MAT 205:  Vector Calculus          Topics for second exam
The exam will cover nearly most of chapters 10 and 11, along with section 12.1 There will be no questions on curvature, nor on the tangential and normal components of acceleration, nor on sections that we did not cover (such as 11.8: Lagrange Multipliers). The exam will be about an hour long, with some very easy questions and some more difficult ones. Expect at least one word problem. You should be familiar with all of the following topics:


Arc Length: Be able to compute the arc length of a vector valued function, as the integral $\int_a^b \vert R'(t)\vert dt$. Know what it means for a curve to be parameterized by arc length, and given a parametric curve, reparameterize it by arc length.


Tangent, Normal, and Binormal vectors: Given a parametric curve R(t), be able to calculate the unit tangent vector $T(t) = \frac{R'(t)}{\vert R'(t)\vert}$, the unit normal $N(t)=\frac{T'(t)}{\vert T'(t)\vert}$, and the unit binormal $B(t) = T(t) \times
N(t)$. Given those, you should be able to find the normal plane to the curve at a given point, and the osculating plane (the latter is the plane with B(t) as a normal vector.


Motion in space: If the position of a particle is given by R(t), its velocity vector is V(t) = R'(t), and the acceleration is given by A(t)=V'(t)=R''(t).


Parametric Surfaces: Be able to match a surface to its parametric representation, and to describe how a parametric surface changes as its domain is adjusted.


Level Curves and Contour plots: Understand what level curves are, what a countour plot is, and their relationship to the graph of a surface.


Limits and Continuity: Be able to compute limits of a function of two variables, and be able to determine when a function of more than one variable is continuous.


Partial derivatives: Be able to compute partial derivatives and second partials. Understand what they represent. Know how to use the chain rule to compute partial and total derivatives of compositions.


Tangent planes, linear approximations: Be able to find the tangent plane to a surface, and use it to approximate a function.


The gradient vector; directional derivatives: Know what the gradient vector is, and its properties. For example, that $\nabla f$ points in the direction of greatest increase of f, and that $\nabla f$ is always perpendicular to level curves of f. Understand what the derivative in the direction of a unit vector u is, and how to compute $D_u f = \nabla f \cdot u$.


Critical points, Maxima and Minima: Be able to locate critical points of functions of several variables. Be able to use them to locate local maxima and minima. Also be able to find extreme values (i.e. absolute maximum and minimum values) of functions defined over closed, bounded regions of the plain. Use the second derivative test where necessary.


Definition of double integrals: Understand the definition of the double integral. Specifically, be able to do problems like those assigned in section 12.1.


 

Scott Sutherland
2000-11-08