Understanding Divergence and Gradient in Vector Fields

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In summary, The divergence is an operation performed on a vector field and results in a scalar function. The gradient, on the other hand, is performed on a scalar field and results in a vector field. The gradient and divergence are related in that the gradient can be applied to a scalar field to get a vector field, which can then be used to calculate the divergence and obtain the Laplacian of the original scalar field.
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asi123
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What is the Divergence? is it only the Partial derivatives?

Lets say I have a vector field: [tex]F=x^2i+y^2j+z^2k[/tex], the divergence is [tex]F=2xi+2yj+2zk[/tex]?

And if it is, than what is the gradient?:confused:
 
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A divergence is evaluated of a vector field, while the gradient (assuming you mean grad) is done for scalar fields. A related operation, the curl is performed on a vector field.

So we have:
curl: vector field -> vector field
div: vector field -> scalar field
grad: scalar field -> vector field

I'm wondering if there is any defined operation such that we can get a scalar field from a scalar field?
 
  • #3


asi123 said:
What is the Divergence? is it only the Partial derivatives?

Lets say I have a vector field: [tex]F=x^2i+y^2j+z^2k[/tex], the divergence is [tex]F=2xi+2yj+2zk[/tex]?
No. the diverence of this vecor field is the scalar function [itex]\nabla\cdot F= 2x+ 2y+ 2z[/itex]. The "[itex]\cdot[/itex]" in that notation is to remind you of a dot product: the result is a scalar.

And if it is, than what is the gradient?:confused:

The gradient is, in effect, the "opposite" of the divergence: it changes a scalar function to a vector field: at each point [itex]\nabla f[/itex] points in the direction of fastest increase and its length is the derivative in that direction.

Notice that if you start with a scalar function, the gradient gives a vector function and you can then apply the divergence to that going back to a scalar function:
[tex]\nabla\cdot (\nabla f)= \nabla^2 f[/itex]
called the "Laplacian" of f. That is a very important operator: it is the simplest second order differential operator that is "invariant under rigid motions".
 
  • #4


Got it, thanks.
 

What is the Divergence?

The divergence is a mathematical concept that describes the rate at which a vector field is spreading or diverging at any given point. It is a measure of the flow of a vector field away from a specific point.

How is the Divergence calculated?

The divergence is calculated by taking the dot product of the vector field and the del operator (∇). This results in a scalar value that represents the amount and direction of flow at a specific point.

What is the physical interpretation of the Divergence?

In physics, the divergence is often used to describe the behavior of a fluid or gas. A positive divergence indicates that the fluid is spreading out, while a negative divergence indicates that the fluid is converging or coming together at a point.

What is the difference between Divergence and Curl?

While the divergence describes the flow of a vector field away from a point, the curl describes the rotation or circulation of the vector field at a point. In other words, the divergence measures the spreading of a field while the curl measures the swirling of a field.

What are some real-world applications of the Divergence?

The divergence is used in many fields, including fluid dynamics, electromagnetism, and weather forecasting. It is also used in computer graphics to simulate fluid flow and in image processing for edge detection. In physics, it is used to describe the behavior of electric and magnetic fields.

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