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expansion & Series representation of Exponential & Trigonometric function\ 6:02 Examples of Taylor series Logarithmic function 18:50 Examples of Taylor series 1(1-x) function 26:54 Series representation of the exponential function 33:21 Series representation of sine & cosine 38:30 Solutions Task 1 - 2 - 3 - 4 - Taylor expansion 1:05:57 Section outro numbers\ 1:06:55 Section intro 1:07:43 What are complex numbers 1:17:09 Addition, subtraction & complex plane 1:30:17 Multiplication & division of complex numbers 1:39:55 Exponentials and polar representation of complex numbers 1:58:16 Solutions - Complex numbers 2:12:27 Factorization of polynomials Fundamental theorem of Algebra 2:18:11 Section outro calculus II Multiple dimensions\ 2:18:27 Section intro 2:19:46 Multidimensional functions 2:25:58 Partial derivatives & Directional derivatives 2:38:43 Total derivatives & Chain rule 2:42:40 Nabla operator Gradient 2:47:32 Nabla operator Divergence 2:58:09 Nabla operator Curl 3:08:08 Laplace operator 3:09:35 Nabla Useful relations 3:18:16 Solutions - Derivatives in multiple dimensions 3:33:49 Finding local extrema in multiple dimensions 3:45:23 Taylor expansion of multidimensional functions 3:52:12 Section outro calculus II Multiple dimensions & Alternative coordinate systems\ 3:52:48 Section intro 3:53:52 Revision Integrals in one dimension 3:57:52 Integrals in two dimensions 4:00:56 Order of integration 4:09:27 Multidimensional integrals 4:12:40 Volume of a pyramid Integration in three dimensions by parametrization 4:20:12 Arc length - Length of a curve 4:25:06 Arc length Examples 4:31:27 Line integrals Scalar functions 4:38:40 Line integrals Vector functions 4:42:24 Difficult example Volume of a sphere in Cartesian coordinates 4:54:26 Alternative coordinate systems Why do we need them 4:57:49 Polar coordinates - Alternative coordinate systems 13 5:01:20 Cylindrical coordinates - Alternative coordinate systems 23 5:04:54 Spherical coordinates - Alternative coordinate systems 33 5:08:26 Volume and surface elements in spherical coordinates 5:16:15 Integration in spherical coordinates Volume & Surface area of a sphere 5:23:16 Volume and surface elements in cylindrical coordinates 5:27:31 Integration in cylindrical coordinates Solids of revolution 5:34:01 Surface elements in polar coordinates 5:37:00 Integration in polar coordinates Gaussian integral 5:47:25 Solutions Task 1 - 2 - 3&4 - Integrals in multiple dimensions 6:12:07 Section outro equations\ 6:12:37 Section intro 6:14:38 What are differential equations 6:27:37 Classification of differential equations Ordinary (ODE) versus partial (PDE) 6:32:35 Classification of ordinary differential equations 6:39:02 Trivial case Direct integration 6:42:50 Homogeneous linear differential equations, superposition & Exponential ansatz 6:55:16 Example for a linear ODE & using the exponential ansatz The harmonic oscillator 7:04:31 Inhomogeneous linear differential equations Homogeneous plus specific solution 7:08:04 Example for solving an inhomogeneous linear differential equation 7:12:32 Integration factor method - Linear ODE with non-constant coefficients 7:19:46 Example Integration factor method 7:25:21 Bernoulli equations Non-linear ODE 7:30:27 Example Bernoulli equations 7:38:49 Separation of variables - Nonlinear ordinary differential equations 7:46:23 Solutions Tasks 1&2 - 3&4 - 5 - Differential equations 8:25:49 Numerical methods - Euler method 8:33:03 Numerically solving higher order differential equations 8:38:14 Add on Solving differential equations numerically 8:39:27 Numerics with Python Solving the differential equations of coupled oscillators 8:56:55 Partial differential equations The heat equation & Eigensystems\ 9:08:12 Section intro 9:10:02 What is a matrix 9:20:18 Matrix addition & subtraction 9:22:29 Matrix multiplication 9:28:13 Example Matrix multiplication 9:36:24 Transpose matrix 9:40:40 Calculating the determinant of a matrix 9:48:14 Example Calculating the determinant of a matrix 9:56:48 Calculating the inverse matrix 10:01:08 Example Calculating the inverse matrix 10:10:31 Inverse matrix Determinant method 10:15:53 Eigensystems Eigenvalues & Eigenvectors of a matrix 10:27:33 Example Calculating eigenvalues & eigenvectors 10:39:27 Trace of a matrix 10:40:52 Special matrices Symmetric & Hermitian matrices 10:49:55 Special matrices Unitary matrices 10:52:48 Special matrices Rotational matrices 10:56:57 Positive definite, negative definite & indefinite matrices 11:02:13 Solutions Task 1&2 - 3 - 4&5 - Matrices 11:32:16 Add on Calculating the eigenvalues of a matrix numerically 11:33:46 Numerics with Python Determining the eigenfrequencies of 3 coupled oscillators transform & Delta distribution\ 11:52:53 Section intro 11:54:36 What is a Fourier
