MATH 251 section 505, Fall 2006, Final TEST
Calculators are OK, but not necessary. There will be 10 problems of weight 5 and 10 pts
for a maximal score 75 pts. For full credit you need to show the whole work.
The problems are aimed to test your knowledge (definitions and main results), under-
standing of the material (main mathematical ideas and techniques) and some basic appli-
cations. The test will be based on your homework assignments. In particular the problems
will cover the following material:
1. Vectors in 2 and 3 dimensions. Dot and cross-products, properties, applications.
2. Equations of lines and planes in 3-D. Tangent planes and normal lines. Quadric sur-
faces: identification, characterization, intersections.
3. Vector functions and space curves: derivatives, integrals. Arc length and curvature
(various parameterizations, normal vector).
4. Functions of many variables, limits and continuity (a must). Partial derivatives, chain
rule. Directional derivatives and gradient vectors.
5. Local extrema of functions of 2 variable. Critical point, saddle point. Absolute mini-
mum and maximum values. Method of Lagrange multipliers for minimizing or maxi-
mizing a function subject to constraints (this whole part is an absolute must).
6. Double integrals over rectangle, iterated integrals, Fubini’s theorem. Triple (volume)
integrals. Application of multiple integrals in mechanics (mass, center of mass, etc).
7. Vector fields in 3-D, conservative vector fields, line integrals of functions (of two and
three variables) and application and line integrals of vector fields (this is an absolute
must).
8. Surface area and surface integrals of functions. Surface integrals of vector fields and
application (an absolute must).
9. Differential operators over vector fields, curl,grad, and div, and their properties.
10. Green’s Theorem and applications (this is an absolute must).
11. Stokes’ Theorem and applications a must.
12. Good luck