## 17Calculus - Divergence of Vector Fields

Using Vectors

Applications

### Partial Integrals

Double Integrals - 2Int

Triple Integrals - 3Int

Practice

### Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Calculus Tools

### Articles

Divergence is also called flux density. 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.

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 $$\displaystyle{ \nabla g = \frac{\partial g }{\partial x}\vhat{i} + \frac{\partial g }{ \partial y}\vhat{j} + \frac{\partial g }{ \partial z}\vhat{k} }$$, which is a vector field. Sometimes $$\nabla g$$ is written grad g.

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) [6mins-29secs]

video by Dr Chris Tisdell

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) [1min-56secs]

video by Dr Chris Tisdell

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) [13mins-38secs]

video by Dr Chris Tisdell

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 [1min-34secs]

video by Dr Chris Tisdell

This is a lot to assimilate. So, it is time to work some practice problems. Once you are done with those, the next topic is conservative vector fields.

Practice

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

Basic

Calculate the divergence of the vector field $$\vec{F} = xyz\vhat{i} + \cos(xyz)\vhat{j} + xy^2z^3\vhat{k}$$

Problem Statement

Calculate the divergence of the vector field $$\vec{F} = xyz\vhat{i} + \cos(xyz)\vhat{j} + xy^2z^3\vhat{k}$$

Solution

### 832 video

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, calculate the divergence of the vector field $$\vec{F} = \langle xy^2, 2xz, 4-z^2y \rangle$$ and at the point $$(1,1,2)$$.

Problem Statement

Unless otherwise instructed, calculate the divergence of the vector field $$\vec{F} = \langle xy^2, 2xz, 4-z^2y \rangle$$ and at the point $$(1,1,2)$$.

Solution

### 833 video

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

Calculate the divergence of the vector field $$\vec{F}(x,y) = x^3y\vhat{i} + yx^2\vhat{j}$$

Problem Statement

Calculate the divergence of the vector field $$\vec{F}(x,y) = x^3y\vhat{i} + yx^2\vhat{j}$$

Solution

### 834 video

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Calculate the divergence of the vector field $$\cos(z)\vhat{i} + \sin(y)\vhat{j}+ \tan(x)\vhat{k}$$

Problem Statement

Calculate the divergence of the vector field $$\cos(z)\vhat{i} + \sin(y)\vhat{j}+ \tan(x)\vhat{k}$$

Solution

### 835 video

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Calculate the divergence of the vector field $$\vec{F} = 2xz \vhat{i} - xy\vhat{j} - z\vhat{k}$$

Problem Statement

Calculate the divergence of the vector field $$\vec{F} = 2xz \vhat{i} - xy\vhat{j} - z\vhat{k}$$

Solution

### 836 video

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Calculate the divergence of the vector field $$\vec{G}(x,y) = x\vhat{j}$$

Problem Statement

Calculate the divergence of the vector field $$\vec{G}(x,y) = x\vhat{j}$$

Solution

### 838 video

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, calculate the divergence of the vector field $$\vec{F} = (x^2-y)\vhat{i} + (y+z)\vhat{j} + (z^2-x)\vhat{k}$$ and at that point $$(1,2,3)$$.

Problem Statement

Unless otherwise instructed, calculate the divergence of the vector field $$\vec{F} = (x^2-y)\vhat{i} + (y+z)\vhat{j} + (z^2-x)\vhat{k}$$ and at that point $$(1,2,3)$$.

Solution

### 837 video

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Calculate the divergence of the vector field $$\vec{F}(x,y,z) = xye^z\vhat{i} + yze^x\vhat{j} + xze^z\vhat{k}$$

Problem Statement

Calculate the divergence of the vector field $$\vec{F}(x,y,z) = xye^z\vhat{i} + yze^x\vhat{j} + xze^z\vhat{k}$$

$$ye^z + ze^x + xe^z(z+1)$$

Problem Statement

Calculate the divergence of the vector field $$\vec{F}(x,y,z) = xye^z\vhat{i} + yze^x\vhat{j} + xze^z\vhat{k}$$

Solution

### 2203 video

video by Michel vanBiezen

$$ye^z + ze^x + xe^z(z+1)$$

Log in to rate this practice problem and to see it's current rating.

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

Problem Statement

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

$$-6z \sin(x) + 5$$

Problem Statement

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

Solution

### 2133 video

video by MIP4U

$$-6z \sin(x) + 5$$

Log in to rate this practice problem and to see it's current rating.

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

Problem Statement

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

Solution

### 2099 video

Log in to rate this practice problem and to see it's current rating.

Intermediate

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

Problem Statement

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

### 1775 video

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

You CAN Ace Calculus

 vectors dot product cross product vector fields curl

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia] Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

### Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

Engineering

Circuits

Semiconductors

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

 The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.
 Calculating Divergence Properties of Divergence Meaning of Divergence Big Picture Practice

Math Word Problems Demystified Shop eBags.com, the leading online retailer of luggage, handbags, backpacks, accessories, and more! Try AmazonFresh Free Trial When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.