prime} $ is in the direction of $ \hat{\bm{e}}_2 - 2 \hat{\bm{e}}_3 $ and $ \hat{\bm{e}}_{2^\prime} = \hat{\bm{e}}_1 $. \question Given the tensor $ \bm{T} $ with components \[ [ T_{ij} ] = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right], \] \begin{parts} \part[2] Find the symmetric and antisymmetric parts of $\bm{T}$. \part[2] Find the dual vector of the antisymmetric part of $\bm{T}$. \end{parts} \question[2] Rank-2 tensors $\bm{T}$, $\bm{R}$, and $\bm{S}$ are related by $\bm{T} = \bm{R} \bm{S}$. Tensors $\bm{R}$ and $\bm{S}$ have the same nonzero eigenvector $\bm{n}$ and corresponding eigenvalues $r_1$ and $s_1$. Find an eigenvalue and nonzero eigenvector of $\bm{T}$. \question Consider the dyadic product $\bm{a} \otimes \bm{b}$. \begin{parts} \part[1] For this dyad, find one nonzero eigenvector that has a nonzero eigenvalue. \part[1] Calculate the first principal scalar invariant of $\bm{a} \otimes \bm{b}$. \part[2] Calculate the third principal scalar invariant of $\bm{a} \otimes \bm{b}$. \end{parts} \question[4] A tensor $\bm{T}$ has a matrix representation \[ \left[ T_{ij} \right] = \left[ \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2 \end{array} \right]. \] Find the eigenvalues and three mutually orthogonal, unit-length eigenvectors. \question[3] If $\bm{B}$ is antisymmetric and $\bm{A}$ is symmetric, calculate and simplify $A_{ij} B_{ij}$. \question[2] What topic on this problem set was easy? What topic was hard? What topic was impossible? \end{questions} \end{document}