Mathematics

Divergence

[latexpage]

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 \overrightarrow{v}(x,y), does “fluid” tend to flow towards (i.e., converge) or away from (i.e., diverge) it?

  • The operator, \nabla \cdot (i.e., the dot product with the gradient operator) measures this!
    • Aside:

\nabla = \begin{bmatrix} \frac { \partial }{ \partial x_{ 1 } } \\ \vdots \\ \frac { \partial }{ \partial x_{ n } } \end{bmatrix}

  • Specifically, \nabla \cdot \overrightarrow{v}(x,y,...) for a particular input (x,y,...) returns a scalar value output indicating whether fluid tends to flow towards that input point or away from it:

divergence_system_diagram

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

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

divergence possibilities.jpg

Constructing the Divergence Formula

Consider the following 2D vector field:

\overrightarrow{v}_x(x,y) = \begin{bmatrix} P(x,y) \\ 0  \end{bmatrix}

where P(x,y) is an arbitrary 2D input → 1D output multivariable function

  • This vector field “reduces” such that there is only an x-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 \overrightarrow{v}_x(x,y)
    • Case 1: xdivergence1
    • Case 2: xdivergence2
    • Case 3: xdivergence3
      • Here, even though the $P(x,y)$ is negative at each individual point, it becomes less negative as x 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 x-direction
      • We can tell that \frac{\partial P}{\partial x} > 0 because the change in P(x,y) with respect to x increases as x increases
  • Conclusions:
    • Recall: a positive change in P(x,y) resulting from positive changes in x corresponds with positive with positive divergence
    • Thus, our formula for divergence will include a term containing \frac{\partial P}{\partial x}

Now, let’s consider another 2D vector:

\overrightarrow{v}_y(x,y) = \begin{bmatrix} 0 \\ Q(x,y)  \end{bmatrix}

where Q(x,y) is an arbitrary 2D input → 1D output multivariable function

  • This vector field “reduces” such that there is only an y-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 \overrightarrow{v}_y(x,y)
    • Case 1: ydivergence1
    • Case 2: ydivergence2
    • Case 3: ydivergence3
      • Here, even though the $Q(x,y)$ is negative at each individual point, it becomes less negative as y 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 y-direction
      • We can tell that \frac{\partial P}{\partial x} > 0 because the change in Q(x,y) with respect to x increases as x increases
    • Conclusions:
      • Recall: a positive change in Q(x,y) resulting from positive changes in y corresponds with positive divergence
      • In addition to our \frac{\partial P}{\partial x}-term, the formula for divergence will include a term containing \frac{\partial Q}{\partial y}

It turns out that \frac{\partial P}{\partial x} and \frac{\partial Q}{\partial y} are the only two quantities we need to know for determining the divergence of a 2D vector field \overrightarrow{v}(x,y)

  • Specifically, if we let

\overrightarrow{v}(x,y) = \begin{bmatrix} P(x,y) \\ Q(x,y)  \end{bmatrix}

where P(x,y) andd Q(x,y) are arbitrary 2D input → 1D output multivariable functions then,

\nabla \cdot \overrightarrow{v}(x,y) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}

Take-home-message:

  • For a 2D vector field,
    • \frac{\partial P}{\partial x} accounts for the x components in the output
    • \frac{\partial Q}{\partial y} accounts for the y 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 x and y 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 x and y 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:

\overrightarrow{v}(x,y,z) = \begin{bmatrix} P(x,y,z) \\ Q(x,y,z) \\ R(x,y,z)  \end{bmatrix}

  • We would write the divergence for a point (x,y,z) in this vector field as

\nabla \cdot \overrightarrow{v}(x,y,z) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}

References:

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