Two-dimensional vector representation

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.
  • #1
Gribouille
8
0
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?
 
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  • #2
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.
 
  • #3
Thanks. a is not constant but depends on x and y, just as b.
 
  • #4
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##.
 
  • #5
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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