• Understanding, analyzing, and applying basic concepts and properties of vector spaces (subspaces, linear (in)dependence, dimension, and basis (transformation)) (k1, k2, k3, k4)
• Knowing the properties of linear and orthogonal mappings and their matrix representation (k1, k2)
• Knowing, determining, and applying eigenvalues, eigenvectors, and eigenspaces of matrices and their properties (k1, k2, k3, k4)
• Understanding the diagonalizability of matrices and the definiteness of symmetric matrices, and applying these concepts to problem-solving (k1, k2, k3, k4)

From the field of function sequences and series:

• Knowing and applying convergence criteria for number series (k1, k2, k3)
• Knowing, investigating, determining, and applying pointwise and uniform convergence of sequences of functions, as well as convergence behavior and properties of function series (k1, k2, k3, k4)
• Knowing, investigating, determining, and applying the convergence behavior, radius of convergence, differentiation, and integration of power series and the development of Taylor series in various applications (e.g, approximation, error estimation) (k1, k2, k3, k4)
• Knowing, investigating, determining, and applying Fourier polynomials/series and their properties (k1, k2, k3, k4)
• Modeling periodic extensions of functions (k2, k3)

From the field of differential and integral calculus:

• Parametrizing (simple) curves, surfaces, and solids in R^3, and knowing and applying essential coordinate transformations (polar, cylindrical, spherical coordinates) (k2, k3, k4).
• Knowing and understanding the fundamentals of multivariable differential and integral calculus (k1, k2).
• Knowing, understanding, applying, and analyzing the basic concepts, properties, and techniques of multivariable differential calculus (k1, k2, k3, k4) (continuity, types of differentiability, local approximation, extremum determination with and without constraints, linear regression, gradient descent methods, vector analysis, multivariate Newton's method).
• Knowing, understanding, applying, and analyzing the basic concepts, properties, and techniques of multivariable integral calculus (k1, k2, k3, k4) (line and surface integrals of scalar and vector fields, volume integrals, fundamental theorems for line integrals, Gauss's, Stokes's, and Green's theorems and their applications).
• Knowing, determining, and applying the properties and solutions of linear differential equations with constant coefficients, and relate them to previously taught mathematical concepts (k1, k2, k3, k4, k5).

• Vector spaces: Linear combination, linear span, row/column space of a matrix, rank of a matrix, representation of vectors with respect to a basis, basis extension theorem, properties of finite-dimensional vector spaces.
• Linear mappings: Matrix representation and change of basis, rank theorem, Gram-Schmidt orthonormalization process, characteristic polynomial of a matrix, principal minors, principal axis system.
• Sequences and series of functions: Weierstrass’s majorant criterion, radius of convergence, Taylor polynomial and remainder terms; orthogonality properties of trigonometric functions, Dirichlet’s convergence theorem.

• Differential calculus: Topological foundations, gradient, Hessian matrix, Jacobian matrix, Taylor series expansion, determination of extrema with and without constraints, Lagrange multipliers method, linear least squares and regression, gradient descent method; generalized chain rule, vector calculus (divergence, curl, Nabla and Laplace operators).

• Integration: Riemann integral, Fubini’s theorem, integration over normal domains, transformation theorem, potential of a gradient field, path independence. • Differential Equations: Fundamental system, solution approach using power series.