3 ms·

I don't see why E = 1/2 k x^2 isn't a quadratic form. It's a quadratic form in a vector space of dimension 1 (here, the real numbers), which matrix is just (1/2

by cpa 15y ago

I don't see why E = 1/2 k x^2 isn't a quadratic form. It's a quadratic form in a vector space of dimension 1 (here, the real numbers), which matrix is just (1/2 k). In fact here, we're dealing with a (degenerated) conic. What's your point? Should the author have written E(x) = ...? Besides, tau is pointless because a tau pie isn't nearly as fun as a pi pie.

scythe 15y ago
At best, it's a cubic form, since the spring constant is also a variable. Similarly 1/2 a t^2 is a cubic form. pi r^2 however is a quadratic form.
- cpa 15y ago
  Nope, check the introduction section of your wikipedia link. It's like saying that f(x) = ax is a function of two variables. Indeed it is, but it's commonly accepted in mathematics to use it as a function of one variable (x), with a parameter a.
  - scythe 15y ago
    Unless you take every spring in the Universe to have the same spring constant (nonsense), k must also be treated as a variable. The energy relation for an individual spring is a quadratic form, but the energy relation for all springs is either a cubic form or a set of quadratic forms (or a bijection from the real line to the set of unary quadratic forms, if you want to be really technical and boring). f(x) = a * x is an example of a function of x which only becomes a function when a value is assigned to the variable 'a'. In other words, it is a type of function. >Indeed it is, but it's commonly accepted in mathematics to use it as a function of one variable (x), with a parameter a. In introductory calculus, sure; in those areas of mathematics where the term 'quadratic form' is actually used, things have to be defined more precisely.
    - cpa 15y ago
      Ah, I think I see your point. You would be happier with E = 1/2k x^2 is a family of quadratic forms, indexed by the real number k, I guess. A bit nitpicky to me (as long as k != 0), but fair.
    - pixcavator 15y ago
      Does this help: f_a (x) = a * x? Is this a function of x in your opinion?
      - fexl 15y ago
        There's my cue to drag in the lambda calculus: \g = (\a\x ...) \f = (g a) Now f is is a function of x.

by cpa 15y ago

scythe 15y ago

At best, it's a cubic form, since the spring constant is also a variable. Similarly 1/2 a t^2 is a cubic form. pi r^2 however is a quadratic form.

cpa 15y ago
Nope, check the introduction section of your wikipedia link. It's like saying that f(x) = ax is a function of two variables. Indeed it is, but it's commonly accepted in mathematics to use it as a function of one variable (x), with a parameter a.
- scythe 15y ago
  Unless you take every spring in the Universe to have the same spring constant (nonsense), k must also be treated as a variable. The energy relation for an individual spring is a quadratic form, but the energy relation for all springs is either a cubic form or a set of quadratic forms (or a bijection from the real line to the set of unary quadratic forms, if you want to be really technical and boring). f(x) = a * x is an example of a function of x which only becomes a function when a value is assigned to the variable 'a'. In other words, it is a type of function. >Indeed it is, but it's commonly accepted in mathematics to use it as a function of one variable (x), with a parameter a. In introductory calculus, sure; in those areas of mathematics where the term 'quadratic form' is actually used, things have to be defined more precisely.
  - cpa 15y ago
    Ah, I think I see your point. You would be happier with E = 1/2k x^2 is a family of quadratic forms, indexed by the real number k, I guess. A bit nitpicky to me (as long as k != 0), but fair.
  - pixcavator 15y ago
    Does this help: f_a (x) = a * x? Is this a function of x in your opinion?
    - fexl 15y ago
      There's my cue to drag in the lambda calculus: \g = (\a\x ...) \f = (g a) Now f is is a function of x.