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## Introduction

Divergence vs. convergence

- Divergence = movement towards a point (i.e., source)
- Convergence = movement away from a point (i.e., sink)

Question: for a given point in a vector field , does “fluid” tend to flow towards (i.e., converge) or away from (i.e., diverge) it?

- The operator, (i.e., the dot product with the gradient operator) measures this!
- Aside:

- Specifically, for a particular input returns a scalar value output indicating whether fluid tends to flow towards that input point or away from it:

Note: for the purposes of this article, we are going to assume is a vector field with a 2D input → 2D output

Three possibilities for interpreting the divergence of a point in a vector field

## Constructing the Divergence Formula

Consider the following 2D vector field:

where is an arbitrary 2D input → 1D output multivariable function

- This vector field “reduces” such that there is only an -component in the output ⇒ all vectors in the vector field either point left or right
- Here are some cases for positive divergence
- Note: assume the three vectors in each diagram sampled from our vector field
- Case 1:
- Case 2:
- Case 3:
- Here, even though the $P(x,y)$ is negative at each individual point, it becomes less negative as increases

- Importantly, all these cases satisfy the following properties
- The only change in input (i.e., origins of the three output vectors) occurs in the -direction
- We can tell that because the change in with respect to increases as increases

- Conclusions:
- Recall: a positive change in resulting from positive changes in corresponds with positive with positive divergence
- Thus, our formula for divergence will include a term containing

Now, let’s consider another 2D vector:

where is an arbitrary 2D input → 1D output multivariable function

- This vector field “reduces” such that there is only an -component in the output ⇒ all vectors in the vector field either point up or down
- Again, here are some cases for positive divergence:
- Note: assume the three vectors in each diagram sampled from our vector field
- Case 1:
- Case 2:
- Case 3:
- Here, even though the $Q(x,y)$ is negative at each individual point, it becomes less negative as increases

- Importantly, all these cases satisfy the following properties
- The only change in input (i.e., origins of the three output vectors) occurs in the -direction
- We can tell that because the change in with respect to increases as increases

- Conclusions:
- Recall: a positive change in resulting from positive changes in corresponds with positive divergence
- In addition to our -term, the formula for divergence will include a term containing

It turns out that and are the only two quantities we need to know for determining the divergence of a 2D vector field

- Specifically, if we let

where andd are arbitrary 2D input → 1D output multivariable functions then,

## Take-home-message:

- For a 2D vector field,
- accounts for the components in the output
- accounts for the components in the output

- For any input point in an arbitrary 2D vector field, we can break down all the outputs associated with neighboring input points into and components
- In the context that our vector field models fluid flow, this can be thought of as determining the net amount of fluid flowing into our input point in the and directions (i.e., the divergence!)

Note: we can also extend these findings to n-dimensional vector fields

- For example, consider an arbitrary 3D vector field:

- We would write the divergence for a point in this vector field as