I thought I would collect here all of the big ideas that I think anyone who successfully completes this course should have a good grasp on.
Vector geometry in 2 and 3 dimensions
Not only should you know all of the different ways to manipulate vectors algebraically but you should know what the manipulations mean. What are you doing when you are adding two vectors? What does the dot product represent? What does the cross-product represent?
Also important is to understand the basic examples of surfaces in 3 dimensions; namely, planes and quadric surfaces. Cylindrical and spherical coordinates give good ways of studying 3 dimensions under certain circumstances.
Ultimately, I hope you learn to think in three dimensions. An important skill is being able to “slice” a solid in three dimensions with a plane and understand what the cross-section looks like.
Basics of vector functions
You should learn the basic idea of plotting a space curve and computing derivatives of space curves (and what this represents).
Functions of several variables and partial derivatives
Partial derivatives arise by treating one variable like it is constant and then taking a usual derivative with respect the remaining variable. You should gain proficiency in computing partial derivatives. You should also understand what they mean geometrically. You should be able to write down the equation of a tangent plane to the graph of a function of two variables. You should gain proficiency in using the chain rule for functions of two variables. The chain rule can be confusing if you aren’t fluent with the notation of partial derivatives. Be able to compute gradient vectors and understand what they mean. Again, if you understand how to slice a graph this is much easier conceptually. The same goes for finding maxima and minima of functions of two variables. Lagrange multipliers is a really useful technique and you should become proficient in using it.
Multiple Integrals
Understand the definition of multiple integrals and be able to compute iterated integrals. Things get tricky when you begin integrating functions over strange regions; you really have to understand three dimensions. Become proficient in computing surface areas, triple integrals, and triple integrals in other coordinates. Understand the basic idea of change of variables for functions of several variables and how to compute multiple integrals using change of variables.
