**This course was developed by Prof. Krishna Garikipati and Dr. Gregory Teichert, at the University of Michigan in partnership with Ansys.**

In this course, we will derive the finite element equations for linear elliptic PDEs with vector variables in three dimensions. One such application case is that of linearized elasticity which is discussed in this course. We begin by discussing the constitutive relation between Cauchy stress and strain and the kinematic relation between strains and displacements. Following this, we discuss several properties of the elasticity tensor. Using this formation, we derive the finite-dimensional weak form of linearized elasticity. We then discuss how the finite-dimensional weak form can be written as a summation of element integrals. Then using the basis functions we transform the weak form to its equivalent matrix-vector form. We finally discuss how Neumann and Dirichlet boundary conditions are handled for such problems.