Integration by Parts

First let’s take a look at the following.

So, that was simple enough.  Now, let’s take a look at,

To do this integral we’ll use the following substitution.

Again, simple enough to do provided you remember how to do substitutions.  By the way make sure that you can do these kinds of substitutions quickly and easily.  From this point on we are going to be doing these kinds of substitutions in our head.  If you have to stop and write these out with every problem you will find that it will take you significantly longer to do these problems.

Paraboloid

The surface of revolution of the parabola which is the shape used in the reflectors of automobile headlights (Steinhaus 1999, p. 242; Hilbert and Cohn-Vossen 1999). It is a quadratic surface which can be specified by the Cartesian equation

(1)

The paraboloid which has radius  at height  is then given parametrically by

			(2)
			(3)
			(4)

where , .
The coefficients of the first fundamental form are given by

			(5)
			(6)

Asymptote

A line that a curve approaches, as it heads towards infinity:

Types

There are three types: horizontal, vertical and oblique:

it can be in a negative direction,

Singular Point

A singular point of an algebraic curve is a point where the curve has "nasty" behavior such as a cusp or a point of self-intersection (when the underlying field  is taken as the reals). More formally, a point  on a curve  is singular if the  and  partial derivatives of  are both zero at the point . (If the field  is not the reals or complex numbers, then the partial derivative is computed formally using the usual rules of calculus.)