MAT 205:  Vector Calculus          Topics for first exam

The exam will cover nearly all of chapter 9 of the text, as well as sections 10.1 and 10.2. There will be no questions on spherical or cylindrical coordinates directly (§9.7), but ability to work with them could be helpful in some contexts.

The exam will be about an hour long, with some very easy questions and some more difficult ones. You should be familiar with all of the following topics:

Arithmetic of Vectors: elementary calculations including addition, subtraction, scalar multiplication, computing length, and so on.

The dot product: Definition ( $A\cdot B = \vert A\vert\vert B\vert\cos\theta$), calculation. Finding angles between vectors, computing a projection (proj_A B), etc. Know that $A\cdot B= 0$ if and only if A and B are perpendicular.

The cross product: Definition ($A\times B$ is perpendicular to both A and B, direction given by the right-hand rule, with $\vert A\times B\vert = \vert A\vert\vert B\vert\sin\theta$). Computation of $A\times B$ as a determinant. The fact that $A\times B = -B \times A$, and that $A\times B = 0$ if and only if A and B are parallel.

Lines: If P = (x₀, y₀, z₀) is a point on the line $\ell$, and V = <a,b,c> is a vector parallel to $\ell$, the line an be written in vector form as $\ell(t) = P + tV$, or equivalently in parametric form as

$$x(t) = x_0 + at \quad y(t) = y_0 +bt \quad z(t) = z_0 + ct,$$

or in symmetric form as

$$\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z - z_0}{c}.$$

You should be able to work with equations of lines. For example, write the equation of a line given a point on it and a vector parallel to it, given two points on it, etc. Furthermore, you should be able to determine if two lines intersect, and if so, find the point of intersection and calculate the angle between the lines. If they don't intersect, you should be able to calculate the distance between them (this is harder than the others).

Planes: A plane is most naturally described given a point on it and a vector perpendicular to it (a normal vector). If P = (x₀, y₀, z₀) is a point on the plane, and N = <a,b,c> is a vector normal to it, then the plane can be described as all points R=<x,y,z> for which $N\cdot (R - P) = 0$, or equvalently

a(x-x₀) + b(y-y₀) + c(y-y₀) = 0.

Any equation of the form ax + by+cz =d is a plane with normal vector <a,b,c>.

Graphs of functions of two variables and surfaces: You should be able to identify the graph of a function. You won't have to draw the graphs, merely identify them. Spherical and Cylindrical coordinates won't be directly used.

Curves as Vector-valued functions; calculus with them: Understand the description of a curve as a vector function R(t). Be able to differentiate and integrate vector functions. Computation of the tangent vector R'(t), the unit tangent T, and the equation of the line tangent to the curve R(t) at some point.