# Lecturer Dmitri Zaitsev

This is the continuation of the course 2E1 lectured by Dr. Timoney in the first semester.

Problem Sheets in PDF: Sheet 13 Sheet 14 Sheet 15 Sheet 16 Sheet 17 Sheet 18 Sheet 19 Sheet 20 Sheet 21 Sheet 22 Sheet 23

Most solutions to the exercises are very similar to those older ones that can be found here and here. Here solutions of the few excercises that are slightly different: Sheet 16 Sheet 23(iii)

Every problem in the calculus of variations has a solution, provided the word solution is suitably understood. -- David Hilbert

Course outline:

Functions in several variables (Chapter 11 in 10th ed. or Chapter 14 in 11th ed. of Thomas' book). Domains and Ranges. Graphs and Level Curves. Level Surfaces. Limits and Continuity. Partial Derivatives. The Chain Rule. Implicit Partial Differentiation. Second-Order and Higher Order Partial Derivatives. Directional Derivatives and Gradient. Tangent lines to Level Curves, Tangent planes to Level Surfaces. Standard Linear Approximation (Linearization). Extreme Values and Saddle Points, Critical Points, Derivative Tests. Finding Local Maxima and Minima and Absolute Maxima and Minima on Closed Bounded Regions. The Method of Lagrange Multipliers. Taylor's Formula and the Error Estimate.

Multiple Integrals (Chapter 12 in 10th ed. or Chapter 15 in 11th ed. of Thomas' book). Double Integrals over Rectangular and more general Bounded Regions. Fubini's Theorems for Calculating Double Integrals. Calculating Area and Center of Mass. Double Integrals in Polar Form. Triple Integral in Rectangular, Cylindrical and Spherical Coordinates. Substitutions in Multiple Integrals.

Linear Algebra (Chapters 4-7 in Anton-Rorres' book). Euclidean n-Space and n-Vectors, Operations with them. Linear Transformations and their Matrices. Subspaces. Linear Combinations of Vectors. Subspaces spanned by a Set of Vectors. Linear Independence of a Set of Vectors. Basis and Dimension. Standard Basis in n-space. Coordinates of Vectors relative to a Basis. General and Particular Solutions for a Linear System. Row, Column and Nullspace of a Matrix. Finding Bases for them using Elementary Row Operations. Rank and Nullity of a Matrix. Inner Products, Lengths, Distances and Angles relative to them. Orthogonal and Orthonormal Bases relative to an Inner Product. Orthogonal projections to Subspaces. Gram-Schmidt Process (see Example 7 in Chapter 6.3). Best Approximation by the Least Squares method. Eigenvalues and Eigenvectors of Square Matrices.

Fourier Analysis (Chapter 10 in Kreyszig' book). Fourier Series for periodic functions. Euler Formulas for the Fourier Coefficients. Even and Odd Functions. Fourier Cosine and Fourier Sine Series for them. Fourier Integral and Fourier Transform.

For your revision I highly recommend the Calculus Textbook by Gilbert Strang that is available online, specifically the Chapters (in PDF) Partial Derivatives (study guide), Multiple Integrals (study guide), and Mathematics after Calculus.

Calculus:
Calculus for Beginners and Artists by Daniel Kleitman.
Multivariable Calculus Online by Jeff Knisley
Graphical representation of functions by Martin J. Osborne
Taylor's formula, linear and quadratic approximations by Eric A. Carlen
Linear Algebra:
Working with Vectors (A Self-Help Workbook for Science and Engineering Students by Jenny Olive)
Java applet introducing 3-vectors by Maths Online
Matrix Algebra Tutorials by S.O.S. MATHematics
A Linear Algebra book by Jim Hefferon (PDF file)
Importance of Linear algebra in Engineering Design Methodology by Mysore Narayanan (PDF file)

Old 2E1 web pages.
2E1 2005-06 by Dmitri Zaitsev with Problem Sheets and their Solutions.
2E1 2004-05 by Dmitri Zaitsev with Problem Sheets and their Solutions.
2E1 2003-04 by Fermin Viniegra with many interesting links.

The exam will have 8 questions, 7 to be answered, just like the last year.

For exam-related problems look in TCD past examination papers and Mathematics department examination papers.

I will appreciate any (also critical) suggestions that you may have for the next term. Let me know your opinion, what can/should be improved, avoided etc. and I will do my best to follow them. Please use the feedback form from where you can also send anonymous messages.