In general, Math 205 deals with the study of "objects" in space. As we have seen, such an object may be a surface. A surface can be represented as a function z = f(x,y). Think of a surface as a landscape with mountains, pits, valleys, and ridges. Now the same questions arise as in Math 115. What is the highest point of a surface? How do we find it? What is the slope as you go in a certain direction? Think about what happens if you stand on a mountain top, i.e., the highest point of a surface. No matter in which direction you go, the slope is initially zero. At such a point the tangent plane to a surface will be horizontal (analogous to the tangent line in Math 115). Once you know how to identify highest and lowest points of a surface this concept can be applied to various optimization problems. If you have mastered the material of Math 115 you will most certainly understand all the ideas of Math 205. However, expect the computations to be a little bit more involved.
One new question arises: Suppose you travel from Boston to Los Angeles and you would like to know what is your highest elevation along the trip? You are not asking for the highest point in the US (which is Mount McKinley in Alaska). You only want to know the highest point along the route you take (which is probably Vail pass in Colorado). This leads to Lagrange multipliers, which deal with the problem of maximizing or minimizing a function along a certain curve and have many applications in other fields such as physics and economics.
In Math 116 you learned how to find volumes of solids of revolution. Most solids, however, are not solids of revolution. In Math 205, methods for finding volumes of arbitrary solids will be presented. This leads to the theory of multiple integrals.
The course finishes with line integrals and Green's theorem. This is a generalization of the Fundamental Theorem of Calculus, which you have encountered in Math 115. Green's theorem is a beautiful theorem that combines parametric curves, derivatives, and multiple integrals. You will see how the various concepts you have learned all come together to form a single theory. Line integrals, which can be used to compute the work done on a moving particle if a force acts on the particle, are one of the major tools in physics and engineering.
Prerequisite: 116, 120, or the equivalent. Not open to students who have completed Math 216/Phys 216.
Distribution: Mathematical Modeling
