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17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > vector fields > divergence

 Calculating Divergence Properties of Divergence Meaning of Divergence Big Picture Practice
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

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

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

Basic Problems

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

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 solution video

video by Dr Chris Tisdell

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

Problem Statement

Calculate the divergence of the vector field $$\vec{F} = \langle xy^2, 2xz, 4-z^2y \rangle$$; $$(1,1,2)$$

Solution

833 solution video

video by MIP4U

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

Problem Statement

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

Solution

834 solution video

video by Dr Chris Tisdell

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

Problem Statement

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

Solution

835 solution video

video by Dr Chris Tisdell

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

Problem Statement

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

Solution

836 solution video

video by Dr Chris Tisdell

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

Problem Statement

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

Solution

838 solution video

video by Dr Chris Tisdell

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

Problem Statement

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

Solution

837 solution video

video by Dr Chris Tisdell

$$\vec{F}(x,y,z) = xye^z\vhat{i} + yze^x\vhat{j} + xze^z\vhat{k}$$

Problem Statement

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

Final Answer

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

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 solution video

video by Michel vanBiezen

Final Answer

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

$$\vec{F} = \langle 6z\cos(x), 7z\sin(x),5z \rangle$$

Problem Statement

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

Final Answer

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

Problem Statement

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

Solution

2133 solution video

video by MIP4U

Final Answer

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

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 $$\langle x+y+z, xyz, 2x+3y+4z \rangle$$.

Final Answer

0

Problem Statement

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

Solution

2099 solution video

Final Answer

0

Intermediate Problems

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 solution video

video by Dr Chris Tisdell