Representing nonlinear functions involving vectors

[Prez Cannady]

I'm having trouble finding textbook material on nonlinear functions on vectors. Just as I could define a function $f$ such that:

$$f(x) = cos(x)$$

I'd like to write something like:

$$f(\vec{x}) = \begin{pmatrix}
f_1(x_1) \\
f_2(x_2) \\
... \\
f_n(x_n)
\end{pmatrix} $$

where $f_i$ is some nonlinear function on $x_i$ like $cos(x_i)$.

Or, if you will...

$$f(\vec{x}) = f_1(x_1) + f_2(x_2) + ... + f_n(x_n)$$

or even more wild linear and non-linear manipulations of $x_i$.Would appreciate any help in finding resources on this topic. I don't even know what the name of this particular area of interest is, assuming it has one.

 

[WWGD]

I'm having trouble finding textbook material on nonlinear functions on vectors. Just as I could define a function $f$ such that:

$$f(x) = cos(x)$$

I'd like to write something like:

$$f(\vec{x}) = \begin{pmatrix}
f_1(x_1) \\
f_2(x_2) \\
... \\
f_n(x_n)
\end{pmatrix} $$

where $f_i$ is some nonlinear function on $x_i$ like $cos(x_i)$.

Or, if you will...

$$f(\vec{x}) = f_1(x_1) + f_2(x_2) + ... + f_n(x_n)$$

or even more wild linear and non-linear manipulations of $x_i$.Would appreciate any help in finding resources on this topic. I don't even know what the name of this particular area of interest is, assuming it has one.

Isn't this done in Standard (multivariate) Calculus?

 

[andrewkirk]

Would appreciate any help in finding resources on this topic. I don't even know what the name of this particular area of interest is, assuming it has one.

The best term to use to look for resources on that is multivariable calculus.

It sounds like you are interested in functions where the domain is $\mathbb R^n$ (ie it is a function of $n$ real variables) and the range is $\mathbb R$. 'Multivariable calculus' is the usual name for that field.

When that extends further to ranges in the vector space $\mathbb R^m$ the term vector calculus is often used. Much of what you'll find there is about 3-dimensional space, but other dimensions are covered too. Things like Jacobians and the Implicit Function Theorem come into play. That then leads on towards differential geometry, which includes vector calculus on non-Euclidean manifolds.

 

[Prez Cannady]

It sounds like you are interested in functions where the domain is $\mathbb R^n$ (ie it is a function of $n$ real variables) and the range is $\mathbb R$. 'Multivariable calculus' is the usual name for that field.

I'm interested in $V \rightarrow W$ nonlinear transformations on vector spaces (where neither need be the same dimension). I've plenty of multivariable calc texts, but none seem to spend any time on this. I've fewer linear algebra texts, but hopes of finding even an honorable mention were in vain.

 

[WWGD]

I'm interested in $V \rightarrow W$ nonlinear transformations on vector spaces (where neither need be the same dimension). I've plenty of multivariable calc texts, but none seem to spend any time on this. I've fewer linear algebra texts, but hopes of finding even an honorable mention were in vain.

But this reference is the standard/canonical example. It may include the linear case ( which you can ignore) but covers the general case, viewing euclidean n-space as a vector space over the reals. That is the only case I am familiar with. Since any n-dimensionsl vector space is, to most effects, the same as euclidean n-space, the advanced calculus approach should cover anything except the very highly specialized.

 

[Prez Cannady]

But this reference is the standard/canonical example. It may include the linear case ( which you can ignore) but covers the general case, viewing euclidean n-space as a vector space over the reals. That is the only case I am familiar with. Since any n-dimensionsl vector space is, to most effects, the same as euclidean n-space, the advanced calculus approach should cover anything except the very highly specialized.

"This reference?"

 

[pasmith]

If you drop the restriction to linear maps then you are essentially talking about arbitrary functions between vector spaces. And since these vector spaces are isomorphic to $\mathbb{R}^n$ or $\mathbb{C}^n$ depending on which scalars you are using you may as well be discussing those spaces. This is the field of real or complex analysis.

Alternatively you can look at the ring of polynomials in $n$ variables or the ring of formal power series in $n$ variables, which would be part of the field of algebra.

Or given an arbitrary $f: V \to V$ you can ask when the group of linear maps $\alpha : V \to V$ such that $f = \alpha \circ f \circ \alpha^{-1}$ is not trivial.

That aside, there's really nothing useful that can be said about arbitrary functions between vector spaces. They're arbitrary; they can do anything.

 

[WWGD]

"This reference?"

Sorry, i meant Advanced Calculus.

 

[Prez Cannady]

Then let me phrase my question this way, and narrowly.

Is there a name for functions that take vectors as arguments and perform non-linear operations on the argument's coefficients? And a method analogous to multiplying square matrices against vectors when performing linear transformations?

And perhaps an example?

 

[RPinPA]

I think I know what you're asking, and "operators" is the closest thing I can think of.

There is analysis, real and complex, aka multivariate analysis, which does have plenty to say about general mappings $R^n \rightarrow R^m$, which would simply use normal function notation, e.g. $\vec v = f(\vec w)$. But I don't think that's what you want. I think you want something that looks like a generalization of $\vec v = A \vec w$. As I said, the only thing I could think of is if $A$ is a nonlinear operator.

Differentiation and Fourier transforming are operators that take functions $R \rightarrow R$ and produce other functions $R \rightarrow R$. Actually both of those are linear operators that act on infinite-dimensional vector spaces. I haven't used operator notation outside those areas, but I think it's a general concept and I don't see any reason why you couldn't have nonlinear operators. I believe quantum mechanics relies heavily on operator notation, some of them nonlinear (I did actually study QM in school but it's too distant a memory for me to say anything useful about it).

I think I could for instance define an operator which took every element of $\vec w$ and squared it. I could even write it as a matrix. Here's a matrix operator which will square each element of a 2-vector:
$\begin {pmatrix}S&0 \\ 0 & S \end {pmatrix}$
where $S$ is an operator that squares a real number. I haven't as I said actually worked with such beasts, but I think you might find fruitful references under "operator theory".

 

[suremarc]

To my knowledge, nonlinear functions are the main study in multivariable calculus (and calculus in general). That appears to be the opinion of a few others here. Calculus is the study of change, often when change happens nonlinearly. Differential operators, the various integral theorems, and things like parameterizations and the Frenet-Serret formulas are most useful as tools of analysis for nonlinear functions. In the linear cases, most of the techniques in multivariable calculus aren’t very interesting, or just don’t work. So I don’t know where your multivariable calculus texts are from, but I’m very suspicious about whether or not they’re what you say they are.

It’s not really accurate to say that multivariable calculus is the study of nonlinear functions, though. Such a topic is far too general, and mathematicians often will restrict their study to special subsets of nonlinear functions. In calculus you’re studying functions differentiable except at isolated points, which allows you to give local descriptions by linear approximation. Without this differentiability requirement, even many continuous functions are not well-behaved (see the Weierstrass function). I think there is some proof that the vast majority of functions behave like the Weierstrass function. Also, there is the Cantor function which is differentiable almost everywhere with derivative zero, but takes on every value between zero and one continuously.

The point is, there’s not really a name for what you’re asking about because it’s too broad. I once heard a joke that went, “Classifying problems into linear and non-linear is like classifying the universe into bananas and non-bananas.” There’s not a general name for the nonlinear functions or the study thereof, much like there’s no name for the study of objects which are not bananas.

 

[WWGD]

Hi, reading this again, I am thinking you, Cannady, may be interested in the Borel or general functional calculus. This is a setup to allow the application of functions to operators, like defining the operation of taking the square root of a matrix, etc.

 

[pasmith]

Then let me phrase my question this way, and narrowly.

Is there a name for functions that take vectors as arguments and perform non-linear operations on the argument's coefficients? And a method analogous to multiplying square matrices against vectors when performing linear transformations?

And perhaps an example?

I still think those concepts are "function" and "evaluation of functions" respectively.

The closest analogue I can think of is tensor algebra, which allows one to write any $f : \mathbb{R}^n \to \mathbb{R}^n$ which is analytic in a neighbourhood of zero as a series $$f(\mathbf{x}) = f(\mathbf{0}) + \sum_{j} A_{ij}x_j + \sum_{j}\sum_{k} B_{ijk}x_jx_k + \sum_{j}\sum_{k}\sum_{l} C_{ijkl} x_jx_kx_l + \cdots$$