Math 13, Multivariable calculus, Course Summary ----------------------------------------------- 24 lectures + 1 bonus lecture. 1. Vectors and vector products. Vector valued functions, limits. 2. Differentiation of vector valued functions, velocity, smoothness, properties of the derivative. 3. Arc length, speed, unit tangent, curvature, principal normal. 4. Tangential and normal components of acceleration. Radius of curvature. Functions of several variables, graphs, level curves, level surfaces. Limits and paths of approach. 5. Properties of limits. Continuity. Partial derivatives. Equality of second partial derivatives. Higher order derivatives. 6. Geometric interpretation of partial derivatives. Equation of tangent plane. Linear approximation, differentials and differentiability. 7. The chain rule: 1 independent variable, 2 independent variables, general case, implicit differentiation. 8. The gradient vector: 2 variable case, steepest ascent, directional derivatives, gradient perpendicular to level curves. 3 variable case, gradient perpendicular to level surfaces, equation of tangent line/plane. 9. Maxima and minima, critical points, saddle points, second derivatives test. 10. Maxima and minima in 3 or more variables, critical points, second derivatives test, completing the square, checking the boundary. Constrained maxima/minima, Lagrange multipliers. 11. Lagrange multipliers in 3 or more variables. 12. Definition of double integrals, calculation of volumes, iterated integrals. 13. Iterated integrals, type I and type II regions, changing the order of integration. Double integrals in Polar coordinates. 14. Polar coordinates, general change of variables formula for double integrals, Jacobians, formula for the area element. 15. Change of variables example. Triple integrals, definition, iterated integrals, regions of different types. The six possible orders of integration. 16. Change of variables in triple integrals. Formula for the volume element. Cylindrical coordinates. Spherical coordinates. Vector calculus: vector fields. 17. Conservative vector fields. Conservation of energy. Line integrals. Physical examples of vector fields. Physical interpretation of line integrals. Properties of line integrals. Example of line integrals that are not path independent. 18. Conservative vector fields and path independence. Method for finding the potential of a conservative vector field. Differential condition for conservative vector fields on simply connected regions in the plane. Parametric surfaces, examples, graphs, surfaces of revolution. 19. Surface area formula. Finding the surface element. Surface area of a sphere. Surface area of a graph. Scalar surface integrals. Orientation of surfaces, orientable and nonorientable surfaces. Vector surface integrals. The surface integral of the point charge field over an enclosing sphere. 20. Interpretation of vector surface integrals. The flux through a surface. Finding the unit normal. Finding the vector surface element. Calculation of vector surface integrals. 21. Divergence and Curl. Definition and basic properties. Interpretation of divergence. Flow per unit volume. Interpretation of curl. Circulation per unit normal surface area. The right hand rule. 22. Advanced properties of divergence and curl. curl grad = 0, div curl = 0, and the converse statements. Stokes theorem. The right hand rule. Orienting the boundary correctly. Using Stokes theorem to calculate surface integrals. 23. Proof that an irrotational vector field on a simply connected region is conservative. Proof of Stokes theorem. Cancellation of internal boundaries. Surfaces with no boundary. The Divergence theorem, correct orientation of the boundary. Using the divergence theorem to compute surface integrals. 24. Divergence theorem examples. Independence of surface integrals on the surface when the divergence is zero. The flux of a point charge through an enclosing surface. The divergence of the point charge field. Cancellation of internal boundaries. Proof of the divergence theorem. 25. Optional bonus lecture: Orthogonal curvilinear coordinates. Formula of the line and volume elements. Formula for gradient, divergence and curl in terms of the line element. Cylindrical coordinates. Spherical coordinates.