MAT 372 Topics

• 5.1: R^n, vector space structure, standard basis, dot product, norm, distance, sup norm, Cauchy-Schwartz Inequality, Triangle Inequality, angle, orthogonality, parallel vectors, line segment, open ball, hyperplane, normal vector, linear function, matrix representation
• 5.2: open set, closed set, unions and intersections, Lindelof Theorem, boundary, interior, closure, relations among these
• 5.3: sequences in R^n, bounded, convergent, compatibility with components, sums, scalar multipl,es, dot products, Bolzano-Weierstrass Theorem, Cauchy Criterion, sequences and closure, sequentially compact, compact, Cantor Intersection Theorem, Heine-Borel Theorem
• 5.4: convex, connected, relatively open or closed, relatively open sets and connectedness, connectedness in R
• 5.5: cluster point, limit of a function, sequences and limits, components and limits, sums, scalar multiples, dot products, norms, quotients
• 5.6: continuous at a point, continuous on a set, continuous, continuity and limits, continuity and sequences, compositions and continuity, compositions and limits, continuity and open sets, continuity and compactness, continuity and connectedness, Extreme Value Theorem, Intermediate Value Theorem, uniformly continuous, uniform continuity and compactness
• 6.1: partial derivatives, partial integrals, C^p, C^infinty, mixed partials, continuity of partial integrals, partial derivatives of partial integrals, uniform convergence of partial improper integrals, Weierstrass M-Test, continuity and partial derivatives of partial improper integrals
• 6.2: (total) derivative, differentiability, Jacobian matrix, Jacobian, "partial Jacobian", matrix representation of derivative, differentiabiltiy and continuity, C^1 and differentiability, tangent vector, tangent plane
• 6.3: error term and differentiability, derivatives of linear functions, algebraic rules for derivatives, chain rule
• 6.4: Mean Value Theorem, Taylor's Formula
• 6.5: Inverse Function Theorem, Implicit Function Theorem
• 7.1: rectangle, volume, grid, finer grid, inner and outer sums, inner and outer volume, Jordan region, volume, volume zero, nonoverlapping, Jordan regions and volume 0 boundary, volume 0 and cubes, set operations and volume, Jordan regions and smooth images
• 7.2: upper and lower sums, upper and lower integrals, Riemann integral, integrability, uniform continuity and integrability, volume and integral, linearity and additivity of integrals, integral and volume 0, positivity and inequalities, Mean Value Theorem for integrals
• 7.3: iterated integral, Fubini's Theorem for rectangles, extension from Jordan region to rectangle, Fubini's Theorem for projectable regions
• 7.4: Change of Variables Theorem
• ? 7.5: compact support, C^infinity-Urysohn's Lemma, C^infinity-Partitions of Unity, locally integrable, integral over bounded open set, Change of Variables for bounded open sets, Change of Variables with singularities