📐 Introduction to Calculus and Vector Analysis

Learn from the definition of differentiation to calculation rules for derivatives and higher-order derivatives, and implement numerical differentiation using NumPy (forward difference, central difference, Richardson extrapolation). Applications to temperature dependence of material properties and reaction rate analysis are also introduced.

Learn the definition of definite integrals, calculation of indefinite integrals, and the relationship between integration and differentiation (fundamental theorem of calculus), and implement numerical integration methods such as the trapezoidal rule, Simpson's rule, and Gaussian quadrature. Applications to heat calculation and spectral analysis are also covered.

Learn partial derivatives, total differentials, chain rule, and Jacobian matrices, and handle extremum problems of multivariable functions (Lagrange multipliers). Multiple integrals (double integrals, triple integrals) and variable transformations (polar, cylindrical, spherical coordinates) are also implemented.

Learn the concept of vector fields, definitions and physical meanings of gradient (grad), divergence (div), and curl (rot). Implementation of Laplacian, vector field visualization, and determination of conservative fields and potential functions.

Learn calculation methods for line integrals (scalar and vector fields) and surface integrals (scalar and vector fields). Understand Green's theorem, Gauss's divergence theorem, and Stokes' theorem, and implement applications to electromagnetism and fluid dynamics.