Linear Algebra and Vector Analysis

Course goals

Content

The course starts with systematically solving systems of linear equations. The following new topics are then discussed: matrix calculus, coordinate transformations, eigenvalues and eigenvectors and vector calculus. New concepts related to differentiation of scalar and vector fields are also introduced, such as gradient, divergence, rotation and Laplacian. Spherical and cylindrical coordinate systems are discussed and applications of line, surface and volume integrals are discussed. The course concludes with the important theorems of Gauss and Stokes, with which surface integrals can be converted to respectively volume and line integrals.
At each stage of the course we endeavor to give examples from Earth science so that the students can form a connection between the mathematics and its practical application in Earth science. Students will also be required to solve a set of geo-relevant numerical exercises in Python.
Planned schedule:
Week 1: Matrices, including: terminology and row reduction, determinants
Week 2: Vectors including: notation, scalar operations, dot and cross products, equations of lines and planes
Week 3: Matrix Calculations and Transformations: matrix equations, multiplication by a constant, addition, multiplication with another matrix, index notation, null matrix, identity matrix, transpose, inverse, operations with determinants, rotation matrices, linearity, linear transformations of matrices, orthogonal transformations, linear (in)dependence in vectors
Week 4:

• Index notation including symmetric/antisymmetric and orthogonal matrices, matrix multiplication with index notation, Kronecker delta, trace of a matrix, Einstein summation convention, dot product in n-dimensional space
• eigenvalues & eigenvectors: including homogeneous linear equations and diagonalisation

Week 5:

• Programming with linear algebra, vectors and matrices: Students will learn how to solve problems involving linear algebra, vectors and matrices with Python, including a guide to the relevant libraries/packages. They will perform various matrix manipulations including multiplication, finding the determinant, inverse and eigenvectors of a matrix. They will also solve some Earth-science based problems including calculating strain rates and deformation in the lithosphere, and mapping plate motions through time.
• MID TERM EXAM

Week 6:

• Multiple integrals: double, triple and line integrals
• 3D coordinate systems: Polar coordinates, cylindrical coordinates, spherical coordinates; length, area & volume elements; basis vectors in polar, cylindrical and spherical coordinates; Jacobians

Week 7: Vector Calculus: Triple dot & cross products, applications of dot & cross products; scalar & vector fields; del operator; grad, div, curl & combinations thereof, including the Laplacian – physical meaning and examples

Week 8:

• Theorems of Vector Calculus: Line integrals of vector fields; conservative fields; potentials; Divergence theorem (Gauss' theorem); Stokes' theorem
• Programming with multiple integrals and vector calculus: Students will calculate the volume and mass of a planet; calculate Earth's density profile, WHAT ELSE?

Week 9:

• Revision class: Wave equation in 2D

• END TERM EXAM

Deze cursus is ingangseis voor (cursussen in specialisaties):
Seismology
Continuum Mechanics & Potential Fields
Lithosphere Dynamics