# Two-dimensional vector representation

• I
• Gribouille
In summary, the conversation discusses representing a two-dimensional vector field with zero divergence and non-zero curl as a gradient, and the steps to calculate the variables a and b. The example of a conservative and rotation-free vector field with a non-constant a is given, as well as a positive non-trivial example involving the magnetic field and electric current density.
Gribouille
Hi,

Is there a method to represent a known two-dimensional vector field w of two coordinates x and y with zero divergence and non-zero curl as
$$\vec{w}(x,y) = a \nabla b \, , \hspace{4mm} \nabla \cdot \vec{w} = 0 \, , \hspace{4mm} \nabla \times \vec{w} = f(x,y) \, ?$$
How would one proceed to calculate a and b?

Gribouille said:
Hi,

Is there a method to represent a known two-dimensional vector field w of two coordinates x and y with zero divergence and non-zero curl as
$$\vec{w}(x,y) = a \nabla b \, , \hspace{4mm} \nabla \cdot \vec{w} = 0 \, , \hspace{4mm} \nabla \times \vec{w} = f(x,y) \, ?$$
How would one proceed to calculate a and b?

No, I don't think so. Assuming a is constant, your vector field can be written as a gradient, which means it is conservative and therefore rotation-free, contradicting your assumption.

Thanks. a is not constant but depends on x and y, just as b.

Unless I made a mistake it is possible sometimes but not in general.

As negative example, consider ##\vec w = \vec c \times \vec r## where r is the position and c is a constant. Consider the unit circle. To get the direction right, ##\nabla b## has to be non-zero but going in a circle. It can't do that without having a rotation, contradiction.

As positive non-trivial example, use the w from above within the unit circle, then continue outside in a symmetric way with zero curl outside, and then add ##d=(10,0)## to it. Now our vector field doesn't have closed circles any more. We can introduce a suitable potential that gets the direction of the gradient right, and then fix the magnitude via a variable ##a##.

if a is a vector and w =a×∇b, then yes. w then is the magnetic field, a is a unit vector normal to x-y plane, b is the magnetic vector potential, and f(x,y) is the electric current density that creates the magnetic field.

Last edited:

## 1. What is a two-dimensional vector representation?

A two-dimensional vector representation is a way of visually representing a mathematical vector in two-dimensional space. It typically consists of an arrow with a certain length and direction, which corresponds to the magnitude and direction of the vector.

## 2. How is a two-dimensional vector represented mathematically?

A two-dimensional vector is typically represented as (x,y) where x is the horizontal component and y is the vertical component. This can also be written as x1i + x2j, where i and j are unit vectors in the x and y direction, respectively.

## 3. What is the difference between a vector and a scalar?

A vector has both magnitude and direction, while a scalar only has magnitude. In two-dimensional vector representation, the length of the arrow corresponds to the magnitude of the vector, and the direction of the arrow corresponds to the direction of the vector.

## 4. How is vector addition and subtraction represented in two-dimensional vector representation?

In two-dimensional vector representation, vector addition and subtraction are performed by adding or subtracting the x and y components separately. For example, to add two vectors v and w, we add their x components and their y components to get the resulting vector v + w = (vx + wx, vy + wy).

## 5. What are some real-life applications of two-dimensional vector representation?

Two-dimensional vector representation is used in a variety of fields, including physics, engineering, and computer graphics. Some examples of its applications include representing forces in a two-dimensional space, calculating the displacement of an object, and creating 2D vector graphics for video games and animations.

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