Syllabus

1. Euclidean Spaces
   1. Vectors and vector algebra in 2 and 3 dimensions.
   2. Inner products
   3. Projection
   4. Lines and planes
   5. The Vector product
2. Functions of several variables
   1. Graphs and level curves
   2. Limits, continuity
   3. Partial derivatives
   4. Differentiability and gradient
   5. Linear approximation
   6. The chain rule
   7. Tangent plane
   8. Directional derivative
   9. Conservation of Energy
3. Curves
   1. The idea of a curve.
   2. Curves in two and three dimensions and examples.
   3. Arclength
   4. Reparametrization of nonsingular curves.
   5. Curvature and torsion. The Frenet formulae.
4. Higher derivatives
   1. Repeated partial derivatives
   2. Partial differential operators
   3. Taylor’s theorem.
5. Optimization
   1. Extrema of real valued functions
   2. Quadratic forms
   3. Lagrange multipliers
6. Multiple integrals
   1. Iteraled integrals
   2. Fubini’s theorem
   3. Geometry of maps from $\RR^2$ to $\RR^2$
   4. Determinants and jacobians
   5. The change of variable theorem
   6. Cartesian, polar, cylindrical, and spherical coordinates
   7. Inverse mappings and implicit functions
7. Line and Surface Integrals
   1. Line integrals
   2. Conservative fields
   3. Parametrization of a surface
   4. Area of a surface
   5. Surface integrals of scalar functions
   6. Surface integrals of vector functions
8. Vector Field Theory
   1. Green’s theorem
   2. Gauss’ theorem
   3. Divergence theorem
   4. Stokes’ theorem