Parametric Surfaces

How do you turn a one-dimensional vector-valued function r(t) into two-dimensional surface? Rhetorically, that's like asking how to turn f(x) into f(x,y)--you just add another variable. But instead of making the vector function r(t,z), the conventional form is r(u,v).

For instance, take <cos(u), sin(u), v>, where u ranges from 0 to 2pi and v ranges from 0 to 3. The x- and y-components dictate that in the xy-plane, you have a circle. And what does that circle do? It gets extruded upwards to 3. The end result is a cylinder with radius 1 and height 3.

What is the purpose of having another way to describe a surface? The primary reason is because the function is easier to interpret.

In the 2D plane, we're familiar with:

as a circle of radius 3. But in 3-space, that becomes a cylinder. But what's to distinguish one from another? In parametric, however, this difference is clear. 

Hyperbolic Trig Functions

And just when you thought that you were rid of them for the rest of your math career, they're back (with a vengeance)! Well...not quite. We should all be familiar with the family of trigonometric functions: sine, cosine, tangent, secant, co-secant, and co-tangent. And we shouldn't forget all of their lovely derivatives. And the derivatives of their inverse functions. Well, don't fret when I tell you that there's more.

Hyperbolic trig functions are a separate class of functions that look like sines and cosines, and similarly, also have somewhat familiar derivatives and other properties. For instance, take sinh (x), the hyperbolic sine

function, pronounced "sin-sh". Though it is defined as:

 it's derivative, like sin (x) is just what you'd think: cosh (x), the hyperbolic cosine, pronounced "cosh". But what's interesting is that the derivative of cosh (x), defined as:

isn't -sinh (x), like how the derivative of cos(x) is -sin(x). Rather, it's just sinh(x). So the sinh(x) and cosh(x) have cyclic derivatives!

Some other "look alike" derivatives of hyperbolic trig functions include:

As of yet, I haven't noticed any pattern to help memorize which derivatives are "mirror-images" of each other, but if anyone does discover something, drop a comment!