Mathematics 226: study topics

Chapter 1, Vectors:

• Vectors in 2, 3, n dimensions.
• Lines: parametric equations, line through 2 given points
• Planes: normal vectors, plane through 3 points, parametric equations.
• Dot product, angle between vectors, projections.
• Cross product, applications: areas, volumes, normal vectors
• Computing distances using projections
• Vectors, hyperplanes, lines, projections in n dimensions.
• Spherical, polar, cylindrical coordinates.

Chapter 2, Differentiation in Several Variables:

• Functions, domain, range, one-one, onto.
• Graphing functions of 2 variables: level curves, contour curves, planar sections. Level sets of functions of 3 variables.
• Open and closed sets.
• Limits and continuity of functions of n variables.
• Partial derivatives, differentiability, linear approximation, tangent planes and hyperplanes, gradient and matrix of partial derivatives.
• Rules of differentiation, including Chain Rule.
• Higher-order partial derivatives, C^k functions.
• Directional derivatives, gradient and steepest ascent.
• Note: we are skipping Newton's method (Sec. 2.4).
• Note on the implicit and inverse function theorems (Sec. 2.6, Theorems 6.5-6.7): this was discussed only very briefly in class and will not be required on the final exam. We will return to these theorems in Math 227 when they are actually needed in applications.

Chapter 4, Maxima and Minima in Several Variables:

• Differentials, Hessian, first-order and second-order Taylor approximations. Formulas for remainder terms (pp. 242-243) are not required.
• Local and global minima and maxima, critical points, second derivative test, global extrema on compact regions.
• Lagrange multipliers (we are skipping Hessian Criterion, pages 266-270)

Chapter 5, Multiple Integration:

• Double and triple integrals: Riemann sums, iterated integrals over elementary regions, Fubini's theorem.
• Basic applications: areas, volumes.
• Changing the order of integration.