The Central Idea
A vector field assigns a vector to every point:
\[ \mathbf{F}(x,y,z). \]We can ask two important local questions:
- Does the field spread outward from the point or converge inward?
- Does the field tend to rotate around the point?
Divergence answers the first question.
Curl answers the second.
From Fields to Differential Operators
The previous lesson introduced the gradient:
\[ \nabla f. \]The gradient acts on a scalar field and produces a vector field.
Divergence and curl instead act on vector fields:
\[ \mathbf{F} = \langle P,Q,R\rangle. \]Their outputs have different mathematical types.
A Type Map
It is useful to keep the three major differential operations organized by input and output type.
\[ \boxed{ f \xrightarrow{\ \nabla\ } \nabla f } \]Gradient:
\[ \text{scalar field} \longrightarrow \text{vector field}. \]Divergence:
\[ \boxed{ \mathbf{F} \xrightarrow{\ \nabla\cdot\ } \nabla\cdot\mathbf{F} } \] \[ \text{vector field} \longrightarrow \text{scalar field}. \]Curl in three dimensions:
\[ \boxed{ \mathbf{F} \xrightarrow{\ \nabla\times\ } \nabla\times\mathbf{F} } \] \[ \text{vector field} \longrightarrow \text{vector field}. \]Divergence
Let
\[ \mathbf{F}(x,y,z) = \langle P,Q,R\rangle. \]The divergence of \(\mathbf{F}\) is
\[ \boxed{ \nabla\cdot\mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} }. \]Divergence is a scalar field.
Divergence in Two Dimensions
For
\[ \mathbf{F}(x,y) = \langle P(x,y),Q(x,y)\rangle, \]the two-dimensional divergence is
\[ \boxed{ \nabla\cdot\mathbf{F} = P_x+Q_y }. \]This quantity measures local expansion or compression of the field in the plane.
Why Divergence Is a Dot Product
Recall the del operator
\[ \nabla = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right\rangle. \]Formally dotting this operator with
\[ \mathbf{F} = \langle P,Q,R\rangle \]gives
\[ \nabla\cdot\mathbf{F} = P_x+Q_y+R_z. \]The notation emphasizes that divergence combines corresponding coordinate directions.
Geometric Meaning of Positive Divergence
If
\[ \nabla\cdot\mathbf{F}>0 \]near a point, the field behaves locally like a source.
More field tends to flow outward from a small region than inward.
In a fluid-flow interpretation, a tiny volume element tends to expand.
Geometric Meaning of Negative Divergence
If
\[ \nabla\cdot\mathbf{F}<0, \]the field behaves locally like a sink.
More field tends to flow inward than outward.
In a fluid interpretation, a small material region tends to compress.
Zero Divergence
If
\[ \nabla\cdot\mathbf{F}=0, \]there is no net local source or sink behavior at that point.
This does not mean that the field itself is zero.
A field can have large vectors and substantial motion while still having zero divergence.
Example: Divergence of a Radial Field
Consider
\[ \mathbf{F}(x,y,z) = \langle x,y,z\rangle. \]Then
\[ P_x=1,\qquad Q_y=1,\qquad R_z=1. \]Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F}=3 }. \]The positive divergence agrees with the visual pattern: vectors point outward from the origin.
Example: Divergence of an Inward Field
Let
\[ \mathbf{F}(x,y,z) = \langle -x,-y,-z\rangle. \]Then
\[ \nabla\cdot\mathbf{F} = -1-1-1 = \boxed{-3}. \]The field points inward and behaves like a sink.
Example: A Divergence-Free Field
Consider
\[ \mathbf{F}(x,y,z) = \langle -y,x,0\rangle. \]Then
\[ P_x=0,\qquad Q_y=0,\qquad R_z=0. \]Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F}=0 }. \]This field rotates around the \(z\)-axis without locally expanding or contracting.
Divergence as Local Flux Density
Divergence can be interpreted as net outward flux per unit volume in the limit of a very small region.
Informally,
\[ \boxed{ \nabla\cdot\mathbf{F} \approx \frac{\text{net outward flux}} {\text{small volume}} }. \]The Divergence Theorem will later turn this local idea into a global relationship between volume and surface integrals.
Divergence and Incompressible Flow
In an idealized incompressible velocity field,
\[ \boxed{ \nabla\cdot\mathbf{v}=0 }. \]This means fluid parcels do not locally expand or contract because of the velocity field.
The condition appears frequently in fluid mechanics.
Curl
Curl measures local rotational tendency of a vector field.
For
\[ \mathbf{F} = \langle P,Q,R\rangle, \]the curl is
\[ \boxed{ \nabla\times\mathbf{F} }. \]In three dimensions, curl is itself a vector field.
The Curl Formula
The three-dimensional formula is
\[ \boxed{ \nabla\times\mathbf{F} = \left\langle R_y-Q_z,\; P_z-R_x,\; Q_x-P_y \right\rangle }. \]Equivalently,
\[ \nabla\times\mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\mathbf{k}. \]Determinant Mnemonic for Curl
Curl is often remembered using the formal determinant
\[ \nabla\times\mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z}\\ P & Q & R \end{vmatrix}. \]Expanding along the first row reproduces the curl formula.
This is a computational mnemonic rather than an ordinary determinant of numerical entries.
Why Curl Is a Vector
Local rotation has both an amount and an axis.
The direction of
\[ \nabla\times\mathbf{F} \]identifies the local axis of rotational tendency.
Its magnitude measures the strength of that tendency.
The Right-Hand Rule
Curl direction is interpreted with the right-hand rule.
Curl the fingers of your right hand in the local direction of rotation. Your thumb points in the curl direction.
Counterclockwise rotation in the \(xy\)-plane corresponds to positive \(z\)-direction curl.
Example: Curl of a Rotational Field
Consider
\[ \mathbf{F}(x,y,z) = \langle -y,x,0\rangle. \]Here
\[ P=-y,\qquad Q=x,\qquad R=0. \]Therefore,
\[ \nabla\times\mathbf{F} = \left\langle 0-0,\; 0-0,\; 1-(-1) \right\rangle. \]Hence,
\[ \boxed{ \nabla\times\mathbf{F} = \langle0,0,2\rangle }. \]Interpreting the Rotational Example
The field
\[ \langle-y,x,0\rangle \]circulates counterclockwise around the \(z\)-axis.
Its curl
\[ \langle0,0,2\rangle \]points in the positive \(z\)-direction, exactly as predicted by the right-hand rule.
A Curl-Free Field
Consider
\[ \mathbf{F}(x,y,z) = \langle2x,2y,2z\rangle. \]Every cross-partial appearing in the curl is zero.
Therefore,
\[ \boxed{ \nabla\times\mathbf{F} = \mathbf{0} }. \]The field expands outward but has no local rotational tendency.
Irrotational Fields
A vector field satisfying
\[ \boxed{ \nabla\times\mathbf{F} = \mathbf{0} } \]is called irrotational on that region.
Irrotational does not mean motion is absent. It means local rotational tendency, as measured by curl, is zero.
Curl in Two Dimensions
For a planar field
\[ \mathbf{F}(x,y) = \langle P,Q\rangle, \]we can embed it in three dimensions as
\[ \langle P,Q,0\rangle. \]Then
\[ \nabla\times\mathbf{F} = \left\langle 0,0,Q_x-P_y \right\rangle. \]The scalar quantity
\[ \boxed{ Q_x-P_y } \]is often called the scalar curl of a two-dimensional vector field.
Positive and Negative Planar Curl
In the \(xy\)-plane:
- \(Q_x-P_y>0\) indicates counterclockwise rotational tendency;
- \(Q_x-P_y<0\) indicates clockwise rotational tendency;
- \(Q_x-P_y=0\) indicates no local planar rotation.
The Paddle-Wheel Interpretation
Imagine placing a tiny paddle wheel in a flowing fluid.
If the wheel tends to spin, the local curl is nonzero.
The axis of the wheel's rotation corresponds to the curl direction.
This is a useful physical picture, although curl is defined mathematically by derivatives rather than by an actual wheel.
Curl and Local Circulation
Curl can also be understood as circulation per unit area in the limit of a very small loop.
Informally,
\[ \boxed{ (\nabla\times\mathbf{F})\cdot\mathbf{n} \approx \frac{\text{circulation around a small loop}} {\text{small area}} }. \]Stokes' Theorem will later turn this local relation into a global one.
Divergence and Curl Measure Different Things
A field can have:
- nonzero divergence and zero curl;
- zero divergence and nonzero curl;
- both divergence and curl nonzero;
- both divergence and curl zero.
The two operators measure independent aspects of local field behavior.
Example: Divergence Without Curl
For
\[ \mathbf{F} = \langle x,y,z\rangle, \]we found
\[ \nabla\cdot\mathbf{F}=3. \]But
\[ \nabla\times\mathbf{F} = \mathbf{0}. \]The field expands outward without rotating locally.
Example: Curl Without Divergence
For
\[ \mathbf{F} = \langle-y,x,0\rangle, \]we found
\[ \nabla\cdot\mathbf{F}=0 \]and
\[ \nabla\times\mathbf{F} = \langle0,0,2\rangle. \]The field rotates without acting locally like a source or sink.
Example: Both Divergence and Curl
Let
\[ \mathbf{F}(x,y,z) = \langle x-y,\;x+y,\;z\rangle. \]The divergence is
\[ \nabla\cdot\mathbf{F} = 1+1+1 = \boxed{3}. \]The curl is
\[ \nabla\times\mathbf{F} = \langle0,0,1-(-1)\rangle = \boxed{\langle0,0,2\rangle}. \]The field simultaneously expands and rotates.
Example: Both Zero
Consider the constant vector field
\[ \mathbf{F} = \langle3,-2,5\rangle. \]Every first partial derivative is zero.
Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F}=0 } \]and
\[ \boxed{ \nabla\times\mathbf{F}=\mathbf{0} }. \]Computing Divergence Carefully
Suppose
\[ \mathbf{F} = \langle x^2y,\;xy^2,\;xyz\rangle. \]Compute only the matching coordinate derivatives:
\[ P_x=2xy, \] \[ Q_y=2xy, \] \[ R_z=xy. \]Thus,
\[ \boxed{ \nabla\cdot\mathbf{F}=5xy }. \]Computing Curl Carefully
Let
\[ \mathbf{F} = \langle yz,\;xz,\;xy\rangle. \]Then
\[ R_y=x,\qquad Q_z=x, \] \[ P_z=y,\qquad R_x=y, \] \[ Q_x=z,\qquad P_y=z. \]Therefore,
\[ \boxed{ \nabla\times\mathbf{F} = \mathbf{0} }. \]Gradient Fields Have Zero Curl
Suppose
\[ \mathbf{F}=\nabla f \]and \(f\) has continuous second partial derivatives.
Then
\[ \boxed{ \nabla\times(\nabla f) = \mathbf{0} }. \]This identity follows from equality of mixed partial derivatives.
Why Curl of a Gradient Vanishes
Write
\[ \nabla f = \langle f_x,f_y,f_z\rangle. \]Then the first curl component is
\[ f_{zy}-f_{yz}=0, \]the second is
\[ f_{xz}-f_{zx}=0, \]and the third is
\[ f_{yx}-f_{xy}=0. \]Therefore the entire curl is zero.
Divergence of a Curl Is Zero
For a sufficiently smooth vector field,
\[ \boxed{ \nabla\cdot(\nabla\times\mathbf{F}) = 0 }. \]This is another fundamental vector-calculus identity.
It says that a vector field created purely as a curl cannot have local source or sink behavior.
The Laplacian Appears from Divergence of a Gradient
If \(f\) is a scalar field, then
\[ \nabla\cdot(\nabla f) \]is called the Laplacian of \(f\).
In Cartesian coordinates,
\[ \boxed{ \nabla^2 f = f_{xx}+f_{yy}+f_{zz} }. \]The Laplacian is a scalar field and appears throughout physics, engineering, probability, and differential equations.
Harmonic Functions
A sufficiently smooth scalar field satisfying
\[ \boxed{ \nabla^2 f=0 } \]is called harmonic on that region.
Harmonic functions arise in steady-state heat flow, electrostatics, fluid flow, and potential theory.
Example: Computing a Laplacian
Let
\[ f(x,y,z)=x^2+y^2+z^2. \]Then
\[ f_{xx}=2,\qquad f_{yy}=2,\qquad f_{zz}=2. \]Therefore,
\[ \boxed{ \nabla^2 f=6 }. \]Divergence in Physical Field Models
Divergence appears whenever local production, removal, expansion, or compression matters.
- fluid flow;
- mass conservation;
- electric-field source models;
- heat and transport equations;
- continuity equations;
- compressible-flow analysis.
Curl in Physical Field Models
Curl appears whenever local circulation or rotational structure matters.
- fluid vorticity;
- electromagnetic induction;
- rotational flow;
- circulation analysis;
- continuum mechanics;
- field decomposition.
Vorticity
In fluid mechanics, the curl of the velocity field
\[ \mathbf{v} \]is called the vorticity:
\[ \boxed{ \boldsymbol{\omega} = \nabla\times\mathbf{v} }. \]Vorticity measures local rotational character of the fluid motion.
Local Rotation Versus Orbital Motion
A path that curves around a point does not automatically imply nonzero curl.
Curl measures local spinning tendency of the field, not merely whether trajectories bend globally.
This distinction becomes important when analyzing flows around obstacles or singularities.
Domain Matters
Divergence and curl are defined only where the vector field and required partial derivatives are defined.
A singularity may interrupt otherwise smooth behavior.
Domain geometry also matters when using curl to infer whether a field has a global potential.
Curl-Free Does Not Always Mean Globally Conservative
If a smooth vector field is a gradient field, then its curl is zero.
The reverse implication requires appropriate domain conditions.
On a region with holes or excluded singularities, a field can have
\[ \nabla\times\mathbf{F}=\mathbf{0} \]everywhere in its domain and still fail to possess one global potential function on that entire domain.
Conservative fields and domain conditions will be developed later.
Divergence and Coordinate Systems
The simple formula
\[ P_x+Q_y+R_z \]is the Cartesian-coordinate formula for divergence.
In cylindrical and spherical coordinates, scale factors change the component formula.
The underlying geometric meaning of divergence remains the same.
Curl and Coordinate Systems
Likewise, the familiar Cartesian formula for curl changes in cylindrical and spherical coordinates.
The coordinate formulas become more elaborate because the basis directions themselves vary with position.
The later lesson on coordinate systems will organize these forms.
Units of Divergence
If a vector field has units
\[ [\mathbf{F}], \]and position has units of length \(L\), then divergence has units
\[ \boxed{ \frac{[\mathbf{F}]}{L} }. \]For a velocity field measured in meters per second, divergence has units of reciprocal seconds.
Units of Curl
Curl also introduces one spatial derivative.
Therefore,
\[ \boxed{ [\nabla\times\mathbf{F}] = \frac{[\mathbf{F}]}{L} }. \]For velocity, vorticity has units of reciprocal time.
Common Error: Confusing Divergence with Gradient
Gradient acts on a scalar field:
\[ \nabla f. \]Divergence acts on a vector field:
\[ \nabla\cdot\mathbf{F}. \]Their outputs are also different:
\[ \nabla f \text{ is a vector}, \]while
\[ \nabla\cdot\mathbf{F} \text{ is a scalar}. \]Common Error: Adding the Wrong Partial Derivatives
Divergence uses matching coordinate derivatives:
\[ \boxed{ P_x+Q_y+R_z }. \]It does not use
\[ P_y+Q_z+R_x \]or another arbitrary combination.
Common Error: Losing a Curl Sign
The second curl component is
\[ \boxed{ P_z-R_x }. \]When using the determinant mnemonic, the middle cofactor carries a negative sign.
This is one of the most common computational errors in curl problems.
Common Error: Treating Curl as a Scalar in Three Dimensions
In three dimensions,
\[ \nabla\times\mathbf{F} \]is a vector with three components.
The scalar expression
\[ Q_x-P_y \]is the \(z\)-component associated with a two-dimensional field.
Common Error: Assuming Zero Divergence Means a Constant Field
The field
\[ \mathbf{F} = \langle-y,x,0\rangle \]has
\[ \nabla\cdot\mathbf{F}=0, \]but it is not constant.
Zero divergence only describes local source-sink balance.
Common Error: Assuming Zero Curl Means Zero Field
The field
\[ \mathbf{F} = \langle x,y,z\rangle \]has
\[ \nabla\times\mathbf{F} = \mathbf{0}, \]but clearly
\[ \mathbf{F}\ne\mathbf{0} \]except at the origin.
A Reliable Procedure for Divergence
- Write the vector field in component form \(\langle P,Q,R\rangle\).
- Differentiate \(P\) with respect to \(x\).
- Differentiate \(Q\) with respect to \(y\).
- Differentiate \(R\) with respect to \(z\).
- Add the three results.
- Simplify.
- Interpret the sign if the problem asks for source-sink behavior.
A Reliable Procedure for Curl
- Write \(\mathbf{F}=\langle P,Q,R\rangle\).
- Compute \(R_y-Q_z\).
- Compute \(P_z-R_x\).
- Compute \(Q_x-P_y\).
- Assemble the three components in that order.
- Simplify.
- Interpret direction with the right-hand rule when appropriate.
Example: Full Divergence Workflow
Let
\[ \mathbf{F}(x,y,z) = \langle x^2z,\;3xy^2,\;2yz^3\rangle. \]Compute the matching partial derivatives:
\[ P_x=2xz, \] \[ Q_y=6xy, \] \[ R_z=6yz^2. \]Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F} = 2xz+6xy+6yz^2 }. \]Example: Full Curl Workflow
Let
\[ \mathbf{F}(x,y,z) = \langle yz,\;x^2+z,\;xy\rangle. \]First component:
\[ R_y-Q_z = x-1. \]Second component:
\[ P_z-R_x = y-y = 0. \]Third component:
\[ Q_x-P_y = 2x-z. \]Therefore,
\[ \boxed{ \nabla\times\mathbf{F} = \langle x-1,\;0,\;2x-z\rangle }. \]Practice 1: Divergence
Find the divergence of
\[ \mathbf{F}(x,y,z) = \langle2x,\;3y,\;4z\rangle. \]Practice 2: Divergence at a Point
Let
\[ \mathbf{F}(x,y,z) = \langle x^2y,\;xy^2,\;z^2\rangle. \]Find
\[ \nabla\cdot\mathbf{F} \]and evaluate it at \((1,2,3)\).
Practice 3: Source or Sink
Consider
\[ \mathbf{F}(x,y) = \langle -2x,-3y\rangle. \]Compute the divergence and classify the local behavior as source-like, sink-like, or divergence-free.
Practice 4: Curl
Find the curl of
\[ \mathbf{F}(x,y,z) = \langle -y,x,0\rangle. \]Practice 5: Curl-Free Field
Determine whether
\[ \mathbf{F}(x,y,z) = \langle yz,xz,xy\rangle \]is curl-free.
Practice 6: Two-Dimensional Curl
Find the scalar curl of
\[ \mathbf{F}(x,y) = \langle x-y,\;x+y\rangle. \]Practice 7: Divergence and Curl Together
For
\[ \mathbf{F}(x,y,z) = \langle x-y,\;x+y,\;2z\rangle, \]compute both divergence and curl.
Practice 8: Curl of a Gradient
Let
\[ f(x,y,z)=x^2y+yz^2. \]Compute
\[ \nabla\times(\nabla f). \]Practice 9: Laplacian
Find the Laplacian of
\[ f(x,y,z)=x^3+y^3+z^3. \]Practice 10: Interpret the Field
Suppose at a point,
\[ \nabla\cdot\mathbf{F}=0 \]and
\[ \nabla\times\mathbf{F} = \langle0,0,5\rangle. \]Describe the local source-sink behavior and rotational tendency.
Practice Answers
Practice 1
\[ \nabla\cdot\mathbf{F} = 2+3+4 = \boxed{9}. \]Practice 2
\[ \nabla\cdot\mathbf{F} = 2xy+2xy+2z = 4xy+2z. \]At \((1,2,3)\),
\[ 4(1)(2)+2(3) = 8+6 = \boxed{14}. \]Practice 3
\[ \nabla\cdot\mathbf{F} = -2-3 = \boxed{-5}. \]The negative divergence indicates sink-like behavior.
Practice 4
\[ \boxed{ \nabla\times\mathbf{F} = \langle0,0,2\rangle }. \]Practice 5
Here
\[ P=yz,\qquad Q=xz,\qquad R=xy. \]The curl components are
\[ R_y-Q_z=x-x=0, \] \[ P_z-R_x=y-y=0, \] \[ Q_x-P_y=z-z=0. \]Therefore,
\[ \boxed{ \nabla\times\mathbf{F} = \mathbf{0} }. \]Practice 6
With
\[ P=x-y, \qquad Q=x+y, \]we have
\[ Q_x=1 \]and
\[ P_y=-1. \]Thus,
\[ \boxed{ Q_x-P_y=2 }. \]Practice 7
Divergence:
\[ \nabla\cdot\mathbf{F} = 1+1+2 = \boxed{4}. \]Curl:
\[ \boxed{ \nabla\times\mathbf{F} = \langle0,0,2\rangle }. \]Practice 8
Because \(f\) is smooth,
\[ \boxed{ \nabla\times(\nabla f) = \mathbf{0} }. \]Practice 9
\[ f_{xx}=6x, \qquad f_{yy}=6y, \qquad f_{zz}=6z. \]Therefore,
\[ \boxed{ \nabla^2f = 6x+6y+6z }. \]Practice 10
Since
\[ \nabla\cdot\mathbf{F}=0, \]there is no net local source or sink behavior.
Since
\[ \nabla\times\mathbf{F} = \langle0,0,5\rangle, \]the field has local rotational tendency about the positive \(z\)-axis, corresponding to counterclockwise rotation in the \(xy\)-plane under the right-hand rule.
What to Remember
For
\[ \mathbf{F} = \langle P,Q,R\rangle, \]divergence is
\[ \boxed{ \nabla\cdot\mathbf{F} = P_x+Q_y+R_z }. \]Curl is
\[ \boxed{ \nabla\times\mathbf{F} = \langle R_y-Q_z,\; P_z-R_x,\; Q_x-P_y \rangle }. \]- Divergence measures local source-sink or expansion-compression behavior.
- Positive divergence is source-like; negative divergence is sink-like.
- Curl measures local rotational tendency.
- The curl direction gives the local axis of rotation.
- Zero divergence does not imply zero field.
- Zero curl does not imply zero field.
- For smooth \(f\), \(\nabla\times(\nabla f)=\mathbf{0}\).
- For smooth \(\mathbf{F}\), \(\nabla\cdot(\nabla\times\mathbf{F})=0\).
- The Laplacian is \(\nabla^2f=\nabla\cdot(\nabla f)\).
Where We Go Next
Divergence and curl describe local behavior. Vector calculus also needs tools that accumulate field effects along paths and across surfaces.
The next lesson, Line Integrals, develops integration along curves. It will connect vector fields with work, circulation, path dependence, and the later theory of conservative fields.