Bibliography

This bibliography provides references for deeper study of the mathematical concepts covered in PyDelt’s theory documentation.

Foundational Calculus Texts

For Intuition and Accessibility

  1. Strang, G. (2010). Calculus. Wellesley-Cambridge Press.

    • Why read it: Free online, excellent visualizations, focuses on understanding over formalism.

    • Best for: Building intuition, seeing connections between concepts.

    • Access: MIT OpenCourseWare

  2. Thompson, S. P. (1914). Calculus Made Easy. Macmillan.

    • Why read it: Classic text, remarkably accessible, still relevant after 100+ years.

    • Best for: Absolute beginners, those intimidated by math.

    • Access: Public domain, freely available online.

For Rigor

  1. Spivak, M. (2008). Calculus (4th ed.). Publish or Perish.

    • Why read it: Rigorous but readable, beautiful proofs, develops mathematical maturity.

    • Best for: Those wanting deep understanding, future mathematicians.

  2. Apostol, T. M. (1967). Calculus, Vol. 1 & 2. Wiley.

    • Why read it: Comprehensive reference, integrates linear algebra with calculus.

    • Best for: Complete coverage, reference work.

  3. Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill.

    • Why read it: The standard for rigorous real analysis.

    • Best for: Graduate-level understanding, proving theorems.

Numerical Methods

  1. Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press.

    • Why read it: Practical algorithms with code, covers everything.

    • Best for: Implementation, understanding trade-offs.

  2. Trefethen, L. N. (2013). Approximation Theory and Approximation Practice. SIAM.

    • Why read it: Modern treatment, connects theory to computation.

    • Best for: Understanding interpolation, polynomial approximation.

  3. Fornberg, B. (1988). “Generation of Finite Difference Formulas on Arbitrarily Spaced Grids.” Mathematics of Computation, 51(184), 699-706.

    • Why read it: Foundational paper for finite difference methods.

    • Best for: Understanding numerical differentiation.

Machine Learning and Deep Learning

  1. Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press.

    • Why read it: The standard deep learning textbook, Chapter 4 covers numerical computation.

    • Best for: Connecting calculus to neural networks.

    • Access: deeplearningbook.org

  2. Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.

    • Why read it: Foundational for understanding optimization, gradients, Hessians.

    • Best for: Optimization theory, understanding gradient descent.

    • Access: stanford.edu/~boyd/cvxbook

  3. Baydin, A. G., Pearlmutter, B. A., Radul, A. A., & Siskind, J. M. (2018). “Automatic Differentiation in Machine Learning: A Survey.” Journal of Machine Learning Research, 18(153), 1-43.

    • Why read it: Comprehensive survey of autodiff, the foundation of modern deep learning.

    • Best for: Understanding how PyTorch/TensorFlow compute gradients.

Functional Data Analysis

  1. Ramsay, J. O., & Silverman, B. W. (2005). Functional Data Analysis (2nd ed.). Springer.

    • Why read it: The definitive text on FDA, basis for PyDelt’s FDA methods.

    • Best for: Understanding spline smoothing, functional representations.

  2. Ramsay, J. O., Hooker, G., & Graves, S. (2009). Functional Data Analysis with R and MATLAB. Springer.

    • Why read it: Practical implementation of FDA concepts.

    • Best for: Hands-on FDA work.

Splines and Interpolation

  1. de Boor, C. (2001). A Practical Guide to Splines (Revised ed.). Springer.

    • Why read it: The authoritative reference on splines.

    • Best for: Deep understanding of spline mathematics.

  2. Wahba, G. (1990). Spline Models for Observational Data. SIAM.

    • Why read it: Connects splines to statistics, optimal smoothing.

    • Best for: Understanding smoothing parameter selection.

Local Regression Methods

  1. Cleveland, W. S. (1979). “Robust Locally Weighted Regression and Smoothing Scatterplots.” Journal of the American Statistical Association, 74(368), 829-836.

    • Why read it: Original LOWESS paper.

    • Best for: Understanding the method PyDelt implements.

  2. Fan, J., & Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman & Hall.

    • Why read it: Comprehensive treatment of local polynomial regression.

    • Best for: Understanding LLA-type methods.

Physics-Informed Machine Learning

  1. Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). “Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations.” Journal of Computational Physics, 378, 686-707.

    • Why read it: Foundational PINN paper.

    • Best for: Understanding physics-informed approaches.

  2. Chen, R. T. Q., Rubanova, Y., Bettencourt, J., & Duvenaud, D. (2018). “Neural Ordinary Differential Equations.” Advances in Neural Information Processing Systems, 31.

    • Why read it: Introduced Neural ODEs.

    • Best for: Understanding continuous-depth networks.

Stochastic Calculus

  1. Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications (6th ed.). Springer.

    • Why read it: Accessible introduction to SDEs.

    • Best for: Understanding Itô calculus, financial applications.

  2. Kloeden, P. E., & Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer.

    • Why read it: Comprehensive treatment of numerical methods for SDEs.

    • Best for: Implementation of stochastic methods.

Complex Analysis

  1. Needham, T. (1997). Visual Complex Analysis. Oxford University Press.

    • Why read it: Beautiful, geometric approach to complex analysis.

    • Best for: Building intuition, seeing the geometry.

  2. Ahlfors, L. V. (1979). Complex Analysis (3rd ed.). McGraw-Hill.

    • Why read it: Standard graduate text.

    • Best for: Rigorous treatment.

Multivariate Calculus

  1. Marsden, J. E., & Tromba, A. (2011). Vector Calculus (6th ed.). W. H. Freeman.

    • Why read it: Clear treatment of gradients, Jacobians, Hessians.

    • Best for: Multivariate calculus foundations.

  2. Hubbard, J. H., & Hubbard, B. B. (2015). Vector Calculus, Linear Algebra, and Differential Forms (5th ed.). Matrix Editions.

    • Why read it: Unified treatment, connects to differential geometry.

    • Best for: Deeper understanding of multivariate calculus.

Online Resources

Video Courses

  1. 3Blue1Brown - “Essence of Calculus” (YouTube)

  2. MIT OpenCourseWare - 18.01 Single Variable Calculus

    • Full course with lectures, notes, problem sets.

    • ocw.mit.edu

  3. Stanford CS231n - Convolutional Neural Networks

Interactive Tools

  1. Desmos - Graphing Calculator

  2. GeoGebra - Dynamic Mathematics

How to Use This Bibliography

If you’re new to calculus:

Start with Thompson’s Calculus Made Easy or 3Blue1Brown videos, then move to Strang.

If you want rigorous foundations:

Work through Spivak, then Rudin for analysis.

If you’re focused on ML applications:

Read Goodfellow et al. Chapter 4, then Baydin et al. on autodiff.

If you’re implementing numerical methods:

Numerical Recipes is your reference, supplemented by Trefethen.

If you’re working with time series:

Ramsay & Silverman for FDA, Cleveland for LOWESS.

If you’re doing physics-informed ML:

Start with Raissi et al., then Chen et al. for Neural ODEs.

Back to: Theory Index