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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 start bold text, v, end bold text, with, vector, on top defined with two components v, start subscript, 1, end subscript and v, start subscript, 2, end subscript,
\begin{aligned} \quad \vec{\textbf{v}}(x, y) = \left[ \begin{array}{c} v_1(x, y) \\ v_2(x, y) \end{array} \right] \end{aligned}
A few vectors near a point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis are sketched below:
• Which of the following describes v, start subscript, 1, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis?

• Which of the following describes v, start subscript, 2, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis?

• Which of the following describes start fraction, \partial, v, start subscript, 1, end subscript, divided by, \partial, x, end fraction, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis?

• Which of the following describes start fraction, \partial, v, start subscript, 2, end subscript, divided by, \partial, y, end fraction, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis?

## Intuition behind the divergence formula

Let's limit our view to a two-dimensional vector field,
$\vec{\textbf{v}}(x, y) = \left[ \begin{array}{c} \blueE{v_1}(x, y) \\ \redE{v_2}(x, y) \end{array} \right]$
Remember, the formula for divergence looks like this:
del, dot, start bold text, v, end bold text, with, vector, on top, equals, start fraction, \partial, start color #0c7f99, v, start subscript, 1, end subscript, end color #0c7f99, divided by, \partial, start color #0c7f99, x, end color #0c7f99, end fraction, plus, start fraction, \partial, start color #bc2612, v, start subscript, 2, end subscript, end color #bc2612, divided by, \partial, start color #bc2612, y, end color #bc2612, end fraction
Why does this have anything to do with changes in the density of a fluid flowing according to start bold text, v, end bold text, with, vector, on top, left parenthesis, x, comma, y, right parenthesis?
Let's look at each component separately.
For example, suppose v, start subscript, 1, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, equals, 0, meaning the vector attached to left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis has no horizontal component. And let's say start fraction, \partial, v, start subscript, 1, end subscript, divided by, \partial, start color #0c7f99, x, end color #0c7f99, end fraction, left parenthesis, start color #0c7f99, x, start subscript, 0, end subscript, end color #0c7f99, comma, y, start subscript, 0, end subscript, right parenthesis happens to be positive. This means that near the point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, the vector field might look something like this.
• The value of v, start subscript, 1, end subscript, left parenthesis, start color #0c7f99, x, end color #0c7f99, comma, y, start subscript, 0, end subscript, right parenthesis increases as start color #0c7f99, x, end color #0c7f99 grows.
• The value of v, start subscript, 1, end subscript, left parenthesis, start color #0c7f99, x, end color #0c7f99, comma, y, start subscript, 0, end subscript, right parenthesis decreases as start color #0c7f99, x, end color #0c7f99 gets smaller.
Therefore, vectors to the left of left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis will point a little to the left, and vectors to the right of left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis 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 start fraction, \partial, v, start subscript, 1, end subscript, divided by, \partial, start color #0c7f99, x, end color #0c7f99, end fraction, left parenthesis, start color #0c7f99, x, start subscript, 0, end subscript, end color #0c7f99, comma, y, start subscript, 0, end subscript, right parenthesis is negative:
• The vectors to the left of left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis will point to the right.
• The vectors to the right of left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis will point to the left.
This indicates an inward fluid flow, according to the x-component.
The same intuition applies if v, start subscript, 1, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis is nonzero. For instance, if v, start subscript, 1, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis is positive and start fraction, \partial, v, start subscript, 1, end subscript, divided by, \partial, start color #0c7f99, x, end color #0c7f99, end fraction, left parenthesis, start color #0c7f99, x, start subscript, 0, end subscript, end color #0c7f99, comma, y, start subscript, 0, end subscript, right parenthesis is also positive, this means all the vectors around left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis point to the right, but they get bigger as we look from left to right. You can imagine the fluid flowing slowly towards left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis 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 start fraction, \partial, v, start subscript, 2, end subscript, divided by, \partial, start color #bc2612, y, end color #bc2612, end fraction is similar. It indicates the change in the vertical component of vectors, v, start subscript, 2, end subscript, as one moves up and down in the vector field, changing y.
For example, suppose v, start subscript, 2, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, equals, 0, meaning the vector attached to left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis has no vertical component. Also suppose start fraction, \partial, v, start subscript, 2, end subscript, divided by, \partial, start color #bc2612, y, end color #bc2612, end fraction, left parenthesis, x, start subscript, 0, end subscript, comma, start color #bc2612, y, start subscript, 0, end subscript, end color #bc2612, right parenthesis is positive, meaning the vertical component of vectors increases as we move upward.
Here's how that might look:
• Vectors below left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis will point slightly downward.
• Vectors above left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis will point slightly upward
This indicates an outward fluid flow, as far as the y-direction is concerned.
Likewise, if start fraction, \partial, v, start subscript, 2, end subscript, divided by, \partial, start color #bc2612, y, end color #bc2612, end fraction, left parenthesis, x, start subscript, 0, end subscript, comma, start color #bc2612, y, start subscript, 0, end subscript, end color #bc2612, right parenthesis is negative, it indicates an inward fluid flow near left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis as far as the y-direction is concerned.

## Divergence adds these two influences

Adding the two components start fraction, \partial, v, start subscript, 1, end subscript, divided by, \partial, x, end fraction and start fraction, \partial, v, start subscript, 2, end subscript, divided by, \partial, y, end fraction brings together the separate influences of the x and y directions in determining whether fluid-density near a given point increases or decreases.
$\Large \nabla \cdot \vec{\textbf{v}} = \!\!\!\!\!\!\!\!\! \overbrace{\dfrac{\partial v_1}{\partial \blueE{x}}}^{ \substack{ \text{Change in density} \\ \text{in the \blueE{x}-direction} } } \!\!\!\!\!\!\!\! + \!\!\!\!\!\!\! \underbrace{\dfrac{\partial v_2}{\partial \redE{y}}}_{ \substack{ \text{Change in density} \\ \text{in the \redE{y}-direction} } }$