scimath
Green Function
A pun on a common path integral Oil on canvas, 12" by 12" original Green functions are a special type of integral. In calculus, integrals provide methods for adding up functions and trends. Everyday single integrals work in 2 dimensions. A function creates a curve on the plane and solving its integral tells you how much total area the function covers. A green function is a path integral. Path integrals are special kinds of integrals. Imagine that the normal integral measures the area under a curve sitting on a plain piece of paper. If the same curve was drawn on top of a paper with a picture on it, a normal integral would still tell you how much of the paper's area was under the function. But what if you wanted to know the "value" of all of the colors the function passed through? That would require a path integral. Path integrals can be very useful for examining functions that exist in fields - where instead of an empty paper or box, each piece of the space holding the function has values associated with it (electric field strength, color, temperature, etc.). Green functions in particular are used to describe particle motion in fields. They can be translated into little squiggles - Feynman Diagrams - which can be worked geometrically to simplify the set of integrals before solving. In polymers, we use an integral called the "Edwards Integral" instead, which is a close relative of the Green function. In this case the "path" follows the actual shape of the polymer backbone. But Edwards integrals don't really suggest a reason for lots of lovely green paint. In the course of learning arithmetic, algebra, calculus and other mathematics I've always pictured the numbers, variables, and equations as animate. Perhaps assigning personalities an backstories to numerals was my way of compensating for not having any dolls to play with. Of course Green functions are crazy loopy beasts in a scintillating blue field. This one looks like it suffered some injury when the first year grad students attempted numerical integration (and is that a failed attempt at renormalization I see?) That's one hairy function!
Artist: Regina Valluzzi
Green Function
A pun on a common path integral Oil on canvas, 12" by 12" original Green functions are a special type of integral. In calculus, integrals provide methods for adding up functions and trends. Everyday single integrals work in 2 dimensions. A function creates a curve on the plane and solving its integral tells you how much total area the function covers. A green function is a path integral. Path integrals are special kinds of integrals. Imagine that the normal integral measures the area under a curve sitting on a plain piece of paper. If the same curve was drawn on top of a paper with a picture on it, a normal integral would still tell you how much of the paper's area was under the function. But what if you wanted to know the "value" of all of the colors the function passed through? That would require a path integral. Path integrals can be very useful for examining functions that exist in fields - where instead of an empty paper or box, each piece of the space holding the function has values associated with it (electric field strength, color, temperature, etc.). Green functions in particular are used to describe particle motion in fields. They can be translated into little squiggles - Feynman Diagrams - which can be worked geometrically to simplify the set of integrals before solving. In polymers, we use an integral called the "Edwards Integral" instead, which is a close relative of the Green function. In this case the "path" follows the actual shape of the polymer backbone. But Edwards integrals don't really suggest a reason for lots of lovely green paint. In the course of learning arithmetic, algebra, calculus and other mathematics I've always pictured the numbers, variables, and equations as animate. Perhaps assigning personalities an backstories to numerals was my way of compensating for not having any dolls to play with. Of course Green functions are crazy loopy beasts in a scintillating blue field. This one looks like it suffered some injury when the first year grad students attempted numerical integration (and is that a failed attempt at renormalization I see?) That's one hairy function!
Artist: Regina Valluzzi