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

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
• Multiplication with scalar
• Scalar product
• Vector product
• Tensors and operations with tensors
• Definition
• 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
• Divergence of tensor field
• Curl of vector 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