The Central Idea
For a vector field, two important local questions are:
- Is the field spreading outward or converging inward?
- Is the field tending to rotate?
Divergence measures the first behavior.
Curl measures the second.
Both are built from derivatives of the vector-field components.
Divergence in Two Dimensions
Let
\[ \mathbf{F}(x,y) = \langle P(x,y),Q(x,y)\rangle. \]The divergence is
\[ \boxed{ \nabla\cdot\mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} }. \]The result is a scalar field.
What Divergence Measures
Divergence measures the local tendency of a vector field to expand away from a point or contract toward it.
Informally:
\[ \boxed{ \nabla\cdot\mathbf{F}>0 \Rightarrow \text{local spreading} } \] \[ \boxed{ \nabla\cdot\mathbf{F}<0 \Rightarrow \text{local convergence} } \] \[ \boxed{ \nabla\cdot\mathbf{F}=0 \Rightarrow \text{no net local expansion to first order} }. \]A Simple Outward Field
Consider
\[ \mathbf{F}(x,y) = \langle x,y\rangle. \]Then
\[ \frac{\partial P}{\partial x}=1 \]and
\[ \frac{\partial Q}{\partial y}=1. \]Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F}=2 }. \]The positive divergence matches the outward-spreading pattern.
A Simple Inward Field
Consider
\[ \mathbf{F}(x,y) = \langle -x,-y\rangle. \]Then
\[ \frac{\partial P}{\partial x}=-1, \qquad \frac{\partial Q}{\partial y}=-1. \]Hence,
\[ \boxed{ \nabla\cdot\mathbf{F}=-2 }. \]The negative divergence reflects local convergence toward the origin.
Zero Divergence Does Not Mean Zero Field
Consider
\[ \mathbf{F}(x,y) = \langle -y,x\rangle. \]Then
\[ \frac{\partial P}{\partial x}=0 \]and
\[ \frac{\partial Q}{\partial y}=0. \]Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F}=0 }. \]The field is not zero. It simply has no net local expansion or contraction to first order.
Divergence in Three Dimensions
Let
\[ \mathbf{F}(x,y,z) = \langle P,Q,R\rangle. \]Then
\[ \boxed{ \nabla\cdot\mathbf{F} = P_x+Q_y+R_z }. \]Divergence still produces a scalar field.
Example: Three-Dimensional Divergence
Let
\[ \mathbf{F}(x,y,z) = \langle x^2,2y,3z\rangle. \]Then
\[ P_x=2x, \qquad Q_y=2, \qquad R_z=3. \]Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F}=2x+5 }. \]Divergence Can Vary from Point to Point
In the previous example,
\[ \nabla\cdot\mathbf{F}=2x+5. \]Therefore:
- the divergence is positive when \(x>-5/2\);
- zero when \(x=-5/2\);
- negative when \(x<-5/2\).
One vector field can therefore contain locally expanding and locally contracting regions.
Source-Like Behavior
Positive divergence is often described as source-like.
Imagine a tiny region surrounding a point.
If more field tendency points outward through the boundary than inward, the point behaves locally like a source.
Divergence quantifies this tendency in the infinitesimal limit.
Sink-Like Behavior
Negative divergence is often described as sink-like.
A tiny region around the point experiences more inward than outward field tendency.
In a velocity-field interpretation, this can correspond to local compression or convergence.
Divergence and Fluid Flow
If
\[ \mathbf{v}(x,y,z) \]is a fluid velocity field, divergence describes local volumetric expansion or contraction of the flow.
In an incompressible flow model, one often has
\[ \boxed{ \nabla\cdot\mathbf{v}=0 }. \]This does not mean the fluid is motionless.
Curl in Three Dimensions
For
\[ \mathbf{F}(x,y,z) = \langle P,Q,R\rangle, \]the curl is
\[ \boxed{ \nabla\times\mathbf{F} = \left\langle R_y-Q_z,\, P_z-R_x,\, Q_x-P_y \right\rangle }. \]The result is a vector field.
Determinant Memory Form for Curl
Curl is often remembered symbolically as
\[ \nabla\times\mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ \partial/\partial x & \partial/\partial y & \partial/\partial z\\ P & Q & R \end{vmatrix}. \]This is a mnemonic for the component formula, not an ordinary numerical determinant.
What Curl Measures
Curl measures local rotational tendency.
Its direction indicates the local axis of rotation according to the right-hand rule.
Its magnitude measures the strength of that local rotational tendency.
A Rotational Field
Consider the planar field
\[ \mathbf{F}(x,y) = \langle -y,x\rangle. \]Embed it in three dimensions:
\[ \mathbf{F}(x,y,z) = \langle -y,x,0\rangle. \]Its curl is
\[ \nabla\times\mathbf{F} = \langle 0,0,2\rangle. \]The positive \(z\)-direction matches counterclockwise rotation in the \(xy\)-plane.
Planar Curl
For a two-dimensional field
\[ \mathbf{F}(x,y) = \langle P,Q\rangle, \]the relevant rotational component is
\[ \boxed{ Q_x-P_y }. \]This can be treated as the \(z\)-component of the three-dimensional curl:
\[ \nabla\times\mathbf{F} = \langle 0,0,Q_x-P_y\rangle. \]Positive and Negative Planar Curl
In the usual \(xy\)-orientation:
\[ Q_x-P_y>0 \]corresponds to a local counterclockwise rotational tendency.
Meanwhile,
\[ Q_x-P_y<0 \]corresponds to a local clockwise rotational tendency.
Zero Curl Does Not Mean Zero Field
Consider
\[ \mathbf{F}(x,y)=\langle x,y\rangle. \]Then
\[ Q_x=0 \]and
\[ P_y=0. \]Therefore,
\[ \boxed{ Q_x-P_y=0 }. \]The field is strongly outward, but it has no local rotational tendency.
Irrotational Fields
A field whose curl is zero is called irrotational on that region.
In three dimensions,
\[ \boxed{ \nabla\times\mathbf{F}=\mathbf{0} }. \]Irrotational does not mean constant, zero, or divergence-free.
Divergence-Free Fields
A field satisfying
\[ \boxed{ \nabla\cdot\mathbf{F}=0 } \]is called divergence-free or solenoidal.
Such a field has no net infinitesimal source or sink behavior.
A Field Can Be Both Divergence-Free and Rotational
For
\[ \mathbf{F}(x,y)=\langle -y,x\rangle, \]we found
\[ \nabla\cdot\mathbf{F}=0 \]but
\[ Q_x-P_y=2. \]The field has no net local expansion, yet it has clear rotational tendency.
A Field Can Be Irrotational but Have Nonzero Divergence
For
\[ \mathbf{F}(x,y)=\langle x,y\rangle, \]we have
\[ \nabla\cdot\mathbf{F}=2 \]and
\[ Q_x-P_y=0. \]The field spreads outward without local rotation.
Divergence and Curl Measure Different Things
Divergence asks about local expansion and contraction.
Curl asks about local rotation.
Neither quantity replaces the other.
A field may have:
- both zero;
- nonzero divergence and zero curl;
- zero divergence and nonzero curl;
- both nonzero.
Example: Both Divergence and Curl Nonzero
Let
\[ \mathbf{F}(x,y) = \langle x-y,x+y\rangle. \]Then
\[ P_x=1, \qquad Q_y=1, \]so
\[ \boxed{ \nabla\cdot\mathbf{F}=2 }. \]Also,
\[ Q_x=1, \qquad P_y=-1, \]so
\[ \boxed{ Q_x-P_y=2 }. \]Local Paddle-Wheel Intuition for Curl
Imagine placing a tiny paddle wheel in a flowing fluid.
If the surrounding velocity field tends to spin the wheel, the local curl is nonzero.
The curl vector points along the axis about which the wheel tends to rotate.
This is an intuition, not a replacement for the derivative definition.
Local Balloon Intuition for Divergence
Imagine a tiny fluid parcel surrounding a point.
Positive divergence tends to make the parcel expand.
Negative divergence tends to make it contract.
Zero divergence means no first-order change in local volume from net expansion or contraction.
Divergence Is Local
A positive value of
\[ \nabla\cdot\mathbf{F} \]at one point says something about behavior near that point.
It does not mean the entire field is globally spreading everywhere.
Divergence can change sign across the domain.
Curl Is Local
A nonzero curl at one point indicates local rotational tendency there.
It does not automatically mean every trajectory forms a closed circle or that the entire field rotates rigidly.
Curl describes infinitesimal behavior, not necessarily global path shape.
Rigid Rotation Example
Consider
\[ \mathbf{F}(x,y) = \langle -\omega y,\omega x\rangle, \]where \(\omega\) is constant.
The planar curl is
\[ Q_x-P_y = \omega-(-\omega) = \boxed{2\omega}. \]The factor \(2\) is characteristic of the relationship between curl and rigid angular rotation.
Why Curl Is Twice the Local Angular Velocity in Rigid Rotation
For ideal rigid-body rotation with angular velocity \(\omega\), the field is
\[ \langle -\omega y,\omega x\rangle. \]The curl magnitude is
\[ 2|\omega|. \]Thus half the curl corresponds to the local angular-velocity vector in this special rigid-rotation setting.
Shear Can Produce Curl
A field does not need circular trajectories to have nonzero curl.
Consider
\[ \mathbf{F}(x,y)=\langle y,0\rangle. \]Then
\[ Q_x=0, \qquad P_y=1. \]Therefore,
\[ \boxed{ Q_x-P_y=-1 }. \]Differential shear alone can create local rotational tendency.
Visual Rotation and Curl Are Related but Not Identical
A vector-field plot can suggest rotation, but visual appearance can be misleading.
The field may curve globally without having nonzero curl everywhere.
Conversely, local shear can create nonzero curl even when the arrows do not form obvious circles.
Derivatives provide the reliable test.
Divergence and the Jacobian Matrix
For
\[ \mathbf{F}(x,y) = \langle P,Q\rangle, \]the Jacobian matrix is
\[ J_{\mathbf{F}} = \begin{pmatrix} P_x & P_y\\ Q_x & Q_y \end{pmatrix}. \]The divergence is the trace:
\[ \boxed{ \nabla\cdot\mathbf{F} = P_x+Q_y }. \]Curl Also Comes from Local Derivative Structure
In two dimensions, the combination
\[ Q_x-P_y \]extracts the antisymmetric part of the local derivative behavior.
This is why curl is closely connected to local rotation rather than local expansion.
Expansion and Rotation Can Be Separated Locally
Near a point, the derivative structure of a vector field can be thought of as combining several effects:
- expansion or contraction;
- rotation;
- shear;
- directional stretching.
Divergence and curl isolate two especially important parts of this local behavior.
Divergence of a Constant Field
Let
\[ \mathbf{F}(x,y,z) = \langle a,b,c\rangle, \]where \(a,b,c\) are constants.
Every spatial derivative is zero, so
\[ \boxed{ \nabla\cdot\mathbf{F}=0 }. \]A uniform field has no local source or sink behavior.
Curl of a Constant Field
For the same constant field,
\[ \mathbf{F} = \langle a,b,c\rangle, \]all component derivatives are zero.
Therefore,
\[ \boxed{ \nabla\times\mathbf{F}=\mathbf{0} }. \]A spatially uniform field has neither local expansion nor local rotation.
Divergence of a Gradient
If
\[ \mathbf{F}=\nabla f, \]then
\[ \nabla\cdot\mathbf{F} = \nabla\cdot(\nabla f). \]This second-order scalar operator is called the Laplacian:
\[ \boxed{ \nabla^2 f = f_{xx}+f_{yy}+f_{zz} }. \]A full study of the Laplacian belongs to later analysis, but the connection begins here.
Curl of a Gradient
For a sufficiently smooth scalar field \(f\),
\[ \boxed{ \nabla\times(\nabla f)=\mathbf{0} }. \]Therefore, gradient fields are locally irrotational under the usual smoothness assumptions.
This connects curl to the earlier study of conservative fields.
Divergence of a Curl
For a sufficiently smooth vector field \(\mathbf{F}\),
\[ \boxed{ \nabla\cdot(\nabla\times\mathbf{F})=0 }. \]This identity is another structural relationship among field operators.
Curl and Conservative Fields
If
\[ \mathbf{F}=\nabla\phi, \]then
\[ \nabla\times\mathbf{F} = \mathbf{0} \]wherever the required second derivatives are sufficiently smooth.
Thus zero curl is a necessary local condition for a smooth conservative field.
Zero Curl Is Not Always Enough Globally
A field can have zero curl everywhere on a domain and still fail to have a single-valued global potential if the domain has a suitable topological obstruction.
On simply connected domains, zero curl provides a much stronger route to global conservativeness under the standard smoothness assumptions.
Local derivative conditions and global topology must be distinguished.
Divergence and Flux
Divergence describes infinitesimal outward flow.
Flux measures field passage through a boundary.
These ideas are linked by the divergence theorem, which connects local divergence throughout a volume to total outward flux through its boundary.
The next flux lesson develops the boundary side of this relationship.
Curl and Circulation
Curl describes infinitesimal rotational tendency.
Circulation measures how strongly a field runs along a closed boundary.
Stokes-type theorems connect local curl over a surface with circulation around the surface boundary.
This is a major local-to-global relationship in vector calculus.
Divergence in Radial Fields
Radial appearance alone does not determine divergence.
The field
\[ \mathbf{F}=\langle x,y\rangle \]has positive divergence.
But other radial fields can have zero divergence away from singularities.
The exact derivative structure matters.
A Radial Inverse-Square Field
In three dimensions, consider
\[ \mathbf{F}(\mathbf{r}) = \frac{\mathbf{r}}{\|\mathbf{r}\|^3}, \qquad \mathbf{r}\neq\mathbf{0}. \]Away from the origin, this field has
\[ \boxed{ \nabla\cdot\mathbf{F}=0 }. \]Yet it points strongly outward and has nonzero flux through spheres.
The missing origin is crucial to the global interpretation.
Singularities Complicate Global Conclusions
The inverse-square example shows that a field can have zero divergence at every ordinary point in its domain while still behaving globally as though a source is concentrated at an excluded singularity.
Local formulas must therefore be interpreted together with domain structure.
Physical Interpretation of Divergence
Depending on the model, divergence can be associated with:
- fluid expansion or compression;
- source density;
- net outward transport;
- local production or depletion of a transported quantity.
The exact physical meaning depends on what the vector field represents.
Physical Interpretation of Curl
Depending on the model, curl can be associated with:
- fluid vorticity;
- local rotational tendency;
- circulation density;
- spatial variation that drives rotational effects.
Again, the operator is mathematical; the physical interpretation depends on the field.
Units of Divergence
Divergence differentiates vector components with respect to distance.
If a velocity field has units
\[ \text{m/s}, \]then its divergence has units
\[ \frac{\text{m/s}}{\text{m}} = \boxed{\text{s}^{-1}}. \]Units help clarify what the quantity means physically.
Units of Curl
Curl also differentiates vector components with respect to distance.
Therefore, for a velocity field measured in meters per second, curl also has units
\[ \boxed{\text{s}^{-1}}. \]This is consistent with its relationship to angular rotation rate.
Common Error: Adding the Wrong Partial Derivatives for Divergence
For
\[ \mathbf{F} = \langle P,Q,R\rangle, \]divergence uses matching coordinate derivatives:
\[ \boxed{ P_x+Q_y+R_z }. \]It does not use every possible derivative.
Common Error: Treating Divergence as a Vector
The divergence of a vector field is a scalar:
\[ \boxed{ \nabla\cdot\mathbf{F} = \text{scalar} }. \]The dot in the operator is a reminder that the result behaves like a scalar combination.
Common Error: Treating Curl as a Scalar in Three Dimensions
In three dimensions,
\[ \boxed{ \nabla\times\mathbf{F} = \text{vector} }. \]Only in a planar setting do we often focus on the single perpendicular component
\[ Q_x-P_y. \]Common Error: Reversing the Planar Curl Sign
For
\[ \mathbf{F}=\langle P,Q\rangle, \]the standard \(z\)-component is
\[ \boxed{ Q_x-P_y }. \]Reversing the order changes the sign and therefore reverses the interpreted rotation direction.
Common Error: Assuming Zero Divergence Means No Motion
The rotational field
\[ \langle -y,x\rangle \]has zero divergence but produces continuous circular motion.
Zero divergence means no net local expansion, not zero velocity.
Common Error: Assuming Zero Curl Means No Change
The outward field
\[ \langle x,y\rangle \]has zero curl but changes strongly from point to point.
Zero curl means no local rotational tendency, not no spatial variation.
Common Error: Reading Global Rotation from Curl Alone
Curl is local.
A nonzero curl value does not guarantee a globally closed orbit.
To determine full trajectory behavior, the entire vector field must be analyzed.
Common Error: Ignoring Singularities and Holes
Local conditions such as
\[ \nabla\times\mathbf{F}=\mathbf{0} \]do not automatically settle global questions about potentials or circulation when the domain has excluded points or holes.
A Reliable Procedure for Divergence
- Write the vector field in component form.
- Differentiate the first component with respect to the first coordinate.
- Differentiate the second component with respect to the second coordinate.
- Continue through all spatial coordinates.
- Add the matching partial derivatives.
- Evaluate at a point if requested.
- Interpret the sign locally.
A Reliable Procedure for Curl in Three Dimensions
- Write \(\mathbf{F}=\langle P,Q,R\rangle\).
- Compute the six cross derivatives needed in the curl formula.
- Form \(R_y-Q_z\).
- Form \(P_z-R_x\).
- Form \(Q_x-P_y\).
- Assemble the three components in order.
- Interpret direction and magnitude when relevant.
A Reliable Procedure for Planar Curl
- Write \(\mathbf{F}=\langle P,Q\rangle\).
- Compute \(Q_x\).
- Compute \(P_y\).
- Subtract \(P_y\) from \(Q_x\).
- Interpret a positive value as counterclockwise local tendency and a negative value as clockwise local tendency under the standard orientation.
Example: Full Divergence Workflow
Let
\[ \mathbf{F}(x,y,z) = \langle x^2y,\ yz,\ xz^2\rangle. \]Compute
\[ P_x=2xy, \] \[ Q_y=z, \] \[ R_z=2xz. \]Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F} = 2xy+z+2xz }. \]Example: Full Curl Workflow
Let
\[ \mathbf{F}(x,y,z) = \langle yz,\ xz,\ xy\rangle. \]Then
\[ R_y=x, \qquad Q_z=x, \] \[ P_z=y, \qquad R_x=y, \] \[ Q_x=z, \qquad P_y=z. \]Therefore,
\[ \boxed{ \nabla\times\mathbf{F} = \langle 0,0,0\rangle }. \]Example: Planar Divergence and Curl Together
Let
\[ \mathbf{F}(x,y) = \langle x^2-y,\ x+y^2\rangle. \]Divergence:
\[ P_x=2x, \qquad Q_y=2y, \] \[ \boxed{ \nabla\cdot\mathbf{F}=2x+2y }. \]Planar curl:
\[ Q_x=1, \qquad P_y=-1, \] \[ \boxed{ Q_x-P_y=2 }. \]Practice 1: Divergence in Two Dimensions
Let
\[ \mathbf{F}(x,y) = \langle 3x,2y\rangle. \]Find \(\nabla\cdot\mathbf{F}\).
Practice 2: Divergence Sign
If
\[ \nabla\cdot\mathbf{F}=-4 \]at a point, does the field behave locally more like a source or a sink?
Practice 3: Planar Curl
Let
\[ \mathbf{F}(x,y) = \langle -2y,2x\rangle. \]Find \(Q_x-P_y\).
Practice 4: Curl Direction
If the planar curl is positive, what local rotational orientation does that indicate under the standard \(xy\)-orientation?
Practice 5: Divergence of a Constant Field
Let
\[ \mathbf{F}(x,y,z)=\langle 4,-1,7\rangle. \]Find its divergence.
Practice 6: Curl of a Constant Field
For
\[ \mathbf{F}(x,y,z)=\langle 4,-1,7\rangle, \]find its curl.
Practice 7: Divergence-Free but Rotational
Verify that
\[ \mathbf{F}(x,y)=\langle -y,x\rangle \]has zero divergence but nonzero planar curl.
Practice 8: Irrotational but Expanding
Verify that
\[ \mathbf{F}(x,y)=\langle x,y\rangle \]has zero planar curl but positive divergence.
Practice 9: Three-Dimensional Divergence
Let
\[ \mathbf{F}(x,y,z) = \langle x^2,xy,z^3\rangle. \]Find \(\nabla\cdot\mathbf{F}\).
Practice 10: Three-Dimensional Curl
Let
\[ \mathbf{F}(x,y,z) = \langle y,z,x\rangle. \]Find \(\nabla\times\mathbf{F}\).
Practice Answers
Practice 1
\[ P_x=3, \qquad Q_y=2. \]Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F}=5 }. \]Practice 2
Negative divergence indicates local convergence, so the field behaves more like
\[ \boxed{\text{a sink}}. \]Practice 3
\[ Q_x=2, \qquad P_y=-2. \]Therefore,
\[ \boxed{ Q_x-P_y=4 }. \]Practice 4
A positive planar curl indicates
\[ \boxed{\text{counterclockwise local rotational tendency}}. \]Practice 5
All component derivatives vanish, so
\[ \boxed{ \nabla\cdot\mathbf{F}=0 }. \]Practice 6
All component derivatives vanish, so
\[ \boxed{ \nabla\times\mathbf{F}=\mathbf{0} }. \]Practice 7
Divergence:
\[ P_x=0, \qquad Q_y=0, \] \[ \boxed{ \nabla\cdot\mathbf{F}=0 }. \]Planar curl:
\[ Q_x=1, \qquad P_y=-1, \] \[ \boxed{ Q_x-P_y=2 }. \]Practice 8
Divergence:
\[ P_x=1, \qquad Q_y=1, \] \[ \boxed{ \nabla\cdot\mathbf{F}=2 }. \]Planar curl:
\[ Q_x=0, \qquad P_y=0, \] \[ \boxed{ Q_x-P_y=0 }. \]Practice 9
\[ P_x=2x, \qquad Q_y=x, \qquad R_z=3z^2. \]Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F} = 3x+3z^2 }. \]Practice 10
Here,
\[ P=y, \qquad Q=z, \qquad R=x. \]Compute:
\[ R_y-Q_z=0-1=-1, \] \[ P_z-R_x=0-1=-1, \] \[ Q_x-P_y=0-1=-1. \]Therefore,
\[ \boxed{ \nabla\times\mathbf{F} = \langle -1,-1,-1\rangle }. \]What to Remember
Divergence and curl measure two different kinds of local vector-field behavior.
Divergence:
\[ \boxed{ \nabla\cdot\mathbf{F} = P_x+Q_y+R_z } \]measures local expansion or convergence.
Curl:
\[ \boxed{ \nabla\times\mathbf{F} = \langle R_y-Q_z,\ P_z-R_x,\ Q_x-P_y\rangle } \]measures local rotational tendency.
- divergence is scalar-valued;
- curl is vector-valued in three dimensions;
- positive divergence is source-like;
- negative divergence is sink-like;
- zero divergence does not mean zero motion;
- zero curl does not mean a constant field;
- a field may have divergence without curl, curl without divergence, both, or neither;
- local derivative behavior must still be interpreted with the global domain in mind.
The central distinction is: divergence measures local spreading, while curl measures local turning.
Where We Go Next
Divergence and curl describe local behavior inside a vector field.
The next lesson, Flux through Curves and Surfaces, measures how much of a vector field passes through a boundary and begins connecting local field behavior to accumulated global quantities.