Problem Sheets in PDF: Sheet 1 Sheet 2 Sheet 3 Sheet 4 Sheet 5 Sheet 6 Sheet 7 Sheet 8 Sheet 9 Sheet 10 Sheet 11 Sheet 12 Sheet 13 Sheet 14 Sheet 15 Sheet 16 Sheet 17 Sheet 18 Sheet 19 Sheet 20

Solutions to the exercises in PDF: Sheet 1 Sheet 2 Sheet 3 Sheet 4 Sheet 5 Sheet 6 Sheet 7 Sheet 8 Sheet 9 Sheet 10 Sheet 11 Sheet 12 Sheet 13 Sheet 14 Sheet 15 Sheet 16 Sheet 17 Sheet 18 Sheet 19 Sheet 20

Course outline:

**Functions in several variables (Chapter 11 in Thomas' book).** Domains and Ranges. Graphs and Level Curves. Level Surfaces. Limits and Continuity. Partial Derivatives. Implicit Partial Differentiation. Second-Order and Higher Order Partial Derivatives. The Chain Rule. 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 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). 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.

Multivariable Calculus Online by Jeff Knisley

Functions of Several Variables by Math Dept of LTCC

Graphical representation of functions by Martin J. Osborne

Taylor's formula, linear and quadratic approximations by Eric A. Carlen

Older 2E1 web page with further helpful links by Fermin Viniegra with many interesting links.

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