Mathematics for Engineers 3

General Data
Academic program	ECAM ENGINEERING PROGRAM			:
Type d'EC	Classes
Lectures : 24h00 Tutorials : 24h00 Total duration : 82h00	Status :	Period : Semester 3	Education language : English

Learning Outcomes
This course is in 2 parts: * Vector calculus: to revise and acquire calculation tools and methods, in order to confidently navigate through multivariable calculus and its applications. In particular, students will: - (outcome C1) be able to confidently use and apply differential operators for functions of several variables (multivariate chain differentiation, implicit differentiation, etc.) - (outcome C2) be able to perform multidimensional integration and line integration. * Linear Algebra: Acquire a good understanding of matrix algebra and methods. In particular, students will - (outcome LA1) understand the fundamental subspaces related to a matrix (as a linear map) - (outcome LA2) acquire fundamental techniques such as orthogonal projections in R^n or diagonalisation of matrices.

Learning Outcomes

This course is in 2 parts:
* Vector calculus: to revise and acquire calculation tools and methods, in order to confidently navigate through multivariable calculus and its applications.
In particular, students will:
- (outcome C1) be able to confidently use and apply differential operators for functions of several variables (multivariate chain differentiation, implicit differentiation, etc.)
- (outcome C2) be able to perform multidimensional integration and line integration.

* Linear Algebra: Acquire a good understanding of matrix algebra and methods.
In particular, students will
- (outcome LA1) understand the fundamental subspaces related to a matrix (as a linear map)
- (outcome LA2) acquire fundamental techniques such as orthogonal projections in R^n or diagonalisation of matrices.

Content
- Multivariable calculus: ° multivariate calculus (coordinate systems, scalar functions, vector functions, conservative vector fields, differential operators, Taylor expansions for functions from R^n to R^p, Implicit function theorem, simple PDE, Local Inverse Theorem) ° multiple and line integral (substitutions, volume integrals, area integrals, curvilinear integrals, Green-Riemann Theorem) - Linear algebra ° Introduction to linear spaces/subspaces in R^n, bases. ° Fundamental subspaces of a matrix - Rank Nullity theorem. ° Orthogonality ° Eigenvalues and Eigenvectors

Content

- Multivariable calculus:
° multivariate calculus (coordinate systems, scalar functions, vector functions, conservative vector fields, differential operators, Taylor expansions for functions from R^n to R^p, Implicit function theorem, simple PDE, Local Inverse Theorem)
° multiple and line integral (substitutions, volume integrals, area integrals, curvilinear integrals, Green-Riemann Theorem)

- Linear algebra
° Introduction to linear spaces/subspaces in R^n, bases.
° Fundamental subspaces of a matrix - Rank Nullity theorem.
° Orthogonality
° Eigenvalues and Eigenvectors

Pre-requisites / co-requisites
Mathematics for engineers 1 and 2, (in particular continuity, differentiation, integration, for functions of 1 variable, and preferably also 2 variables, simple matrix algebra.)

Bibliography
Course lecture Notes. G. Strang: Linear algebra and its applications (Wellesley-Cambridge Press) N. Piskunov: Differential and integral calculs. (Mir, Moscow, 1969) J. Stewart: Calculus, early transcendentals.

Assessment(s)
N°	Nature	Coefficient	Observable objectives
1	Mid Term	0,25	Midterm : C1, partially C2
2	online quizzes.	0,20	Continuous Assessment
3	Final exam:	0,30	Midterm : C2, LA1
4	Midterm : C2, LA1	0,25	Written exam