# Showing a vector field is irrational on

• tamintl
In summary, a vector field is considered irrational if the magnitude of the vector at any point cannot be expressed as a rational number. To determine if a vector field is irrational, one can use the Pythagorean theorem to calculate the magnitude at different points, and if it always results in a non-terminating, non-repeating decimal, the field is irrational. A vector field cannot be both rational and irrational and examples of irrational vector fields include the radial field and unit vector field. It is important to show that a vector field is irrational as it can help us understand its behavior and can have practical applications in fields such as physics.
tamintl

## Homework Statement

Let F = ( -y/(x2+y2) , x/(x2+y2) ) Show that this vector field is irrotational on ℝ2 - {0}, the real plane less the origin. Then calculate directly the line integral of F around a circle of radius 1.

## The Attempt at a Solution

To show F is irrotational we must show curl v = 0. Although I'm unsure what it means by finding it on ℝ2 - {0} ?

It just means find the curl at a general point (x,y) where x and y aren't both 0. The vector field is undefined at (0,0).

okay thanks.. I got it

## What does it mean for a vector field to be irrational?

A vector field is said to be irrational if the magnitude of the vector at any given point cannot be expressed as a rational number. In other words, the length of the vector cannot be written as a ratio of two integers.

## How can I determine if a vector field is irrational?

To show that a vector field is irrational, you can use the Pythagorean theorem to calculate the magnitude of the vector at different points. If the magnitude is always a non-terminating, non-repeating decimal, then the vector field is irrational.

## Can a vector field be both rational and irrational?

No, a vector field can only be either rational or irrational. A vector field cannot have some vectors with rational magnitudes and others with irrational magnitudes.

## What are some examples of irrational vector fields?

One example of an irrational vector field is a field where the magnitude of the vector at any point is equal to the distance from the origin to that point. This is known as the radial field and is commonly used in physics and engineering.

Another example is a vector field where the magnitude of the vector at any point is equal to the length of the vector itself. This is known as the unit vector field and is often used in mathematics and computer graphics.

## Why is it important to show that a vector field is irrational?

Showing that a vector field is irrational can help us understand the behavior and characteristics of the field. It can also have practical applications in fields such as physics, where irrational vector fields can represent physical quantities such as force or velocity.

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