From Points to Regions

The divergence div F of a vector field F gives us information at each point. In this section we show how the divergence combined with integration can be used to determine what is happening in an entire region. We look at one, two, and three dimensions.

One dimension
Two dimensions
Three dimensions
Integration in Cylindrical Coordinates
Integration in Spherical Coordinates

One Dimension

If traffic density in an interval, [a, b], is a constant, rho, then the total number of cars in the interval is determined by simple multiplication

Number of cars = rho (b - a)

If the traffic density is not constant and is given by a function, rho(x) then simple multiplication is replaced by integration

Similarly, if the rate, div F, at which traffic density is decreasing in an interval, [a, b], is a constant, k, then the rate at which the number of cars in the interval is decreasing is determined by simple multiplication

Rate of decrease of number of cars = k (b - a)

and if the rate at which the density is decreasing is not constant then simple multiplication is replaced by integration.

Discuss the units involved in each element of the formula
Suppose the traffic density along a road is described by the function
rho(x) = 500 x (1 - x)
measured in units of cars per mile. How many cars are there in the section of the road between the points x = 0 and x = 1?
Suppose the traffic flow along a road is described by the function
F(x) = 20 x (1 - x)
measured in units of cars per minute. Find the rate at which the number of cars in the section of the road between x = 0 and x = 1 is decreasing? Compute F(0) and F(1). Do you have any comments?
Suppose the traffic flow along a road is described by the function
F(x) = 20 x (1 - x)
measured in units of cars per minute. Find the rate at which the number of cars in the section of the road between x = 1/2 and x = 1 is decreasing? Compute F(1/2) and F(1). Do you have any comments?
Suppose the traffic flow along a road is described by the function
F(x) = 20 x (1 - x)
measured in units of cars per minute. Find the rate at which the number of cars in the section of the road between x = 0 and x = 1/2 is decreasing? Compute F(0) and F(1/2). Do you have any comments?

Two Dimensions

In R² we are primarily interested in regions like those described in the section on volume that can be described by

E = { (x, y) : a <= x <= b and f(x) <= y <= g(x)}

E = { (x, y) : c <= y <= d and p(y) <= x <= q(y)}

Recall that in R² a vector field

representing migration is measured in units like individuals per unit of time per unit of length and

which represents the rate at which population density at a point is decreasing is measured in units like individuals per unit of area per unit of time -- that is, individuals per unit of length² per unit of time.

If the population density in a region is constant then the total population is obtained by simple multiplication

Total population = density * area

and if the population density is given by a funtion then simple multiplication is replaced by integration.

This is often computed by an iterated integral

Similarly, if the rate at which the population density is decreasing is constant then the rate at which the total population is decreasing is obtained by simple multiplication and if the rate at which the population density is decreasing is not constant then simple multplication is replaced by integration

which is often computed by an iterated integral

Discuss the units involved in each element of the formula
Discuss the units involved in each element of the formula
Consider a circular habitat described by
D = { (x, y) : x² + y² <= 4 }
Suppose that the popluation density in this habitat is described by the function
rho(x, y) = x² + y²
Find the total population in this habitat.
Consider another circular habitat described by
D = { (x, y) : x² + y² <= R² }
where x and y are measured in kilometers. Suppose that population migration is described by
and is measured in individuals per year per kilometer.
- Does this describe immigration or emigration?
- At what rate is the total population of the habitat decreasing?
- At what rate are individuals crossing the boundary
  D = { (x, y) : x² + y² = R² }
- Do you have any coments?

Three Dimensions

In R³ we are usually interested in regions that can be described by

E = { (x, y, z) : (x, y) is in D and p(x, y) <= z <= q(x, y) }

where D is a region that can be described by

D = { (x, y) : a <= x <= b and f(x) <= y <= g(x)}

We can also work with regions that are made up of several pieces that can be described in this way.

Recall that if a vector field

represents migration and is measured in units, for example, of individuals per unit of area per unit of time then

represents the rate at which the population density at each point is decreasing and is measured in units of individuals per unit of volume per unit of time.

If population density is contant then the total population in a region is just density multiplied by volume and if population density is not constant then simple multiplication is replaced by multiple integration and often evaluated by an iterated integral.

Similarly if the rate at which the population density is decreasing is constant then the rate at which the total population is decreasing is obtained by simple multiplication and if the rate at which the population density is decreasing is not constant then the rate at which the total population is decreasing is obtained by multiple integration and often evaluated by an iterated integral.

Discuss the units involved in each element of the equation
Discuss the units involved in each element of the equation
The reservoir behind a small dam is described by
E = { (x, y, z) : 0 <= x <= 100, 0 <= y <= 100 - x, and 0.1 x + 0.1 y - 10 <= z <= 0 }
where x, y, and z are measured in feet.
The population density of fish in the reservoir is described by
rho(x, y, z) = 20 + 2z
measured in fish per cubic foot.
Find the total number of fish in the reservoir.

Heat and Conduction

Heat is often measured in calories -- one calorie is the amount of energy required to raise one gram of water one degree Celsius. The temperature of a body, T(x, y, z), is a measure of the density of heat at that point. If the temperature of a body varies from point to point then heat will flow from hotter areas to cooler areas by a process called conduction. This flow is described by the vector field

where alpha² is a positive constant called the thermal diffusivity of the material of which the object is made. Notice that heat is flowing "downhill" from points at which the temperature is high toward points at which it is lower. The rate at which the temperature at a point is decreasing is, as usual, div F and, thus, the rate at which the temperature at a point is increasing is given by

This last equation is called the Heat Equation and describes the way in which heat "diffuses" through an object whose composition is uniform. It describes other kinds of diffusion as well and is also called the Diffusion Equation. The heat equation can be written in three, two, or one dimensions.

In one dimension it describes the way heat diffuses through a thin wire and in two dimensions it describes the way in which heat diffuses through a thin plate.

The temperature of a uniform solid block, C, described mathematically by
C = { (x, y, z) : 0 <= x <= 1, 0 <= y <= 2, and 0 <= z <= 3 }
is given by
T(x, y, z) = 10 x + 20 y + 30 z
The block is thermally isolated, so that no heat flows into or out of the block. Heat will flow within the block by conduction and the block will approach an equilibrium in which its temperature is uniform. What is its temperature at equilibrium?
The temperature of a uniform solid tetrahedron, E, described mathematically by
E = { (x, y, z) : 0 <= x <= 1, 0 <= y <= 1 - x, and 0 <= z <= 1 - x - y }
is given by
T(x, y, z) = 2 x + 4 y - z
The tetrahedron is thermally isolated, so that no heat flows into or out of it. Heat will flow within the tetrahedron by conduction and it will approach an equilibrium in which its temperature is uniform. What is its temperature at equilibrium?

Integration in Cylindrical Coordinates

Missing figure In many situations cylindrical coordinates are easier to use than Cartesian coordinates. For example, the figure at the right shows a cylindrical block of radius R and height H that is heated by a red wire running through its center. In this situation the temperature of a point in the block depends on its distance from the wire. If we set up our coordinate system so that the wire is running along the z-axis then the cylinder can be described in cylindrical coordinates by
Missing equation
and the temperature of each point might be described by a function like
In this situation we are often interested in the integral

which can be evaluated by the iterated integral

One of the striking qualities of mathematics and, perhaps, the most important theme of this course is that the exact same mathematics can describe very different or apparently very different physical situations. For example, a function like the function

might describe the temperature at each point in a uniform cylinder. In this case it might be measured in degrees Celsius and is indirectly a measure of the amount of heat energy or calories per unit of volume at each point in the cylinder. It can be thought of as the density of heat energy.

The same function or a similar function might represent the density or concentration of a pollutant in a cylindrical aquifer. The same function or a similar function might represent the density of mosquitos in a cylindrical region around a fluorescent bulb. For each of these three situations describe the units in each element of the three formulas below and describe what the two integrals measure.

Suppose that the concentration of a particular pollutant in a cylindrical aquifer whose length is 100 meters and whose radius is 10 meters is given by
What is the total amount of the pollutant in the aquifer? What is the average concentration of pollutant in the aquifer?
Suppose that the temperature of a uniform cylindrical bar of metal of length ten centimeters and radius one centimeter is given by
Suppose that the bar is thermally isolated so that no heat flows into or out of the bar and that heat flows within the bar by conduction. What will the temperature of the bar be at equilibrium -- that is, when its temperature is uniform?

Integration in Spherical Coordinates

In many situations it is more natural to use spherical coordinates than either Cartesian coordinates of cylindrical coordinates. For example, when a spherical object of the form

is heated by a source at its center, its temperature at each point might be described by a function like

In this situation we are often interested in the integral

which can be evaluated by

might describe the temperature at each point in a uniform sphere. In this case it might be measured in degrees Celsius and is indirectly a measure of the amount of heat energy or calories per unit of volume at each point in the cylinder. It can be thought of as the density of heat energy.

The same function or a similar function might represent the density or concentration of a pollutant in a spherical aquifer. The same function or a similar function might represent the density of mosquitos in a spherical region around an incandescent bulb. For each of these three situations describe the units in each element of the three formulas below and describe what the two integrals measure.

Suppose that the concentration of a particular pollutant in a spherical aquifer whose radius is 10 meters is given by
What is the total amount of the pollutant in the aquifer? What is the average concentration of pollutant in the aquifer?
Suppose that the temperature of a uniform metal sphere of radius one centimeter is given by
Suppose that the sphere is thermally isolated so that no heat flows into or out of the sphere and that heat flows within the sphere by conduction. What will the temperature of the sphere be at equilibrium -- that is, when its temperature is uniform?