University of Tennessee, Knoxville MATH 537
Professor Wise · University of Tennessee, Knoxville · 21 lectures with notes
Every lecture with its full transcript and chapters, and one search across all 21 lectures.
Lecture 01: Vectors and Index Notation
Steven M. Wise builds the index-notation toolkit for vector operations in continuum mechanics.
Lecture 02: Tensor Algebra
Second-order tensors are linear maps whose frame components obey matrix algebra and can be derived using dyadic products.
Lecture 03: Change of Basis
Tensor algebra foundations connect skew-symmetric tensors to vectors and establish orthogonal change-of-basis rules.
Lecture 04: Eigenvalues and Special Decompositions
Symmetric tensors diagonalize in orthogonal eigenvector frames, enabling square roots, polar decompositions, and higher-order tensor operations.
Lecture 05: Fourth-Order Tensors
Fourth-order tensor operations and symmetries lead to the two-parameter isotropic linear elasticity law.
Lecture 06: Derivatives and Gradients
Scalar-field gradients are vectors, vector-field gradients are second-order tensors, and divergence is the trace of that tensor.
Lecture 07: Divergence, Curl, and Integral Theorems
Invariant definitions connect divergence, curl, and Laplacians to their component formulas and integral theorems.
Lecture 08: Units and Sample Calculations
Physical units provide a consistency check for tensor-calculus operations and continuum-mechanics equations.
Lecture 09: Derivatives of Tensor Functions
Tensor-function derivatives raise tensor order: scalar outputs yield second-order gradients, while second-order tensor outputs yield fourth-order derivatives.
Lecture 10: Masses, Forces, and Stresses
Cauchy's stress tensor converts continuum force and torque balance into local equilibrium equations.
Lecture 11: Simple States of Stress
Stress on any plane resolves into normal and shear components, while principal directions identify planes with zero shear traction.
Lecture 12: Deformations
Every admissible homogeneous deformation can be reduced to translations, rotations, and simple extensions.
Lecture 13: Finite and Infinitesimal Strain
The right Cauchy-Green tensor measures finite stretch and shear, while small displacement gradients yield the infinitesimal strain tensor.
Lecture 14: Motions and the Total Time Derivative
The material derivative tracks changes along moving particles, adding a velocity-transport term to spatial time derivatives.
Lecture 15: The Reynolds Transport Theorem
The Reynolds transport theorem converts changing-domain integrals into local time variation plus flux through the moving boundary.
Lecture 16: Balance Laws: Mass, Momentum, and Energy
Mass conservation becomes the continuity equation, while Reynolds transport theorem links integral balances to local continuum laws.
Lecture 17: Local Balance Laws
Localizing continuum balance laws yields momentum, stress symmetry, energy, and entropy equations.
Lecture 18: Entropy Production and Lagrangian Balance Laws
The Lagrangian balance laws follow from transforming integral balances, with mass conservation and the Piola-Kirchhoff stresses connecting material and spatial descriptions.
Lecture 19: Frame Indifference and Material Constraints
Frame indifference and incompressibility constrain constitutive laws and generate the pressure term used in Navier-Stokes.
Lecture 20: Ideal Fluids and Euler Equations
Ideal-fluid pressure enforces incompressibility while the Euler equations evolve velocity without shear stress or viscosity.
Lecture 21: Potential Flow and Elastic Fluids
Irrotational fluid motion leads to Bernoulli relations and scalar-potential equations in both ideal and elastic fluid models.