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