1D mathematical tools

The aim of this course is to give students the mathematical tools needed to set up and use modeling tools in mechanics.
By the end of this course, they will have mastered the use of functions of one variable, sequences, series, differential equations and parametric curves.

1. Functions of one real variable

Limits, continuity, derivatives, tangents, limited development, asymptotes.
Reciprocal function.
Derivatives of a product, of composite functions, of a reciprocal function.
Usual functions: polynomials, trig, exponential, log, hyperbolic trig.

2. Integration

Definition using primitives: where F is a primitive of f.
Interpretation in terms of surface.
Properties: linearity, positivity. Change of variables. Integration by parts.
Extensions: , , with f with f defined on [a, b[...
Usual integrals.

3. Sequences

Convergent, bounded and monotonic sequences.
Geometric and arithmetic sequences.
Linear recurrent sequences _with one or two terms.

In TD: recurrent sequences of order 1.
4. Series

Convergent and absolutely convergent series, comparison.
Geometric series.

5. differential equations

Notation: . Notion of solution.
First- and second-order linear equations with constant coefficients.
TD: boundary problems.
First-order linear equations with non-constant coefficients.
Linear systems, eigenvalues, stability.

In TD: some non-linear examples.
6. Parametric curves

Studies of parametric curves. Curvature.

7. Fourier series

and .
Computation of Fourier coefficients. Plancherel.
Solving the diffusion equation.
Fourier transforms.

Engineering Sciences and Techniques