Please see the attached PDF file for a nicely typeset syllabus.

Topics/notions printed in italics are not a part of the exam. All other topics/notions are expected to mastered at the level of a reflex action.

Preliminaries.
- Linear algebra.
  - Scalar product, vector product, mixed product, tensor product. Transposed matrix.
  - Tensors.
  - Cofactor matrix cof�� and determinant det��. Geometrical interpretation.
  - Cayley–Hamilton theorem, characteristic polynomial, eigenvectors, eigenvalues.
  - Trace of a matrix.
  - Invariants of a matrix and their relation to the eigenvalues and the mixed product.
  - Properties of proper orthogonal matrices, angular velocity.
  - Polar decomposition. Geometrical interpretation.
- Elementary calculus.
  - Matrix functions. Exponential of a matrix.
  - Representation theorem for scalar valued isotropic tensorial functions and tensor valued isotropic tensorial functions.
  - Gâteaux derivative, Fréchet derivative. Derivatives of the invariants of a matrix.
  - Operators ∇, div and rot for scalar and vector fields. Operators div and rot for tensor fields. Abstract definitions and formulae in Cartesian coordinate system. Identities in tensor calculus.
- Line, surface and volume integrals.
  - Line integral of a scalar valued function ∫_γφ dX, line integral of a vector valued function ∫_γv • dX.
  - Surface integral of a scalar valued function ∫_Sφ dS, surface integral of a vector valued function ∫_Sv • dS, surface Jacobian.
  - Volume integral, Jacobian matrix.
- Stokes theorem and its consequences.
  - Potential vector field, path independent integrals, curl free vector fields. Characterisation of potential vector fields.
  - Korn equality.
- Elementary concepts in classical physics.
  - Newton laws.
  - Galilean invariance, principle of relativity, non-inertial reference frame.
  - Fictitious forces (Euler, centrifugal, Coriolis).
Kinematics of continuous medium.
- Basic concepts.
  - Notion of continuous body. Abstract body, placer, configuration.
  - Reference and current configuration. Lagrangian and Eulerian description.
  - Deformation/motion χ.
  - Local and global invertibility of the motion/deformation, condition det�� > 0.
  - Deformation gradient �� and its geometrical interpretation. Polar decomposition �� = ℝ�� of the deformation gradient and its geometrical interpretation.
  - Relative deformation gradient.
  - Deformation gradient and polar decompostion for simple shear.
  - Displacement U.
  - Deformation of infinitesimal line, surface a volume elements. Concept of isochoric motion.
  - Lagrangian velocity field V, Eulerian velocity field v. Material time derivative ${\ensuremath{\frac{{\mathrm{d}}{}}{{\mathrm{d}}{t}}}}$ of Eulerian quantities.
  - Streamlines and pathlines (trajectories).
  - Spatial velocity gradient ��, its symmetric part �� and skew-symmetric part ��.
- Strain measures.
  - Left and right Cauchy–Green tensor, �� and ℂ. Hencky strain.
  - Green–Saint-Venant strain tensor ��, Euler–Almansi strain tensor ��. Geometrical interpretation.
  - Linearised strain $\linstrain$.
- Compatibility conditions for linearised strain $\linstrain$ in ${\ensuremath{{\mathbb R}}}^2$. Compatibility conditions for linearised strain $\linstrain$ in ${\ensuremath{{\mathbb R}}}^3$.
- Rate quantities.
  - Rate of change of Green–Saint-Venant strain, rate of change of Euler–Almansi strain and their relation to the symmetric part of the velocity gradient ��.
  - Rate of change of infinitesimal line, surface and volume elements. Divergence of the Eulerian velocity field and its relation to the change of volume.
  - Objective derivatives of tensorial quantities (Oldroyd derivative, Truesdell derivative).
- Kinematics of moving surfaces.
  - Lagrange criterion for material surfaces.
- Reynolds transport theorem.
  - Reynolds transport theorem for the volume moving with the medium.
  - Reynolds transport theorem in the presence of surface discontinuities.
Dynamics and thermodynamics of continuous medium.
- Mechanics.
  - Balance laws for continuous medium as counterparts of the classical laws of Newtonian physics of point particles.
  - Concept of contact/surface forces. Existence of the Cauchy stress tensor �� (tetrahedron argument).
  - Pure tension, pure compression, tensile stress, shear stress.
  - Balance of mass, linear momentum and angular momentum in Eulerian description.
  - Balance of angular momentum and its implications regarding the symmetry of the Cauchy stress tensor. Proof of the symmetry of the Cauchy stress tensor.
  - Balance of mass, linear momentum and angular momentum in Lagrangian description.
  - First Piola–Kirchhoff stress tensor ��_R and its relation to the Cauchy stress tensor ��. Piola transformation.
  - Formulation of boundary value problems in Eulerian and Lagrangian descripition, transformation of traction boundary conditions from the current to the reference configuration.
- Elementary concepts in thermodynamics of continuous medium.
  - Specific internal energy e, energy/heat flux j_q.
  - Balance of total energy in the Eulerian and Lagrangian description.
  - Balance of internal energy in the Eulerian and Lagrangian description.
  - Referential heat flux J_q.
  - Specific Helmholtz free energy, specific entropy.
- Boundary conditions.
- Geometrical linearisation. Incompressibility condition in the linearised setting. Specification of the boundary conditions in the linearised setting.
- Balance laws in the presence of discontinuities.
Simple constitutive relations.
- Pressure and thermodynamic pressure, engineering equation of state. Derivation of compressible and incompressible Navier–Stokes fluid model via the representation theorem for tensor valued isotropic tensorial functions. Complete thermodynamical description of a compressible viscous heat conducting fluid – Navier–Stokes–Fourier equations.
- Cauchy elastic material. Derivation via the representation theorem for tensor valued isotropic tensorial functions.
- Green elastic material. (Hyperelastic solid.) Relation between the specific Helmholtz free energy and the Cauchy stress tensor for an elastic solid. Rate type formulation of constiutive relations for a hyperelastic solid.
- Physical units, dimensionless quantities, Reynolds number.
Simple problems in the mechanics of continuous medium.
- Archimedes law.
- Deformation of a cylinder (linearised elasticity). Hooke law.
- Inflation of a hollow cylinder made of an incompressible isotropic elastic solid. (Comparison of the linearised elasticity theory and fully nonlinear theory.)