# User:Danenberg

## Provenance

Of Gens Danenbergorum; Peter Charles Danenberg, Jr.

## *Officium*

MediaWiki Development; namely WikiTeX

## Sandbox

### Direction field

Reproduced from William E. Boyce and Richard C. DiPrima, *Elementary Differential Equations and Boundary Problems* (Hoboken: John Wiley, 2005), 35.

### Vector field

Gradient vector field of along with its contour map, courtesy of a beautiful hack by Thomas Sefzick.

### Tangent planes

At increasing levels of detail, the surface approaches its tangent plane around P.

#### Falling leaf

#### Anti-intuition

### Linear independence

#### Problem

If for , show that is linearly independent on every interval for all fixed .

#### Wronskian

#### Solution

has a non-zero Wronskian on ; and is therefore linearly independent on that interval.

### Clairaut's Theorem

That does not contradict Clairaut's Theorem , where and are permuted indices of with replacement, since and are not continuous.

### Violent curiosity

Tangent at a point of plane-ellipsoid intersection:

### Partial derivatives

### Limit

That--doesn't-exist's graphical representation.

### Family of curves

### Curve with contour

### Erotic paraboloid

Context from `#math` on freenode:

00:57 <klutometis> I've come up with a new nomination: the "erotic paraboloid" 00:57 <klutometis> http://wikisophia.org/wiki/User:Danenberg#Erotic_paraboloid 00:58 <thermoplyae> you appear to have a lot of free time 00:59 <klutometis> thermoplyae: me? it corresponds roughly with calculus; so no expenditure of free time 00:59 <TRWBW> klutometis: hehe, n1, something just like it at http://www.youtube.com/watch?v=eBGIQ7ZuuiU 01:00 <thermoplyae> you say that, but all the busy people i know don't draw graphs of "co-sinusoidal demi-toroid"s 01:00 <klutometis> damn it, TRWBW; I can't believe I got rick rolled 01:00 <thermoplyae> just sayin' 01:00 <thermoplyae> haha 01:01 <klutometis> thanks, boys; good night

### Osculating cirles

### Tangent, normal, binormal

### Curvature of e

### <t, t², t³> ∩ <sin(t), sin(2t), t>

### Cylinder ∩ paraboloid

### Co-sinusoidal demi-toroid

### Toroidal spiral

### Orthogonal matrices

#### Problem

#### Inverse matrix

#### Solution

### Surfaces

### Fugale Gelehrtheit

### Index conjecture

Let be the index set to the n-ary Cartesian product of set of cardinalities . Therefore, ; let .

Let, furthermore, be the index of set corresponding to .