Introduction to Mathematical Physics I | Department of Physics

Introduction to Mathematical Physics I

(a) Linear transformations of the plane    
i. Affine planes and vector spaces    
ii. Vector spaces and their affine spaces    
iii. Euclidean and affine transformations  
 iv. Representing linear transformations by matrices    
v. Areas and determinants  

(b) Eigenvectors and eigenvalues    
i. Conformal linear transformations    
ii. Eigenvectors and eigenvalues  
 iii. Markov processes  

(c) Linear differential equations in the plane    
i. Functions of matrices    
ii. Computing the exponential of a matrix    
iii. Differential equation and phase portraits    
iv. Applications of differential equations  
 (d) Scalar products    
i. The Euclidean scalar product    
ii. Quadratic forms and symmetric matrices    
iii. Normal modes 
iv. Normal modes in higher dimensions    
v. Special relativity: The Poincare’ group and the Galilean group   
(e) Calculus in the plane
i. The differential calculus and the examples of the chain rule: the Born approximation and Kepler motion  
ii. Partial derivatives and differential forms.  
iii. The pullback notation  
iv. Taylor’s formula  
v. Lagrange multiplier  

(f) Double integrals  
i. Exterior derivative  
ii. Two-forms
iii. Pullback and integration for two-forms  
iv. Two-forms in three space  
v. Green’s theorem in the plane

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