Advanced Calculus

In this module, you will learn the mathematical foundations for handling real-world problems with multiple free parameters and boundary conditions:

• Optimization,
• Data analysis, and
• Signal processing

These skills are essential for data science applications such as process optimization, function approximation, signal analysis and manipulation, the structure and functioning of neural networks, and the selection of appropriate loss and evaluation functions for their training.

In order to understand these applications, differential calculus in multiple variables is fundamental. It enables the precise description and analysis of functions in complex systems and forms the basis for many optimization methods and algorithms in machine learning, such as the calculation of gradients and the implementation of backpropagation in neural networks, which you will explore in detail for a simple network using numpy.

Another focus is the Fourier transformation, which allows time- or location-dependent data to be decomposed into frequencies. You will learn how to use these transformations to evaluate data series, edit music, or compress images. This involves complex numbers, integrals, and the theory of function spaces to enable efficient and elegant data analysis.

Differential Calculus of Functions in Multiple Variables

Students can explain the differentiability of functions in multiple variables and calculate and geometrically interpret relevant quantities. Students can identify local extrema and solve optimization problems with constraints (Lagrange function). Students can apply gradient methods in practical examples and implement them using numpy.

Application of optimization to neural networks

Students understand how the backpropagation algorithm works in 'Vanilla Gradient Descent' for training neural networks. Students are able to implement this algorithm in numpy in simple cases.

Fourier Transformation

Students can identify and apply appropriate basis functions to approximate functions. Students can use the Fourier series to describe and approximate functions. Students understand complex numbers for a more elegant description of the Fourier series and can use this form. Students can analyze and interpret amplitude and phase spectra (spectral analysis). Students can apply this understanding with the numerical implementation of the Fourier transformation (FFT) in simple cases.

Foundation in Calculus

Linear Algebra