Divergence and Curl

In this section, we examine two important operations on a vector field: divergence and curl. They are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-dimensional versions of the Fundamental Theorem of Calculus. In addition, curl and divergence appear in mathematical descriptions of fluid mechanics, electromagnetism, and elasticity theory, which are important concepts in physics and engineering. We can also apply curl and divergence to other concepts we already explored. For example, under certain conditions, a vector field is conservative if and only if its curl is zero.

In addition to defining curl and divergence, we look at some physical interpretations of them, and show their relationship to conservative and source-free vector fields.

Divergence

Divergence is an operation on a vector field that tells us how the field behaves toward or away from a point. Locally, the divergence of a vector field F in

ℝ^{2}

at a particular point P is a measure of the “outflowing-ness” of the vector field at P. If F represents the velocity of a fluid, then the divergence of F at P measures the net rate of change with respect to time of the amount of fluid flowing away from P (the tendency of the fluid to flow “out of” P). In particular, if the amount of fluid flowing into P is the same as the amount flowing out, then the divergence at P is zero.

Note the divergence of a vector field is not a vector field, but a scalar function. In terms of the gradient operator

\nabla = 〈 \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} 〉,

Note this is merely helpful notation, because the dot product of a vector of operators and a vector of functions is not meaningfully defined given our current definition of dot product.

To illustrate this point, consider the two vector fields in [link]. At any particular point, the amount flowing in is the same as the amount flowing out, so at every point the “outflowing-ness” of the field is zero. Therefore, we expect the divergence of both fields to be zero, and this is indeed the case, as

in [link]. At any given point, more fluid is flowing in than is flowing out, and therefore the “outgoingness” of the field is negative. We expect the divergence of this field to be negative, and this is indeed the case, as

div (R) = \frac{\partial}{\partial x} (− x) + \frac{\partial}{\partial y} (− y) = −2 .

To get a global sense of what divergence is telling us, suppose that a vector field in

ℝ^{2}

represents the velocity of a fluid. Imagine taking an elastic circle (a circle with a shape that can be changed by the vector field) and dropping it into a fluid. If the circle maintains its exact area as it flows through the fluid, then the divergence is zero. This would occur for both vector fields in [link]. On the other hand, if the circle’s shape is distorted so that its area shrinks or expands, then the divergence is not zero. Imagine dropping such an elastic circle into the radial vector field in [link] so that the center of the circle lands at point (3, 3). The circle would flow toward the origin, and as it did so the front of the circle would travel more slowly than the back, causing the circle to “scrunch” and lose area. This is how you can see a negative divergence.

One application for divergence occurs in physics, when working with magnetic fields. A magnetic field is a vector field that models the influence of electric currents and magnetic materials. Physicists use divergence in Gauss’s law for magnetism, which states that if B is a magnetic field, then

\nabla \cdot B = 0;

Another application for divergence is detecting whether a field is source free. Recall that a source-free field is a vector field that has a stream function; equivalently, a source-free field is a field with a flux that is zero along any closed curve. The next two theorems say that, under certain conditions, source-free vector fields are precisely the vector fields with zero divergence.

Proof

The converse of [link] is true on simply connected regions, but the proof is too technical to include here. Thus, we have the following theorem, which can test whether a vector field in

ℝ^{2}

where C is a simple closed curve and D is the region enclosed by C. Since

P_{x} + Q_{y} = div F,

Therefore, Green’s theorem can be written in terms of divergence. If we think of divergence as a derivative of sorts, then Green’s theorem says the “derivative” of F on a region can be translated into a line integral of F along the boundary of the region. This is analogous to the Fundamental Theorem of Calculus, in which the derivative of a function

f

Using divergence, we can see that Green’s theorem is a higher-dimensional analog of the Fundamental Theorem of Calculus.

We can use all of what we have learned in the application of divergence. Let v be a vector field modeling the velocity of a fluid. Since the divergence of v at point P measures the “outflowing-ness” of the fluid at P,

div v (P) > 0

implies that more fluid is flowing out of P than flowing in. Similarly,

div v (P) < 0

implies the more fluid is flowing in to P than is flowing out, and

div v (P) = 0

Curl

The second operation on a vector field that we examine is the curl, which measures the extent of rotation of the field about a point. Suppose that F represents the velocity field of a fluid. Then, the curl of F at point P is a vector that measures the tendency of particles near P to rotate about the axis that points in the direction of this vector. The magnitude of the curl vector at P measures how quickly the particles rotate around this axis. In other words, the curl at a point is a measure of the vector field’s “spin” at that point. Visually, imagine placing a paddlewheel into a fluid at P, with the axis of the paddlewheel aligned with the curl vector ([link]). The curl measures the tendency of the paddlewheel to rotate.

Consider the vector fields in [link]. In part (a), the vector field is constant and there is no spin at any point. Therefore, we expect the curl of the field to be zero, and this is indeed the case. Part (b) shows a rotational field, so the field has spin. In particular, if you place a paddlewheel into a field at any point so that the axis of the wheel is perpendicular to a plane, the wheel rotates counterclockwise. Therefore, we expect the curl of the field to be nonzero, and this is indeed the case (the curl is

2 k) .

To see what curl is measuring globally, imagine dropping a leaf into the fluid. As the leaf moves along with the fluid flow, the curl measures the tendency of the leaf to rotate. If the curl is zero, then the leaf doesn’t rotate as it moves through the fluid.

The definition of curl can be difficult to remember. To help with remembering, we use the notation

\nabla \times F

Thus, this matrix is a way to help remember the formula for curl. Keep in mind, though, that the word determinant is used very loosely. A determinant is not really defined on a matrix with entries that are three vectors, three operators, and three functions.

is a vector field in a plane, then

curl F \cdot k = (Q_{x} - P_{y}) k \cdot k = Q_{x} - P_{y} .

where C is a simple closed curve and D is the region enclosed by C. Therefore, the circulation form of Green’s theorem can be written in terms of the curl. If we think of curl as a derivative of sorts, then Green’s theorem says that the “derivative” of F on a region can be translated into a line integral of F along the boundary of the region. This is analogous to the Fundamental Theorem of Calculus, in which the derivative of a function

f

Using curl, we can see the circulation form of Green’s theorem is a higher-dimensional analog of the Fundamental Theorem of Calculus.

We can now use what we have learned about curl to show that gravitational fields have no “spin.” Suppose there is an object at the origin with mass

m_{1}

Recall that the gravitational force that object 1 exerts on object 2 is given by field

Using Divergence and Curl

Now that we understand the basic concepts of divergence and curl, we can discuss their properties and establish relationships between them and conservative vector fields.

Therefore, we can take the divergence of a curl. The next theorem says that the result is always zero. This result is useful because it gives us a way to show that some vector fields are not the curl of any other field. To give this result a physical interpretation, recall that divergence of a velocity field v at point P measures the tendency of the corresponding fluid to flow out of P. Since

div curl (v) = 0,

the net rate of flow in vector field curl(v) at any point is zero. Taking the curl of vector field F eliminates whatever divergence was present in F.

Proof

With the next two theorems, we show that if F is a conservative vector field then its curl is zero, and if the domain of F is simply connected then the converse is also true. This gives us another way to test whether a vector field is conservative.

Proof

Since conservative vector fields satisfy the cross-partials property, all the cross-partials of F are equal. Therefore,

Since a conservative vector field is the gradient of a scalar function, the previous theorem says that

curl (∇ f) = 0

This equation makes sense because the cross product of a vector with itself is always the zero vector. Sometimes equation

\nabla \times \nabla (f) = 0

Proof

Therefore, F satisfies the cross-partials property on a simply connected domain, and [link] implies that F is conservative.

The same theorem is also true in a plane. Therefore, if F is a vector field in a plane or in space and the domain is simply connected, then F is conservative if and only if

curl F = 0 .

We have seen that the curl of a gradient is zero. What is the divergence of a gradient? If

f

is a function of two variables, then

div (∇ f) = \nabla \cdot (∇ f) = f_{x x} + f_{y y} .

This operator is called the Laplace operator, and in this notation Laplace’s equation becomes

\nabla^{2} f = 0 .

Therefore, a harmonic function is a function that becomes zero after taking the divergence of a gradient.

Using this notation we get Laplace’s equation for harmonic functions of three variables:

Harmonic functions arise in many applications. For example, the potential function of an electrostatic field in a region of space that has no static charge is harmonic.

Key Concepts

Key Equations

For the following exercises, determine whether the statement is true or false.

If the coordinate functions of $F : ℝ^{3} \to ℝ^{3}$

have continuous second partial derivatives, then $curl (div (F))$

equals zero.

\nabla \cdot (x i + y j + z k) = 1 .

False

All vector fields of the form $F (x, y, z) = f (x) i + g (y) j + h (z) k$

are conservative.

If $curl F = 0,$

then F is conservative.

True

If F is a constant vector field then $div F = 0 .$

If F is a constant vector field then $curl F = 0 .$

True

For the following exercises, find the curl of F.

F (x, y, z) = x y^{2} z^{4} i + (2 x^{2} y + z) j + y^{3} z^{2} k

F (x, y, z) = x^{2} z i + y^{2} x j + (y + 2 z) k

curl F = i + x^{2} j + y^{2} k

F (x, y, z) = 3 x y z^{2} i + y^{2} sin z j + x e^{2 z} k

F (x, y, z) = x^{2} y z i + x y^{2} z j + x y z^{2} k

curl F = (x z^{2} - x y^{2}) i + (x^{2} y - y z^{2}) j + (y^{2} z - x^{2} z) k

F (x, y, z) = (x cos y) i + x y^{2} j

F (x, y, z) = (x - y) i + (y - z) j + (z - x) k

curl F = i + j + k

F (x, y, z) = x y z i + x^{2} y^{2} z^{2} j + y^{2} z^{3} k

F (x, y, z) = x y i + y z j + x z k

curl F = − y i - z j - x k

F (x, y, z) = x^{2} i + y^{2} j + z^{2} k

F (x, y, z) = a x i + b y j + c k

for constants a, b, c

curl F = 0

For the following exercises, find the divergence of F.

F (x, y, z) = x^{2} z i + y^{2} x j + (y + 2 z) k

F (x, y, z) = 3 x y z^{2} i + y^{2} sin z j + x e^{2 z} k

div F = 3 y z^{2} + 2 y sin z + 2 x e^{2 z}

F (x, y) = (sin x) i + (cos y) j

F (x, y, z) = x^{2} i + y^{2} j + z^{2} k

div F = 2 (x + y + z)

F (x, y, z) = (x - y) i + (y - z) j + (z - x) k

F (x, y) = \frac{x}{\sqrt{x^{2} + y^{2}}} i + \frac{y}{\sqrt{x^{2} + y^{2}}} j

div F = \frac{1}{\sqrt{x^{2} + y^{2}}}

F (x, y) = x i - y j

F (x, y, z) = a x i + b y j + c k

for constants a, b, c

div F = a + b

F (x, y, z) = x y z i + x^{2} y^{2} z^{2} j + y^{2} z^{3} k

F (x, y, z) = x y i + y z j + x z k

div F = x + y + z

For the following exercises, determine whether each of the given scalar functions is harmonic.

u (x, y, z) = e^{− x} (cos y - sin y)

w (x, y, z) = {(x^{2} + y^{2} + z^{2})}^{− 1 / 2}

Harmonic

If $F (x, y, z) = 2 i + 2 x j + 3 y k$

and $G (x, y, z) = x i - y j + z k,$

find $curl (F \times G) .$

If $F (x, y, z) = 2 i + 2 x j + 3 y k$

and $G (x, y, z) = x i - y j + z k,$

find $div (F \times G) .$

div (F \times G) = 2 z + 3 x

Find $div F,$

given that $F = ∇ f,$

where $f (x, y, z) = x y^{3} z^{2} .$

Find the divergence of F for vector field $F (x, y, z) = (y^{2} + z^{2}) (x + y) i + (z^{2} + x^{2}) (y + z) j + (x^{2} + y^{2}) (z + x) k .$

div F = 2 r^{2}

Find the divergence of F for vector field $F (x, y, z) = f_{1} (y, z) i + f_{2} (x, z) j + f_{3} (x, y) k .$

For the following exercises, use $r = \| r \|$

and $r = (x, y, z) .$

Find the $curl r .$

curl r = 0

Find the $curl \frac{r}{r} .$

Find the $curl \frac{r}{r^{3}} .$

curl \frac{r}{r^{3}} = 0

Let $F (x, y) = \frac{− y i + x j}{x^{2} + y^{2}},$

where F is defined on ${(x, y) \in ℝ \| (x, y) \neq (0, 0)} .$

Find $curl F .$

For the following exercises, use a computer algebra system to find the curl of the given vector fields.

[T] $F (x, y, z) = arctan (\frac{x}{y}) i + ln \sqrt{x^{2} + y^{2}} j + k$

curl F = \frac{2 x}{x^{2} + y^{2}} k

[T] $F (x, y, z) = sin (x - y) i + sin (y - z) j + sin (z - x) k$

For the following exercises, find the divergence of F at the given point.

F (x, y, z) = i + j + k

at $(2, −1, 3)$

div F = 0

F (x, y, z) = x y z i + y j + z k

at $(1, 2, 3)$

F (x, y, z) = e^{− x y} i + e^{x z} j + e^{y z} k

at $(3, 2, 0)$

div F = 2 - 2 e^{−6}

F (x, y, z) = x y z i + y j + z k

at (1, 2, 1)

F (x, y, z) = e^{x} sin y i - e^{x} cos y j

at (0, 0, 3)

div F = 0

For the following exercises, find the curl of F at the given point.

F (x, y, z) = i + j + k

at $(2, −1, 3)$

F (x, y, z) = x y z i + y j + x k

at $(1, 2, 3)$

curl F = j - 3 k

F (x, y, z) = e^{− x y} i + e^{x z} j + e^{y z} k

at (3, 2, 0)

F (x, y, z) = x y z i + y j + z k

at (1, 2, 1)

curl F = 2 j - k

F (x, y, z) = e^{x} sin y i - e^{x} cos y j

at (0, 0, 3)

Let $F (x, y, z) = (3 x^{2} y + a z) i + x^{3} j + (3 x + 3 z^{2}) k .$

For what value of a is F conservative?

a = 3

Given vector field $F (x, y) = \frac{1}{x^{2} + y^{2}} (− y, x)$

on domain $D = \frac{ℝ^{2}}{{(0, 0)}} = {(x, y) \in ℝ^{2} \| (x, y) \neq (0, 0)},$

is F conservative?

Given vector field $F (x, y) = \frac{1}{x^{2} + y^{2}} (x, y)$

on domain $D = \frac{ℝ^{2}}{{(0, 0)}},$

is F conservative?

F is conservative.

Find the work done by force field $F (x, y) = e^{− y} i - x e^{− y} j$

in moving an object from P(0, 1) to Q(2, 0). Is the force field conservative?

Compute divergence $F = (sinh x) i + (cosh y) j - x y z k .$

div F = cosh x + sinh y - x y

Compute curl $F = (sinh x) i + (cosh y) j - x y z k .$

For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity $ω = 〈 a, b, c 〉 .$

If P is a point in the body located at $r = x i + y j + z k,$

the velocity at P is given by vector field $F = ω \times r .$

A three dimensional diagram of an object rotating about the x axis in a counterclockwise manner with constant angular velocity w = <a,b,c>. The object is roughly a sphere with pointed ends on the x axis, which cuts it in half. An arrow r is drawn from (0,0,0) to P(x,y,z) and down from P(x,y,z) to the x axis.

Express F in terms of i, j, and k vectors.

(b z - c y) i (c x - a z) j + (a y - b x) k

Find $div F .$

Find $curl F$

curl F = 2 ω

In the following exercises, suppose that $\nabla \cdot F = 0$

and $\nabla \cdot G = 0 .$

Does $F + G$

necessarily have zero divergence?

Does $F \times G$

necessarily have zero divergence?

F \times G

does not have zero divergence.

In the following exercises, suppose a solid object in $ℝ^{3}$

has a temperature distribution given by $T (x, y, z) .$

The heat flow vector field in the object is $F = − k ∇ T,$

where $k > 0$

is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is $\nabla \cdot F = − k \nabla \cdot ∇ T = − k \nabla^{2} T .$

Compute the heat flow vector field.

Compute the divergence.

\nabla \cdot F = −200 k [1 + 2 (x^{2} + y^{2} + z^{2})] e^{− x^{2} + y^{2} + z^{2}}

[T] Consider rotational velocity field $v = 〈 0, 10 z, −10 y 〉 .$

If a paddlewheel is placed in plane $x + y + z = 1$

with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit time.

A three dimensional diagram of a rotational velocity field. The arrows are showing a rotation in a clockwise manner. A paddlewheel is shown in plan x + y + z = 1 with n extended out perpendicular to the plane.

Divergence and Curl

Divergence

Proof

Curl

Using Divergence and Curl

Proof

Proof

Proof

Key Concepts

Key Equations

Glossary