MULTIVARIABLE CALCULUS 09

Divergence and Curl

Divergence measures local spreading or compression in a vector field, while curl measures local rotational tendency.

The main idea

Suppose we have a two-dimensional vector field

\[ \mathbf{F}(x,y) = \begin{pmatrix} P(x,y)\\ Q(x,y) \end{pmatrix}. \]

At every point, the field gives a direction and magnitude.

Two important questions are:

\[ \boxed{ \text{Is the field locally spreading outward or flowing inward?} } \] \[ \boxed{ \text{Does the field have a local tendency to rotate?} } \]

Divergence answers the first question. Curl answers the second.

1. Divergence

For

\[ \mathbf{F} = \begin{pmatrix} P\\ Q \end{pmatrix}, \]

the divergence is

\[ \boxed{ \nabla\cdot\mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} } \]

The result is a scalar.

2. Reading the notation

The expression

\[ \nabla\cdot\mathbf{F} \]

is read as

“the divergence of \(\mathbf{F}\)”

or sometimes

“del dot \(\mathbf{F}\)”.

3. What does divergence measure?

Imagine a very small region surrounding a point in a flowing fluid.

If more fluid flows out of the region than flows into it, the point behaves locally like a source.

If more flows in than out, it behaves locally like a sink.

\[ \boxed{ \nabla\cdot\mathbf{F}>0 \quad\Rightarrow\quad \text{local outward spreading} } \] \[ \boxed{ \nabla\cdot\mathbf{F}<0 \quad\Rightarrow\quad \text{local inward compression} } \] \[ \boxed{ \nabla\cdot\mathbf{F}=0 \quad\Rightarrow\quad \text{no net local expansion} } \]

4. Example of positive divergence

Consider

\[ \mathbf{F}(x,y) = \begin{pmatrix} x\\ y \end{pmatrix}. \]

Here,

\[ P=x, \qquad Q=y. \]

Therefore,

\[ \frac{\partial P}{\partial x}=1 \]

and

\[ \frac{\partial Q}{\partial y}=1. \]

Hence,

\[ \boxed{ \nabla\cdot\mathbf{F}=2 } \]

The positive value agrees with the fact that the vectors point outward from the origin.

5. Outward-flow diagram

6. Example of negative divergence

Consider

\[ \mathbf{F}(x,y) = \begin{pmatrix} -x\\ -y \end{pmatrix}. \]

Then

\[ \frac{\partial(-x)}{\partial x}=-1 \]

and

\[ \frac{\partial(-y)}{\partial y}=-1. \]

Therefore,

\[ \boxed{ \nabla\cdot\mathbf{F}=-2 } \]

The vectors point inward, so the field has negative divergence.

7. Zero divergence does not mean zero field

Consider

\[ \mathbf{F}(x,y) = \begin{pmatrix} -y\\ x \end{pmatrix}. \]

Then

\[ \frac{\partial(-y)}{\partial x}=0 \]

and

\[ \frac{\partial x}{\partial y}=0. \]

\[ \boxed{ \nabla\cdot\mathbf{F}=0 } \]

The field is not zero. It simply has no net local expansion or contraction.

8. Curl

Curl measures local rotational tendency.

For a two-dimensional field

\[ \mathbf{F} = \begin{pmatrix} P\\ Q \end{pmatrix}, \]

the scalar rotational component is

\[ \boxed{ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} } \]

This is the component of the three-dimensional curl perpendicular to the \(xy\)-plane.

9. Example of rotation

Again consider

\[ \mathbf{F}(x,y) = \begin{pmatrix} -y\\ x \end{pmatrix}. \]

Here,

\[ P=-y, \qquad Q=x. \]

Therefore,

\[ \frac{\partial Q}{\partial x}=1 \]

and

\[ \frac{\partial P}{\partial y}=-1. \]

Hence,

\[ \boxed{ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2 } \]

The nonzero result agrees with the rotational behaviour of the field.

10. Rotational-field diagram

11. Curl in three dimensions

For a three-dimensional vector field

\[ \mathbf{F} = \begin{pmatrix} P\\ Q\\ R \end{pmatrix}, \]

the curl is a vector:

\[ \boxed{ \nabla\times\mathbf{F} = \begin{pmatrix} R_y-Q_z\\ P_z-R_x\\ Q_x-P_y \end{pmatrix} } \]

12. Determinant notation for curl

The same expression is often remembered using

\[ \nabla\times\mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ \partial/\partial x & \partial/\partial y & \partial/\partial z\\ P & Q & R \end{vmatrix}. \]

This is a useful mnemonic for calculating the components of the curl.

13. Physical intuition for curl

Imagine placing a tiny paddle wheel in a flowing fluid.

If the local flow tends to make the wheel rotate, the curl is nonzero.

The direction of the curl vector gives the axis of local rotation, according to the right-hand rule.

14. Divergence and curl are different

A field can spread without rotating.

For example,

\[ \mathbf{F}=(x,y) \]

has positive divergence but zero two-dimensional curl.

A field can also rotate without spreading.

For example,

\[ \mathbf{F}=(-y,x) \]

has zero divergence but nonzero curl.

15. Comparison

\[ \boxed{ \nabla\cdot\mathbf{F} \quad \text{measures local expansion or compression} } \] \[ \boxed{ \nabla\times\mathbf{F} \quad \text{measures local rotation} } \]

16. The operator \(\nabla\)

In three dimensions,

\[ \nabla = \begin{pmatrix} \partial/\partial x\\ \partial/\partial y\\ \partial/\partial z \end{pmatrix}. \]

Applying it to a scalar field gives the gradient:

\[ \boxed{ \nabla f = \text{gradient} } \]

Taking its dot product with a vector field gives divergence:

\[ \boxed{ \nabla\cdot\mathbf{F} = \text{divergence} } \]

Taking its cross product with a vector field gives curl:

\[ \boxed{ \nabla\times\mathbf{F} = \text{curl} } \]

17. Notice the different outputs

If \(f\) is a scalar field,

\[ \nabla f \]

is a vector.

If \(\mathbf{F}\) is a vector field,

\[ \nabla\cdot\mathbf{F} \]

is a scalar.

In three dimensions,

\[ \nabla\times\mathbf{F} \]

is a vector.

\[ \boxed{ \text{scalar} \xrightarrow{\nabla} \text{vector} } \] \[ \boxed{ \text{vector} \xrightarrow{\nabla\cdot} \text{scalar} } \] \[ \boxed{ \text{vector} \xrightarrow{\nabla\times} \text{vector} } \]

18. Divergence in conservation models

Suppose

\[ \rho(x,y,z,t) \]

represents the density of some conserved quantity and

\[ \mathbf{J} \]

represents its flux.

A basic conservation equation is

\[ \boxed{ \frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{J} = 0 } \]

If material flows out of a small region, the amount remaining inside decreases.

19. Why this matters in mathematical modelling

Divergence and curl allow us to describe local properties of flows without following every individual particle.

They appear in fluid dynamics, electromagnetism, diffusion, conservation laws and partial differential equations.

20. A useful warning

A picture of arrows can suggest whether a field is spreading or rotating, but divergence and curl are local derivative quantities.

They are calculated from how the vector components change near each point.

So the mathematical definitions, rather than visual appearance alone, determine their values.

21. The central idea

\[ \boxed{ \mathbf{F} \longrightarrow \nabla\cdot\mathbf{F} \longrightarrow \text{local expansion} } \] \[ \boxed{ \mathbf{F} \longrightarrow \nabla\times\mathbf{F} \longrightarrow \text{local rotation} } \]

Together, divergence and curl reveal two fundamental local behaviours hidden inside a vector field.