Contact

Ruhr University Bochum

Mechanics of Materials
Building IC Room 03-713
Universitaetsstrasse 150
44801 Bochum
Germany

Phone: +49 (02 34) 32 - 26025
Email: mechmat(at)rub.de

Postal Address

Ruhr University Bochum
Mechanics of Materials
Building IC Postbox 15
44780 Bochum
Germany



Variational Calculus (WiSe 2023/24)

Lecturer
Dr. Ulrich Hoppe
Course ID, hours/week, CP
125 505, 3 h/week, 5 CP
Study programmes
MSc-CE, MSc-BI, MSc-MB
Frequency of occurrence
every winter term
Time/place
Fri, 8:30 - 10:00 Uhr, HZO 90
Fri, 12:15 - 13:45 Uhr, HZO 90
For the first time
13.10.2023
Examination
Written exam, 90 min
Next exam dates:
06.03.2024
E-Learning-Platform
Supplementary material can be found on the E-Learning-Platform Moodle

Content

  • Vectors and operation with vectors
    • Definition
    • Addition
    • Multiplication with scalar
    • Scalar product
    • Vector product
  • Tensors and operations with tensors
    • Definition
    • Addition
    • Multiplication with scalar
    • Contraction
    • Product of tensors and vectors
    • Tensor product
    • Examples from continuum mechanics
  • Vector and tensor fields
    • Definition
    • Nabla operator
    • Divergence of vector field
    • Gradient of vector field
    • Divergence of tensor field
    • Curl of vector field
    • Gradient of tensor field
    • Examples from continuum mechanics
  • Integral theorems
    • Line integral
    • Gauss theorem
    • Stokes theorem
  • Curvilinear coordinates
    • Cylindrical coordinates
    • Spherical coordinates
    • Examples from continuum mechanics
  • One-dimensional variational problem
    • Functional
    • Variational problem
    • First variation
    • Necessary condition and Euler equation
    • Examples from mechanics
  • Extended 1-D variational problems
    • Variable end-point
    • n unknown functions
    • Functional depending on higher order derivatives
    • Variational problems with constraints
    • Examples from continuum mechanics
  • Sufficient conditions
    • Second variation
    • Necessary condition
    • Sufficient condition
  • Multi-dimensional variational problems
    • 2-dimensional variational problems
    • 3-dimensional variational problems
    • 4-dimensional variational problems
    • Examples from continuum mechanics
  • Direct methods in the calculus of variation
    • Rayleigh-Ritz method
    • Method of finite differences and finite elements
    • Dual variational principles
    • Variational-asymptotic method

Literature

can be found on the E-Learning-Platform Moodle