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## Multivariable calculus

### Course: Multivariable calculus>Unit 2

Lesson 11: Divergence and curl (articles)

# Intuition for divergence formula

Why does adding up certain partial derivatives have anything to do with outward fluid flow?

## Warmup for the intuition

In the last article, I showed you the formula for divergence, as well as the physical concept it represents. However, you might still be wondering how these two are connected. Before we dive into the intuition, the following questions should help us warm up by thinking of partial derivatives in the context of a vector field.
Reflection question: A two-dimensional vector field is given by a function $\stackrel{\to }{\mathbf{\text{v}}}$ defined with two components ${v}_{1}$ and ${v}_{2}$,
$\begin{array}{r}\phantom{\rule{1em}{0ex}}\stackrel{\to }{\mathbf{\text{v}}}\left(x,y\right)=\left[\begin{array}{c}{v}_{1}\left(x,y\right)\\ {v}_{2}\left(x,y\right)\end{array}\right]\end{array}$
A few vectors near a point $\left({x}_{0},{y}_{0}\right)$ are sketched below:
• Which of the following describes ${v}_{1}\left({x}_{0},{y}_{0}\right)$?

• Which of the following describes ${v}_{2}\left({x}_{0},{y}_{0}\right)$?

• Which of the following describes $\frac{\partial {v}_{1}}{\partial x}\left({x}_{0},{y}_{0}\right)$?

• Which of the following describes $\frac{\partial {v}_{2}}{\partial y}\left({x}_{0},{y}_{0}\right)$?

## Intuition behind the divergence formula

Let's limit our view to a two-dimensional vector field,
$\stackrel{\to }{\mathbf{\text{v}}}\left(x,y\right)=\left[\begin{array}{c}{v}_{1}\left(x,y\right)\\ {v}_{2}\left(x,y\right)\end{array}\right]$
Remember, the formula for divergence looks like this:
$\mathrm{\nabla }\cdot \stackrel{\to }{\mathbf{\text{v}}}=\frac{\partial {v}_{1}}{\partial x}+\frac{\partial {v}_{2}}{\partial y}$
Why does this have anything to do with changes in the density of a fluid flowing according to $\stackrel{\to }{\mathbf{\text{v}}}\left(x,y\right)$?
Let's look at each component separately.
For example, suppose ${v}_{1}\left({x}_{0},{y}_{0}\right)=0$, meaning the vector attached to $\left({x}_{0},{y}_{0}\right)$ has no horizontal component. And let's say $\frac{\partial {v}_{1}}{\partial x}\left({x}_{0},{y}_{0}\right)$ happens to be positive. This means that near the point $\left({x}_{0},{y}_{0}\right)$, the vector field might look something like this.
• The value of ${v}_{1}\left(x,{y}_{0}\right)$ increases as $x$ grows.
• The value of ${v}_{1}\left(x,{y}_{0}\right)$ decreases as $x$ gets smaller.
Therefore, vectors to the left of $\left({x}_{0},{y}_{0}\right)$ will point a little to the left, and vectors to the right of $\left({x}_{0},{y}_{0}\right)$ will point a little to the right (see the diagram above). This suggests an outward fluid flow, at least as far as the $x$-component is concerned.
In contrast, here's how it looks if $\frac{\partial {v}_{1}}{\partial x}\left({x}_{0},{y}_{0}\right)$ is negative:
• The vectors to the left of $\left({x}_{0},{y}_{0}\right)$ will point to the right.
• The vectors to the right of $\left({x}_{0},{y}_{0}\right)$ will point to the left.
This indicates an inward fluid flow, according to the $x$-component.
The same intuition applies if ${v}_{1}\left({x}_{0},{y}_{0}\right)$ is nonzero. For instance, if ${v}_{1}\left({x}_{0},{y}_{0}\right)$ is positive and $\frac{\partial {v}_{1}}{\partial x}\left({x}_{0},{y}_{0}\right)$ is also positive, this means all the vectors around $\left({x}_{0},{y}_{0}\right)$ point to the right, but they get bigger as we look from left to right. You can imagine the fluid flowing slowly towards $\left({x}_{0},{y}_{0}\right)$ from the left, but flowing fast away from it to the right. Since more is leaving than is coming in, the density at this point decreases.
Analyzing the value $\frac{\partial {v}_{2}}{\partial y}$ is similar. It indicates the change in the vertical component of vectors, ${v}_{2}$, as one moves up and down in the vector field, changing $y$.
For example, suppose ${v}_{2}\left({x}_{0},{y}_{0}\right)=0$, meaning the vector attached to $\left({x}_{0},{y}_{0}\right)$ has no vertical component. Also suppose $\frac{\partial {v}_{2}}{\partial y}\left({x}_{0},{y}_{0}\right)$ is positive, meaning the vertical component of vectors increases as we move upward.
Here's how that might look:
• Vectors below $\left({x}_{0},{y}_{0}\right)$ will point slightly downward.
• Vectors above $\left({x}_{0},{y}_{0}\right)$ will point slightly upward
This indicates an outward fluid flow, as far as the $y$-direction is concerned.
Likewise, if $\frac{\partial {v}_{2}}{\partial y}\left({x}_{0},{y}_{0}\right)$ is negative, it indicates an inward fluid flow near $\left({x}_{0},{y}_{0}\right)$ as far as the $y$-direction is concerned.

## Divergence adds these two influences

Adding the two components $\frac{\partial {v}_{1}}{\partial x}$ and $\frac{\partial {v}_{2}}{\partial y}$ brings together the separate influences of the $x$ and $y$ directions in determining whether fluid-density near a given point increases or decreases.

## Want to join the conversation?

• When will there be practice exercises for this content? I feel like I am learning content but will not be able to say, pass a test. Answers will be appreciated.