# Analysis of vector fields, fourier and harmonics

In summary: I have searched around on the internet and I still can't find a good answer.SirAskalot: In summary, SirAskalot is working on a problem involving vector fields and wants a way to measure the properties of the vector field without using the time domain. He is looking for a method that uses the entire vector space.
Hi

I am working on a optimization problem involving vector fields. In order to define a objective function I need a measure (scalar quantity) of some properties of the vector field. The vector field comes from a finite element analysis, that is the vector field is calculated on a discretized domain. If possible I want to exclude the time domain.

The properties I am interested in are the path/direction of the vector field. Due to the geometry and material properties the vector field (or flux) travels in an unwanted path.
Until now I have used a 1-dimensional measure( flux through a surface), calculated the Fourier transform and total harmonic distortion to describe the problem. But I feel this is not a good enough description of the behavior.

I have tried searching for a better solution but having difficulties finding the proper solution. From all the math class I have had up until now I don't recall a method having such a function. I have thought of a 2D Fourier transform but i lack the knowledge of analyzing the results. In essence I need a scalar quantity in the end. I don't know if one can describe the vector field by harmonic content as with the 1D case.

I don't need a actual unit on the measure, but a quantity who describe the field and how the field changes when the geometry of the domain is changed.

I would be very happy if any of you guys and girls could guide me in the right direction and suggest some methods I can look into.

Thanks,

(if this thread fits better in another sub-category, let me know)

TBH, I'm not entirely sure what you're asking. What follows is an attempt to answer what I *think* you're asking.

A path integral of a force field represents work (i component) and flux (j component). If you only have flux, you can only describe half the force field.

Maybe my inquiry was unclear. I should have attached a figure showing my intention. Unfortunately I don't have access to one at the moment, maybe i'll make one tomorrow.

A line integral would be a possibility and I have thought of doing such, but I am unsure on what line/path/geometry to choose. Maybe along some streamline, but then again choosing a similar streamline in the next trial would be hard.

What I was hoping for was a method utilizing the complete vector space. Not depending on the geometry of the domain.

joeblow: Thanks for the input, maybe you got some more if I describe the problem a little better.

If any of you got another ideas I would be very happy, anything goes.

## What is a vector field?

A vector field is a mathematical concept used to represent the direction and magnitude of a physical quantity, such as velocity or force, at every point in space. It is typically visualized as a collection of arrows or lines, with the length and direction of each arrow corresponding to the magnitude and direction of the vector at that point.

## What is Fourier analysis?

Fourier analysis is a mathematical technique used to decompose a complex function into a series of simpler functions, known as Fourier series. This allows for the representation of a function as a sum of sinusoidal components, making it easier to study and manipulate.

## What are harmonics?

Harmonics are sinusoidal components that make up a complex function in Fourier analysis. They are integral multiples of a fundamental frequency and contribute to the overall shape and behavior of the function.

## How are vector fields and Fourier analysis related?

Vector fields and Fourier analysis are related through the concept of harmonic functions. A harmonic function is a solution to the Laplace's equation, which involves the use of vector fields. Additionally, Fourier analysis can be used to decompose a vector field into its harmonics, providing insight into its behavior.

## What are some applications of vector fields, Fourier analysis, and harmonics?

Vector fields, Fourier analysis, and harmonics have many practical applications in various fields such as physics, engineering, and signal processing. Some examples include analyzing fluid flow in aerodynamics, studying the behavior of electrical circuits, and processing audio and image signals in technology.

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