Vector calculus

“A man can have many minds and many lovers, but only one normalised unit vector.”

~ Oscar Wilde on vectors

“I know not what darkness lurks in the hearts of men, but it can escape only if the instantaneous div is strictly positive.”

~ Joseph Conrad on vector calculus

“It's what put the $\nabla$ in Del Taco!”

~ Professor Frink

Vector calculus is a branch of mathematics invented by Gibbs and Heaviside in the Good Old Days. Vector calculus lets us predict the behavior of magnetism, gravity, and the X factor discovered by the Fantastic Four.

Vector calculus is applicable throughout Euclidean space (at least the portion that is zoned Residential, $\mathbf {R} ^{3}$). (Also the portion in Soviet Russia zoned Я₃.) A more generalized approach called geometric algebra applies to other spaces, including the following:

• Warp Space, where the laws of physics are extended to get our heroes out of tight spots, and
• Parking spaces in Latin American capitals, where a tiny calculus (the calculito) is used, along with a shoe-horn.

Basic objects

Vector calculus operates on a vector field, which the reader can visualize by taking a drive through Eastern Washington state and admiring the amber waves of grain. This can be either a wheat field or a dental-floss farm. Each individual stalk is like a vector. Any breeze blowing through the field sufficient to create those lyrical waves will illustrate vectors at play. Readers elsewhere can visualize vector fields using LSD, with only slight loss of relevance.

Vector calculus, the soup

The vector calculus with which the reader will be most familiar is a variety of fish soup known for its salty aftertaste. That it is part of haute cuisine ("fancy cooking") is a given, but vector calculus asks us to consider how fast it is getting hauteur, and in what direction all this is pointing.

The soup is served in three flavors, although only a true master of the form differentiates them.

Grad

Grad is the simplest flavor to prepare.

Grad is the simplest flavor to prepare, and is often denoted by an inverted circle, which also denotes the number of added ingredients. To prepare grad, simply take a can opener, pour the contents over scalars, and heat until the soup rises to the level of grad. One can then slurp down the dot product, though if the chef is having a bad day, it may be a cross product.

Div

Just you quit staring at her div.

“It's my law and order now, biatch.”

~ James "Cocky" Maxwell on classifying Gauss' Law into one of the Four Food Groups

Div measures whether stuff is going into the pot or coming out. If overheated and the phone rings, Div generally comes out at a high volume density.

Div requires more complex ingredients than grad, including a carefully prepared "multidimensional" soup. The soup is available from various vectors. Chop the soup finely so that no piece is wider than Δx; then delicately mix the pieces back together, thickening with gum Arabic or gum Chewing; serve hot.

As to how many people it feeds, simply feast your eyes on this:

$\operatorname {div} \,\mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {\partial U}{\partial x}}+{\frac {\partial V}{\partial y}}+{\frac {\partial W}{\partial z}}.$

That is pretty impressive, although most of the letters are either upside-down or backwards.

Curl

Curl is the most difficult soup to prepare.

Main article: Curling

Curl is by far the most difficult soup to prepare, as it is involves the crossing of two exquisitely structured soups into a single whole. This soup has a tendency to curl around a point. The recipe is outside the scope of this article, but the soup is stirred during cooking using a technique called the right-hand rule. In the Southern Hemisphere, it curls in the opposite direction, and local law thus specifies a left-hand rule.

Curl does not generalize to the non-Euclidean spaces mentioned at the start of this article. If you are out there, it may be best to simply stop in at a Taco Bell.

The term Curl was first applied by Manfred Mann in his 19th-Century treatise, Blinded by the Light. Kelvin Stokes, known on-stage as eminently talented rapper Twista, consumes Curl before making serious artistic decisions, and numerous dances (notably, the Twist) have come from this soup. Apparently Richard Nixon was under the influence of Curl when he notoriously opined that his adversaries should "twist slowly, slowly in the wind" before getting deep-sixed (itself a term steeped in mathematics).