# Components of Curl, Divergence

• Contingency
In summary, you can approximate a field as constant over each side by dividing the field into orthogonal component functions.
Contingency

## Homework Statement

I'm trying to understand where the Cartesian components of the rotor and the divergence of a vector field derived.

I read that the divergence of a vector field is defined by:
$\vec { \nabla } \cdot \vec { F } =\lim _{ V\rightarrow \left\{ P \right\} }{ \frac { 1 }{ \left| V \right| } \iint _{ \partial V }^{ }{ \vec { F } \cdot \vec { dA } } }$

Also, that components of the rotor are defined by:
$\vec { \nabla } \times \vec { F } \cdot \hat { u } =\lim _{ A(\Gamma )\rightarrow 0 }{ \frac { 1 }{ \left| A(\Gamma ) \right| } \int _{ \Gamma }^{ }{ \vec { F } \cdot \vec { dr } } }$

I'm trying to understand where the standard "sum of partial derivatives", and the mnemonic determinant with a row of unit vectors come from. I don't see a correlation between the definitions and these simple representations. How are they derived from the definition?

## Homework Equations

$\vec { \nabla } \times \vec { F } \cdot \hat { u } =\lim _{ A(\Gamma )\rightarrow 0 }{ \frac { 1 }{ \left| A(\Gamma ) \right| } \int _{ \Gamma }^{ }{ \vec { F } \cdot \vec { dr } } }$

$\vec { \nabla } \cdot \vec { F } =\lim _{ V\rightarrow \left\{ P \right\} }{ \frac { 1 }{ \left| V \right| } \iint _{ \partial V }^{ }{ \vec { F } \cdot \vec { dA } } }$

## The Attempt at a Solution

No clue

You can choose a cube as volume (if the general limit exists, it is the same for a special type of volume), and get the coordinate representation in the limit.
In the same way, you can use a square as path.

This is obviously not a precalculus question, so I am moving it to the appropriate section.

My bad, didn't double check. Sorry

mfb said:
You can choose a cube as volume (if the general limit exists, it is the same for a special type of volume), and get the coordinate representation in the limit.
In the same way, you can use a square as path.

Can you please be a bit more specific?
I divided my field into orthogonal component functions, and I'm trying to look at the circulation of the field over a square with sides parallel to the axis. In order to get my result i'd have to approximate the field as constant over each side.. Why can I do that?

## 1. What are the two main components of curl?

The two main components of curl are the tangential and normal components, which represent the rotation and stretching of a vector field, respectively.

## 2. What is the physical interpretation of divergence?

Divergence represents the net flow of a vector field out of or into a given point in space, and can be interpreted as the strength of a source or sink at that point.

## 3. What is the relationship between curl and rotation?

There is a direct relationship between curl and rotation, where the magnitude of curl represents the rate of rotation at a given point in a vector field.

## 4. How is divergence related to the flux of a vector field?

Divergence is related to the flux of a vector field through a closed surface, where the divergence at a point represents the net flow of the vector field through a small volume surrounding that point.

## 5. What are some real-world applications of curl and divergence?

Curl and divergence have many real-world applications in fields such as fluid dynamics, electromagnetism, and computer graphics. They can be used to analyze the flow of fluids, the behavior of electric and magnetic fields, and the generation of visual effects in animation and video games.

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