Course Outline. 2. Vectors in V 3.

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1 1. Vectors in V 2. Course Outline a. Vectors and scalars. The magnitude and direction of a vector. The zero vector. b. Graphical vector algebra. c. Vectors in component form. Vector algebra with components. The standard basis vectors ı, j. 2. Vectors in V 3. a. The component form of a vector in V 3. The standard basis vectors ı, j and k. b. The magnitude of a vector in V 3. The distance between two points in R 3. c. Vectors in V 3 in component form. Vector algebra with components. The standard basis vectors ı, j and k. d. Parallel vectors. Unit vectors. 3. The dot product. a. The definition and basic properties of the dot product. b. The angle between two vectors. Orthogonal vectors. c. The Cauchy-Schwarz and triangle inequalities. d. Components and projections. 4. The cross product of vectors in V 3. a. 2 2 and 3 3 determinants. b. The definition and basic properties of the cross product. c. The right-hand rule for the direction of the cross product. d. The magnitude of the cross product. The area of a parallelogram. The volume of a parallelpiped. e. The torque vector. 5. Planes in R 3. a. The equation of the plane through a given point normal to a given vector. b. The equation of the plane through three (non-colinear) points. c. The equation of the plane, parallel to another, through a given point.

2 6. Vector-valued functions. a. Functions from R to V 2 or V 3. b. Curves traced by vector-valued functions. c. Limits, derivatives and integrals of vector-valued functions. The tangent (derivative) vector to a curve traced by a vector-valued function. d. The arclength integral. 7. Motion in space. a. Position, velocity, speed and acceleration. b. Newton s second law. c. Projectile motion. 8. Functions of several variables. a. Functions taking R 2 or R 3 to R. Their domains and ranges. b. Graphs, contours, density plots and traces of functions of two variables. c. Level surfaces of functions of three variables. 9. Limits and continuity for functions of several variables. a. Limits of functions of several variables. b. Non-existence of the limit when the value of the function at a point depends on the path to the point. c. Calculation of limits. d. Continuity at a point. 10. Partial derivatives. a. The definition of a partial derivative of a function of two variables. b. Calculation of partial derivatives. c. Higher-order partial derivatives. d. Equality of certain mixed partial derivatives. e. Partial derivatives of functions of three variables. 11. Linear approximation. a. The tangent plane to the graph of a function of two variables. b. Differentials. The principle of (local) linear approximation, versions I and II. c. Linear approximation and differentials of functions of three variables. d. The definition of differentiability for functions of several variables.

3 12. The chain rule. a. The chain rule for various compositions: z(t) = f(x(t), y(t)), u(x, t) = f(θ(x, t)), w(p, q) = f(x(p, q), y(p, q)), etc. Tree diagrams. b. Implicitly defined functions. Partial derivatives of implicit functions. 13. Gradients and directional derivatives. a. The gradient of a function of several variables. b. The derivative of a function at a point p in a unit direction u. c. Properties of the gradient. Direction and rate of most rapid increase. The orthogonality of g to the contour surfaces (or curves) of g. 14. Extrema of functions of several variables. a. Relative (or local) extrema and saddle points. b. Critical points. c. The second derivative test for functions of two variables. d. Absolute extrema. 15. Constrained optimization. a. The Lagrange multiplier method for identifying constrained extrema. b. Finding absolute extrema over a region bounded by a curve or surface. c. The Lagrange multiplier method with several constraints. 16. Double integrals. a. The double integral over a rectangular and nonrectangular regions. b. Iterated integrals over rectangles and regions bounded by curves. Fubini s theorem. Changing the order of integration. c. Applications of the double integral: Area, volume, moments, mass and center of mass. d. The surface area integral. 17. Triple integrals. a. Triple integrals over boxes and more general regions. b. Iterated integrals with dv = dx da, dv = dy da and dv = dz da. Fubini s theorem. Changing the order of integration. c. Applications of the triple integral: Volume, moments mass and center of mass.

4 18. Change of variable in double and triple integrals. a. The Jacobian determinant. The change of variable formulae dx dy = (x, y) (u, v) du dv, and dx dy dz = (x, y, z) (u, v, w) du dv dw. b. Integrals in polar, cylindrical and spherical coordinates. c. Examples of nonstandard changes of variable. Ellipsoidal coordinates. 19. Vector fields. a. Two and three dimensional vector fields. b. The flowlines of a vector field. c. Conservative, or gradient fields. Potential functions. d. The curl of a vector field. The curl test for conservative fields. e. Finding the the potential of a conservative field. 20. Line integrals. a. Line integrals of functions. b. Line integrals of vector fields over oriented curves. Calculation of the work done traversing a curve in a force field. c. Evaluation of line integrals by parametrization. 21. Path independent line integrals. a. Path independent line integrals and conservative vector fields. b. The fundamental theorem of line integration. c. Line integrals over closed paths. 22. Surface integrals. a. Integrals of functions over surfaces. b. Integrals of vector fields over surfaces (flux integrals). c. Calculation of flux integrals.

5 23. The divergence theorem. a. The divergence of a vector field. b. The physical interpretation: Divergence as the infinitesimal flux per unit volome. c. The statement and justification of the divergence theorem. d. Applications of the divergence theorem. Computing flux integrals over closed surfaces, or by replacing a surface with a closed surface. Flux integrals of inverse square vector fields. 24. Stokes theorem and Green s theorem. a. The physical interpretation of curl F n as the infinitesimal circulation per unit area normal to n. b. Stokes theorem. c. Applications of Stokes theorem: Computing line and surface integrals. Showing that F is conservative if curl F = 0. d. Green s theorem as the two-dimensional version of Stokes theorem.