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## Multivariable calculus

### Course: Multivariable calculus>Unit 5

Lesson 1: Formal definitions of div and curl (optional reading)

# Formal definition of divergence in three dimensions

Learn how surface integrals and 3D flux are used to formalize the idea of divergence in 3D.

## Background

It is a short step between these two prerequisites, and understanding the formal definition of divergence in three dimensions. For that reason, I'm going to keep this article relatively short, assuming that you have the intuition behind both of those pieces of background knowledge.

## What we're building to

• The goal is to capture the intuition of outward fluid flow at a point in a mathematical formula.
• In three-dimensions, divergence is defined using the following limit:
\begin{aligned} \text{div}\,\blueE{\textbf{F}}\goldE{(x, y, z)} = \lim_{|\redE{R}_{\goldE{(x, y, z)}}| \to 0} \!\!\!\! \overbrace{ \dfrac{1}{|\redE{R}_{\goldE{(x, y, z)}}|} \!\!\!\!\! \underbrace{ \iint_\redE{S} \blueE{\textbf{F}} \cdot \greenE{\hat{\textbf{n}}}\, \redE{d\Sigma} }_{\text{Flux through the surface of \redE{R}}} }^{\text{Average outward flow from \redE{R} per unit volume}} \end{aligned}
There is quite a lot going on in this definition, but most of the complexity lies in that flux integral. If you understand that part, the rest comes from taking the limit with respect to a region shrinking around a point.

## From a region to a point

Let's say you have a three-dimensional vector field.
start color #0c7f99, F, end color #0c7f99, left parenthesis, x, comma, y, comma, z, right parenthesis, left arrow, start text, T, h, r, e, e, negative, d, i, m, e, n, s, i, o, n, a, l, space, v, e, c, t, o, r, space, f, i, e, l, d, end text
As always, think of this vector field as representing a fluid flow. The divergence start text, d, i, v, end text, start color #0c7f99, F, end color #0c7f99 tries to measure the "outward flow" of this fluid at each point. However, it doesn't quite make sense to talk about what it means for fluid to flow out of a point.
What does make sense is the idea of fluid flowing out of region. Specifically, picture some region start color #bc2612, R, end color #bc2612 in the vector field.
Let's name the surface of this region "start color #bc2612, S, end color #bc2612". In the article on flux in three dimensions, I showed how you can measure the rate at which fluid is leaving this region by taking the flux of start color #0c7f99, start bold text, F, end bold text, end color #0c7f99 over the surface start color #bc2612, S, end color #bc2612:
start underbrace, minus, start fraction, d, left parenthesis, start text, f, l, u, i, d, space, m, a, s, s, space, i, n, space, start color #bc2612, R, end color #bc2612, end text, right parenthesis, divided by, d, t, end fraction, end underbrace, start subscript, start text, R, a, t, e, space, a, t, space, w, h, i, c, h, space, f, l, u, i, d, space, e, x, i, t, s, space, start color #bc2612, R, end color #bc2612, end text, end subscript, equals, start underbrace, \iint, start subscript, start color #bc2612, S, end color #bc2612, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, start color #bc2612, d, \Sigma, end color #bc2612, end underbrace, start subscript, start text, F, l, u, x, space, s, u, r, f, a, c, e, space, i, n, t, e, g, r, a, l, end text, end subscript
Here, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, left parenthesis, x, comma, y, comma, z, right parenthesis is a vector-valued function which returns the outward facing unit normal vector at each point on start color #bc2612, S, end color #bc2612.
Divergence itself is concerned with the change in fluid density around each point, as opposed mass. We can get the change in fluid density of start color #bc2612, R, end color #bc2612 by dividing the flux integral by the volume of start color #bc2612, R, end color #bc2612. To denote the volume of start color #bc2612, R, end color #bc2612, put bars around it:
vertical bar, start color #bc2612, R, end color #bc2612, vertical bar, left arrow, start text, V, o, l, u, m, e, space, o, f, space, start color #bc2612, R, end color #bc2612, end text
So here's what rate at which fluid density changes inside start color #bc2612, R, end color #bc2612 looks like:
minus, start fraction, d, left parenthesis, start text, f, l, u, i, d, space, start color #0c7f99, start text, d, e, n, s, i, t, y, end text, end color #0c7f99, space, i, n, space, start color #bc2612, R, end color #bc2612, end text, right parenthesis, divided by, d, t, end fraction, equals, start fraction, 1, divided by, vertical bar, start color #bc2612, R, end color #bc2612, vertical bar, end fraction, \iint, start subscript, start color #bc2612, S, end color #bc2612, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, start color #bc2612, d, \Sigma, end color #bc2612
The divergence of start color #0c7f99, start bold text, F, end bold text, end color #0c7f99 at a point start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05 is defined as the limit of this change-in-fluid-density expression as the region shrinks around the point start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05.
start text, d, i, v, end text, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, equals, start underbrace, limit, start subscript, start color #bc2612, R, end color #bc2612, \to, start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, end subscript, end underbrace, start subscript, start color #bc2612, R, end color #bc2612, start text, space, s, h, r, i, n, k, s, space, a, r, o, u, n, d, space, end text, start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, end subscript, start fraction, 1, divided by, vertical bar, start color #bc2612, R, end color #bc2612, vertical bar, end fraction, \iint, start subscript, start color #bc2612, S, end color #bc2612, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, start color #bc2612, d, \Sigma, end color #bc2612
In that equation, I wrote start color #bc2612, R, end color #bc2612, \to, start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05 to communicate the idea of start color #bc2612, R, end color #bc2612 shrinking around the point start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05. At the end of the day, all this notation is just a desperate attempt to communicate a heavily visual idea with symbols. You will see different authors use different notation. If you prefer, you could alternatively start by saying start color #bc2612, R, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, end subscript is a region which contains the point start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, then write the following:
start text, d, i, v, end text, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, equals, limit, start subscript, vertical bar, start color #bc2612, R, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, end subscript, vertical bar, \to, 0, end subscript, start fraction, 1, divided by, vertical bar, start color #bc2612, R, end color #bc2612, start subscript, start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05, end subscript, vertical bar, end fraction, \iint, start subscript, start color #bc2612, S, end color #bc2612, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, start color #bc2612, d, \Sigma, end color #bc2612
I have a slight preference for this last notation, just because it makes it a bit easier to see the connection between start color #a75a05, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #a75a05 on the left hand side and the right hand side without relying so heavily on the context in which all the terms are defined.

## Congratulations!

If you are at the point where you can understand this (rather complicated) definition, it is a good sign that you have a solid mental grasp of both divergence and surface integrals. It also means you are in a strong position to understand the divergence theorem, which connects this idea to that of triple integrals.

## Want to join the conversation?

• Is R(x,y,z) supposed to indicate surface area or volume? I believe it should be surface area... • R is a 3D region, but it is neither area nor volume. Consider this: is Mike weight or height? He is a person, not a quantity. He has a weight and a height, but he himself is not equal to any number.

R(x,y,z) just emphasizes that the region R must contain the point (x,y,z). To denote the volume, we can use |R| or |R(x,y,z)|, which you can sort of think of as the "magnitude" of the region. But strictly this is just a definition, just like "+" is defined as plus.

We can ask, however, whether R represents a 3D solid region or a 2D closed surface. I personally think R is a 3D region and S is the surface enclosing it. And I think |R| is the volume and Σ the surface area. dΣ is a tiny bit of the surface area (but still a quantity).

Therefore, R and S are things and |R| and Σ are numbers (though you could argue that numbers are also things).

Of course, it is important to note that different people use different notation, so don't take this too seriously. Always pay attention to context.
• Since S is a "closed" surface, why is no circle-sign in the middle of the double sigma? Is the circle-sign to be included only with single integrals? • You can put a circle-sign around a double integral to indicate a closed surface, but you don't have to. The circle sign just makes it explicit that the surface is closed, which is helpful when you want to express integral equations conceptually without having to get too into the math. A good example of this are Maxwell's equations. People rarely use the full equations for computations, but instead use them to concisely describe electromagnetism.

My guess as to why there's no circle sign here is that this article is concerned with a formal definition, not a conceptual explanation, and so the circle-sign is a little too hand-wavy.
(1 vote)
• What are the dimensions of the physical quantity F(x,y,z) represents?
they should be mass/unit area.... according to me.
(1 vote) • What does d sigma in the expression imply?
is it the same as dxdy?
(1 vote) • Is there a way to go back and forth between a surface and its volume? For example, assuming I was given a parametrization for that red blob surface, how would i set up a triple integral to find the contained volume?
(1 vote) 