Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Radius of Convergence
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Gradients
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
17calculus on YouTube
More Math Help
Tutoring
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > vector fields > divergence

### Calculus Main Topics

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

general learning tools

additional tools

### Related Topics and Links

Divergence of Vector Fields

The divergence of a vector field is a measure of how a vector field diverges. The result of calculating the divergence will be a function. This function can be evaluated at a point to give a number that tells us how the vector field diverges at that point. First, let's look at the gradient to refresh our memories on the del operator.

Divergence is also called flux density.

The result of calculating the divergence of a vector field is a scalar function.

Remember from your study of gradients that the del operator is $$\displaystyle{ \nabla = \frac{\partial }{\partial x}\vhat{i} + \frac{\partial }{ \partial y}\vhat{j} + \frac{\partial }{ \partial z}\vhat{k} }$$. If we are given a function $$g(x,y,z)$$, the gradient of $$g$$ is $$\displaystyle{ grad ~ g = \nabla g = \frac{\partial g }{\partial x}\vhat{i} + \frac{\partial g }{ \partial y}\vhat{j} + \frac{\partial g }{ \partial z}\vhat{k} }$$, which is a vector field.

Calculating Divergence

To calculate the divergence, we use the same del operator in a little different way and, as you would expect, the notation looks different. The divergence is given by the equation $$\vec{ \nabla } \cdot \vec{F}$$ where $$\vec{F}$$ is a vector field and '$$\cdot$$' indicates the dot product. The notation gets a bit strange but here is what this means. If we have a vector field $$\vec{F}(x,y,z) = f_i(x,y,z)\vhat{i} +$$ $$f_j(x,y,z)\vhat{j} +$$ $$f_k(x,y,z)\vhat{k}$$, then the divergence of the vector field $$\vec{F}$$ is

 $$\displaystyle{ \vec{ \nabla } \cdot \vec{F} = }$$ $$\displaystyle{ \left[ \frac{\partial }{\partial x} \vhat{i} + \frac{\partial }{ \partial y}\vhat{j} + \frac{\partial }{ \partial z}\vhat{k} \right] \cdot }$$ $$\displaystyle{ \left[ f_i(x,y,z)\vhat{i} + \right. }$$ $$\displaystyle{ \left. f_j(x,y,z)\vhat{j} +f_k(x,y,z)\vhat{k} \right] = }$$ $$\displaystyle{ \frac{\partial f_i }{\partial x} + \frac{\partial f_j }{ \partial y} + \frac{\partial f_k }{ \partial z} }$$

Things To Notice
1. We do not have a true dot product in the above equation since $$\vec{ \nabla }$$ is not a vector, it is an operator. However, we stretch the notation here to think of the del operator as a vector. In actuality, we can think of the dot product here as a way to 'distribute' the partial derivatives to each term in $$\vec{F}$$.
2. When writing the divergence, mathematicians often write the del operator with the vector sign above it to emphasize that we need to think of the del operator as a vector. We have done this here too. However, $$\vec{ \nabla }$$ and $$\nabla$$ both refer to the same del operator shown above. [ As usual, check with your instructor to see what they expect. ]
3. In the gradient equation $$\nabla g$$, there is no dot for a dot product. It would be incorrect to have a dot for a gradient since a dot product is an operation on two vectors.
4. Another way to write the divergence is to write $$div ~ \vec{F}$$, so $$div ~ \vec{F} = \vec{ \nabla } \cdot \vec{F}$$.

Okay, so let's watch a video clip that explains the divergence in more detail.

 Dr Chris Tisdell: What is the divergence? (Part 1)

Before we go on, let's work a few practice problems.

Practice 1

Find the divergence of the vector field $$\vec{F} = \langle 6z\cos(x), 7z\sin(x),5z \rangle$$.

answer

solution

Properties of Divergence

It may not be obvious from the equations we use to calculate divergence but divergence is a linear operator. That means most of the rules of algebra and calculus that you already know apply also to divergence. Here is a list of a few of them.

In the table below, $$\vec{F}$$ and $$\vec{G}$$ are vector fields, $$a$$ is a scalar and $$\varphi$$ is a scalar function.

equation

notes

$$\vec{ \nabla } \cdot \left( a\vec{F} \right) = a \left( \vec{ \nabla } \cdot \vec{F} \right)$$

$$\vec{ \nabla } \cdot \left( \vec{F} + \vec{G} \right) = \vec{ \nabla } \cdot \vec{F} + \vec{ \nabla } \cdot \vec{G}$$

$$\vec{ \nabla } \cdot \left( \varphi \vec{F} \right) = \left( \nabla \varphi \right) \cdot \vec{F} + \varphi \left( \vec{ \nabla } \cdot \vec{F} \right)$$

product rule involving a scalar function

$$\vec{ \nabla } \cdot \left( \vec{F} \times \vec{G} \right) = \left( \vec{ \nabla } \times \vec{F} \right) \cdot \vec{G} -$$ $$\vec{F} \cdot \left( \vec{ \nabla } \times \vec{G} \right)$$

product rule involving the curl

$$\vec{ \nabla } \cdot \left( \nabla \varphi \right) = \vec{ \nabla }^2 \varphi$$

Laplacian of a scalar function

$$\vec{ \nabla } \cdot \left( \vec{ \nabla } \times \vec{F} \right) = 0$$

identity

Here is a quick video clip discussing a few of these properties.

 Dr Chris Tisdell: What is the divergence? (Part 2)

Now it's time for a practice problem.

Practice 2

Calculate the divergence of the curl of $$\langle x+y+z,xyz,2x+3y+4z\rangle$$.

answer

solution

Meaning of Divergence

Whew! That is a lot of math. Now that you know how to calculate divergence of a vector field, you may be asking yourself, what does it mean and how do I use this? This video clip gives great examples and explanation on how to understand the result of a divergence calculation.

 Dr Chris Tisdell: What is the divergence? (Part 3)

Big Picture

 The divergence measures the net outflow of a vector field. If the divergence is positive everywhere, then there is a net outflow over every closed curve/surface. This is sometimes referred to as a source. If the divergence is negative everywhere, then there is a net inflow over every closed curve/surface. This is sometimes referred to as a sink. A vector field with zero divergence everywhere is called 'incompressible' with zero net outflow over every closed curve/surface.

Okay, so after all that math and stuff, your head is probably spinning. Lets get an idea of the big picture. The list above is adapted from the notes of Dr Chris Tisdell. In this video clip, he discusses these items. [ We highly recommend that you go to his website and download the notes for this topic. ]

 Dr Chris Tisdell: Divergence of vector fields
 There is a lot to get your head around as far as divergence goes. So, it is time to work some practice problems. Once you are done with those, the next topic is conservative vector fields. next: conservative vector field →

### Search 17Calculus

Practice Problems

Instructions - - Unless otherwise instructed, calculate the divergence of these vector fields. If a point is given, find the divergence at that point also.

 Level A - Basic

Practice A01

$$\vec{F} = xyz\vhat{i} + \cos(xyz)\vhat{j} + xy^2z^3\vhat{k}$$

solution

Practice A02

$$\vec{F} = \langle xy^2, 2xz, 4-z^2y \rangle$$; $$(1,1,2)$$

solution

Practice A03

$$\vec{F}(x,y) = x^3y\vhat{i} + yx^2\vhat{j}$$

solution

Practice A04

$$\cos(z)\vhat{i} + \sin(y)\vhat{j}+ \tan(x)\vhat{k}$$

solution

Practice A05

$$\vec{F} = 2xz \vhat{i} - xy\vhat{j} - z\vhat{k}$$

solution

Practice A06

$$\vec{G}(x,y) = x\vhat{j}$$

solution

Practice A07

$$\vec{F} = (x^2-y)\vhat{i} + (y+z)\vhat{j} + (z^2-x)\vhat{k}$$; $$(1,2,3)$$

solution

 Level B - Intermediate

Practice B01

Compute the divergence and the curl of $$\vec{F}(x,y,z) = (\sinh x)\vhat{i} + (\cosh y)\vhat{j} -xyz\vhat{k}$$ and verify that $$\vec{\nabla} \cdot ( \vec{\nabla} \times \vec{F}) = 0$$.

solution

8