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Syllabus for Math 53. Theory (1) Functions and their graphs. Functions studied in Multivariable Calculus. (2) Vectors and their linear combinations. (3) Coordinates. (4) The dot-product and its bilinear property. (5) Areas and 2x2-determinants. (6) Volumes and 3x3-determinants. (7) Cross-product and its properties. (8) Linear functions, lines and planes. (9) Quadratic functions in 2 variables. (10) Classification of quadratic curves. (11) Quadratic surfaces. (12) Polar, cyclindrical and spherical coordinates. (13) Vector-functions and parametric curves. (14) Functions in several variables, their graphs and level sets. (15) Limits and continuity. (16) Partial derivatives and differentiability. (17) Can a discontinuous function be differentiable? have partial derivarives? (18) Taylor’s formula, and Clairaut’s theorem. (19) Composite functions and the chain rule. (20) Gradient and directional derivatives. (21) Maxima, minima and critical points. (22) Classification of critical points in 2 variables. (23) Constrained extrema, and Lagrange’s method. (24) lagrange’s method in the case of two (or more) constraints. (25) Integration over rectangles and boxes, and Fubini’s theorem. (26) Change of variables in multiple integrals. (27) Moments of inertia. (28) Vector fields and their flow lines. 1 2 (29) Conservative vector fields, and the Newton-Leibnitz theorem. (30) Green’s theorem. (31) Proof of Green’s theorem: additivity with respect to regions. (32) Proof of Green’s theorem: invariance with respect to reparameterization. (33) Proof of Green’s theorem: reduction to the Fundamental Theorem of Calculus. (34) The del-operator, and its properties. (35) Newton’s gravitation force field. (36) Are curl-free vector fields conservative? (37) Rotation, its velocity vector field, and its curl. (38) Dynamical interpretation of curl. (39) Parametric surfaces and their areas. (40) Circulation and flux. (41) Stokes’ theorem. (42) Gauss’ theorem. (43) Dynamical interpretation of divergence. (44) Is a divergence-free vector field a curl? (45) Electrostatic fields and Faraday’s ”force lines”.