Curl and Grad
Div and Grad
Div and Curl
The Relationship Between Divergence,
Curl and the Gradient
Bernd Schröder
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Introduction
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Introduction
1. Gradient ~∇f
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Introduction
1. Gradient ~∇f : Points in the direction of steepest ascent.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Introduction
1. Gradient ~∇f : Points in the direction of steepest ascent.
2. Divergence ~∇ · ~F
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Introduction
1. Gradient ~∇f : Points in the direction of steepest ascent.
2. Divergence ~∇ · ~F: Measures source strength.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Introduction
1. Gradient ~∇f : Points in the direction of steepest ascent.
2. Divergence ~∇ · ~F: Measures source strength.
3. Curl ~∇ × ~F
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Introduction
1. Gradient ~∇f : Points in the direction of steepest ascent.
2. Divergence ~∇ · ~F: Measures source strength.
3. Curl ~∇ × ~F: Measures vorticity.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Introduction
1. Gradient ~∇f : Points in the direction of steepest ascent.
2. Divergence ~∇ · ~F: Measures source strength.
3. Curl ~∇ × ~F: Measures vorticity.
What else can we say?
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Introduction
1. Gradient ~∇f : Points in the direction of steepest ascent.
2. Divergence ~∇ · ~F: Measures source strength.
3. Curl ~∇ × ~F: Measures vorticity.
What else can we say? Let’s look at the combinations of any
two of these operators with each other
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Introduction
1. Gradient ~∇f : Points in the direction of steepest ascent.
2. Divergence ~∇ · ~F: Measures source strength.
3. Curl ~∇ × ~F: Measures vorticity.
What else can we say? Let’s look at the combinations of any
two of these operators with each other (except for the gradient
of a divergence).
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Gradient Fields have Zero Curl
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Gradient Fields have Zero Curl
(Assuming continuous second partial derivatives, so we have
Clairaut’s Theorem.)
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Gradient Fields have Zero Curl
(Assuming continuous second partial derivatives, so we have
Clairaut’s Theorem.)
curl grad(f )
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Gradient Fields have Zero Curl
(Assuming continuous second partial derivatives, so we have
Clairaut’s Theorem.)

  ∂f 
∂
∂x
 ∂∂x   ∂ f 
curl grad(f ) =  ∂ y  ×  ∂ y 
∂
∂z
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
∂f
∂z
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Gradient Fields have Zero Curl
(Assuming continuous second partial derivatives, so we have
Clairaut’s Theorem.)

  ∂f 
∂
∂x
 ∂∂x   ∂ f 
curl grad(f ) =  ∂ y  ×  ∂ y 
∂
∂z


= 

Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
∂f
∂z
∂ 2f
∂ 2f
∂ y∂ z − ∂ z∂ y
∂ 2f
∂ 2f
∂ z∂ x − ∂ x∂ z
∂ 2f
∂ 2f
−
∂ x∂ y
∂ y∂ x




Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Gradient Fields have Zero Curl
(Assuming continuous second partial derivatives, so we have
Clairaut’s Theorem.)

  ∂f 
∂
∂x
 ∂∂x   ∂ f 
curl grad(f ) =  ∂ y  ×  ∂ y 
∂
∂z


= 

Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
∂f
∂z
∂ 2f
∂ 2f
∂ y∂ z − ∂ z∂ y
∂ 2f
∂ 2f
∂ z∂ x − ∂ x∂ z
∂ 2f
∂ 2f
−
∂ x∂ y
∂ y∂ x



0

 =  0 .

0
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Gradient Fields have Zero Curl
(Assuming continuous second partial derivatives, so we have
Clairaut’s Theorem.)

  ∂f 
∂
∂x
 ∂∂x   ∂ f 
curl grad(f ) =  ∂ y  ×  ∂ y 
∂
∂z


= 

∂f
∂z
∂ 2f
∂ 2f
∂ y∂ z − ∂ z∂ y
∂ 2f
∂ 2f
∂ z∂ x − ∂ x∂ z
∂ 2f
∂ 2f
−
∂ x∂ y
∂ y∂ x



0

 =  0 .

0
How about the converse?
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Gradient Fields have Zero Curl
(Assuming continuous second partial derivatives, so we have
Clairaut’s Theorem.)

  ∂f 
∂
∂x
 ∂∂x   ∂ f 
curl grad(f ) =  ∂ y  ×  ∂ y 
∂
∂z


= 

∂f
∂z
∂ 2f
∂ 2f
∂ y∂ z − ∂ z∂ y
∂ 2f
∂ 2f
∂ z∂ x − ∂ x∂ z
∂ 2f
∂ 2f
−
∂ x∂ y
∂ y∂ x



0

 =  0 .

0
How about the converse?
Does zero curl imply the field is a gradient field?
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Consider the essence of the magnetic field generated by a
current along the z-axis.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Consider the essence of the magnetic field generated by a
current along the z-axis.


y
− x2 +y
2


x
curl  x2 +y
2 
0
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Consider the essence of the magnetic field generated by a
current along the z-axis.

 



y
y
∂
− x2 +y
− x2 +y
2
2
∂x

 



x
x
=  ∂∂y  ×  x2 +y
curl  x2 +y
2 
2 
∂
0
0
∂z
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Consider the essence of the magnetic field generated by a
current along the z-axis.

 
 



y
y
∂
− x2 +y
− x2 +y
0
2
2
∂x

 
 


x
x
0 
=  ∂∂y  ×  x2 +y
curl  x2 +y
2  =
2 
y
x
∂
∂
∂
0
0
∂ x x2 +y2 + ∂ y x2 +y2
∂z
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Consider the essence of the magnetic field generated by a
current along the z-axis.

 
 



y
y
∂
− x2 +y
− x2 +y
0
2
2
∂x

 
 


x
x
0 
=  ∂∂y  ×  x2 +y
curl  x2 +y
2  =
2 
y
x
∂
∂
∂
0
0
∂ x x2 +y2 + ∂ y x2 +y2
 ∂z

0

0 

= 
 1·(x2 +y2 )−x·2x 1·(x2 +y2 )−y·2y 
+
2
2
(x2 +y2 )
(x2 +y2 )
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Consider the essence of the magnetic field generated by a
current along the z-axis.

 
 



y
y
∂
− x2 +y
− x2 +y
0
2
2
∂x

 
 


x
x
0 
=  ∂∂y  ×  x2 +y
curl  x2 +y
2  =
2 
y
x
∂
∂
∂
0
0
∂ x x2 +y2 + ∂ y x2 +y2
 ∂z

0

0 

= 
 1·(x2 +y2 )−x·2x 1·(x2 +y2 )−y·2y 
+
2
2
(x2 +y2 )
(x2 +y2 )


0

0 
=  2 2

y −x
x2 −y2
+
2
2
(x2 +y2 )
(x2 +y2 )
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Consider the essence of the magnetic field generated by a
current along the z-axis.

 
 



y
y
∂
− x2 +y
− x2 +y
0
2
2
∂x

 
 


x
x
0 
=  ∂∂y  ×  x2 +y
curl  x2 +y
2  =
2 
y
x
∂
∂
∂
0
0
∂ x x2 +y2 + ∂ y x2 +y2
 ∂z

0

0 

= 
 1·(x2 +y2 )−x·2x 1·(x2 +y2 )−y·2y 
+
2
2
(x2 +y2 )
(x2 +y2 )

  
0
0

  
0
=  2 2
= 0
y −x
x2 −y2
+
0
2
2
(x2 +y2 )
(x2 +y2 )
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
6
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
I
6
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
+
I
I
6
~
3
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
+
+
I
I
I
6
~
~
3
3
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
=
+
+
}
I
I
I
6
~
~
~
3
3
3
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
are surrounded
=
+
+
}
I
I
I
6
~
~
~
3
3
3
by circular
magnetic
fields
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
are surrounded


curl 
=
+
+
}
I
y
− x2 +y
2



0
  
x
= 0
x2 +y2 
0
0
I
I
6
~
~
~
3
3
3
by circular
magnetic
fields
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
are surrounded


curl 
=
+
+
}
I
y
− x2 +y
2



0
  
x
= 0
x2 +y2 
0
0
I
I
The problem is with the singularity along the z-axis.
6
~
~
~
3
3
3
by circular
magnetic
fields
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
are surrounded


curl 
=
+
+
}
I
y
− x2 +y
2



0
  
x
= 0
x2 +y2 
0
0
I
I
6
~
~
~
3
3
3
by circular
magnetic
fields
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
The problem is with the singularity along the z-axis. If
we try to apply Stokes’ Theorem, the interpolating surface
would contain the singularity.
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Zero Curl Alone is not Enough
Currents
are surrounded


curl 
=
+
+
}
I
y
− x2 +y
2



0
  
x
= 0
x2 +y2 
0
0
I
I
6
~
~
~
3
3
3
by circular
magnetic
fields
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
The problem is with the singularity along the z-axis. If
we try to apply Stokes’ Theorem, the interpolating surface
would contain the singularity.
So: The curl is a local measure of vorticity.
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
9
:
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
9
:
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
Simply connected means that
closed curves can be interpolated with surfaces that stay
inside the region.
9
:
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
:
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
:
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
In a region that is not simply
connected
:
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Simply Connected Regions
In a region that is not simply
connected, some curves cannot be interpolated with surfaces.
:
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives on a simply connected region
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives on a simply connected region is a gradient field
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives on a simply connected region is a gradient field if
and only if
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives on a simply connected region is a gradient field if
and only if ~∇ × ~F = ~0
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives on a simply connected region is a gradient field if
and only if ~∇ × ~F = ~0
Proof.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives on a simply connected region is a gradient field if
and only if ~∇ × ~F = ~0
Proof. We have already proved that the curl of a gradient field
is zero.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives on a simply connected region is a gradient field if
and only if ~∇ × ~F = ~0
Proof. We have already proved that the curl of a gradient field
is zero. Conversely, because the region is simply connected
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives on a simply connected region is a gradient field if
and only if ~∇ × ~F = ~0
Proof. We have already proved that the curl of a gradient field
is zero. Conversely, because the region is simply connected,
Stokes’ Theorem can be applied to curves inside the region.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives on a simply connected region is a gradient field if
and only if ~∇ × ~F = ~0
Proof. We have already proved that the curl of a gradient field
is zero. Conversely, because the region is simply connected,
Stokes’ Theorem can be applied to curves inside the region. So
it’s a simple consequence of Stokes’ Theorem that zero curl on
a simply connected region implies that integrals over closed
curves are zero.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
Definition. A connected region E in three-dimensional space is
called simply connected if and only if, for every simple closed
curve C contained in E, there is a surface S that is contained in
E and which has C as its boundary.
Theorem. A vector field ~F with continuous second partial
derivatives on a simply connected region is a gradient field if
and only if ~∇ × ~F = ~0
Proof. We have already proved that the curl of a gradient field
is zero. Conversely, because the region is simply connected,
Stokes’ Theorem can be applied to curves inside the region. So
it’s a simple consequence of Stokes’ Theorem that zero curl on
a simply connected region implies that integrals over closed
curves are zero.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
]
JJ
@
I
@
PP
i
1
)
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
2. Heat flux is proportional to −gradu, where u is the
temperature.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
2. Heat flux is proportional to −gradu, where u is the
temperature.
H
O
T
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
2. Heat flux is proportional to −gradu, where u is the
temperature.
H
O
T
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
C
O
L
D
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
2. Heat flux is proportional to −gradu, where u is the
temperature.
C
H
O
T
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
O
-
L
D
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
2. Heat flux is proportional to −gradu, where u is the
temperature.
C
H
O
T
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
O
-
thermal
flux
L
D
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
2. Heat flux is proportional to −gradu, where u is the
temperature.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
2. Heat flux is proportional to −gradu, where u is the
temperature.
3. Consider the net heat transfer through the surface S (per
time unit).
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
2. Heat flux is proportional to −gradu, where u is the
temperature.
3. Consider the net heat transfer through the surface S (per
time unit). It is proportional to the surface integral
ZZ
− grad(u) · d~S.
S
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
2. Heat flux is proportional to −gradu, where u is the
temperature.
3. Consider the net heat transfer through the surface S (per
time unit). It is proportional to the surface integral
ZZ
− grad(u) · d~S.
S
4. The net heat transfer through S (per time unit) is the rate of
change of the net heat content of B (per time unit),
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation
1. Consider a small ball B centered at~r with radius a and
surface S.
2. Heat flux is proportional to −gradu, where u is the
temperature.
3. Consider the net heat transfer through the surface S (per
time unit). It is proportional to the surface integral
ZZ
− grad(u) · d~S.
S
4. The net heat transfer through S (per time unit) is the rate of
change of the net heat content
of B (per time unit), which
ZZZ
∂
is proportional to −
u dV.
∂t
B
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation.
ZZ
∂
− grad(u) · d~S = −k
∂t
S
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
ZZZ
u dV
B
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation.
ZZ
∂
− grad(u) · d~S = −k
∂t
S
ZZ
grad(u) · d~S =
S
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
ZZZ
k
B
ZZZ
u dV
B
∂u
dV
∂t
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation.
ZZ
∂
− grad(u) · d~S = −k
∂t
S
ZZ
grad(u) · d~S =
div grad(u) dV =
B
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
∂u
dV
∂t
k
∂u
dV
∂t
ZZZ
B
u dV
B
k
ZZZ
B
S
ZZZ
ZZZ
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation.
ZZ
∂
− grad(u) · d~S = −k
∂t
S
ZZ
grad(u) · d~S =
ZZZ
k
B
S
ZZZ
u dV
B
∂u
dV
∂t
∂u
dV
B
B ∂t
ZZZ
ZZZ
1
1
∂u
lim 4
div grad(u) dV = lim 4
k
dV
a→0 πa3
a→0 πa3
∂
t
B
B
3
3
ZZZ
div grad(u) dV =
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
ZZZ
k
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Heat Equation.
ZZ
∂
− grad(u) · d~S = −k
∂t
S
ZZ
grad(u) · d~S =
ZZZ
k
B
S
ZZZ
u dV
B
∂u
dV
∂t
∂u
dV
B
B ∂t
ZZZ
ZZZ
1
1
∂u
lim 4
div grad(u) dV = lim 4
k
dV
a→0 πa3
a→0 πa3
∂
t
B
B
3
3
∂u
div grad(u) (~r, t) = k (~r, t)
∂t
ZZZ
div grad(u) dV =
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
ZZZ
k
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
ZZZ
ZZZ
div (grad(u)) dV =
B
k
B
Div and Curl
∂u
dV
∂t
q P (~r)
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
ZZZ
ZZZ
div (grad(u)) dV =
B
k
B
Div and Curl
∂u
dV
∂t
q P (~r)
@
@
R
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
qP (~r)
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
ZZZ
ZZZ
div (grad(u)) dV =
B
k
B
Div and Curl
∂u
dV
∂t
q P (~r)
@
As the radius a shrinks, the approximations
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
@
R
qP (~r)
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
ZZZ
ZZZ
div (grad(u)) dV =
B
k
B
Div and Curl
∂u
dV
∂t
q P (~r)
@
@
R
As the radius a shrinks, the approximations
ZZZ
ZZZ
4
∂u
4
∂u
div (grad(u)) dV ≈ πa3 div (grad(u)) ,
k
dV ≈ πa3 k
3
3
∂t
B
B ∂t
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
qP (~r)
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
ZZZ
ZZZ
div (grad(u)) dV =
B
k
B
Div and Curl
∂u
dV
∂t
q P (~r)
@
@
R
As the radius a shrinks, the approximations
ZZZ
ZZZ
4
∂u
4
∂u
div (grad(u)) dV ≈ πa3 div (grad(u)) ,
k
dV ≈ πa3 k
3
3
∂t
B
B ∂t
improve towards equality.
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
qP (~r)
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
ZZZ
ZZZ
div (grad(u)) dV =
B
k
B
Div and Curl
∂u
dV
∂t
q P (~r)
@
@
R
As the radius a shrinks, the approximations
ZZZ
ZZZ
4
∂u
4
∂u
div (grad(u)) dV ≈ πa3 div (grad(u)) ,
k
dV ≈ πa3 k
3
3
∂t
B
B ∂t
improve towards equality.
q
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
qP (~r)
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
ZZZ
ZZZ
div (grad(u)) dV =
k
B
B
Div and Curl
∂u
dV
∂t
q P (~r)
@
@
R
As the radius a shrinks, the approximations
ZZZ
ZZZ
4
∂u
4
∂u
div (grad(u)) dV ≈ πa3 div (grad(u)) ,
k
dV ≈ πa3 k
3
3
∂t
B
B ∂t
improve towards equality.
q
qP (~r)
Q
Q
QQ
s
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
q
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
ZZZ
ZZZ
div (grad(u)) dV =
k
B
B
Div and Curl
∂u
dV
∂t
q P (~r)
@
@
R
As the radius a shrinks, the approximations
ZZZ
ZZZ
4
∂u
4
∂u
div (grad(u)) dV ≈ πa3 div (grad(u)) ,
k
dV ≈ πa3 k
3
3
∂t
B
B ∂t
improve towards equality.
q
qP (~r)
Q
Q
QQ
s
div (grad(u)) (~r, t) = k
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
∂u
(~r, t)
∂t
q
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Divergence of a Curl
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Divergence of a Curl
div curl ~F
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Divergence of a Curl


div curl ~F = 
∂
∂x
∂
∂y
∂
∂z
 
 
 · 
∂
∂x
∂
∂y
∂
∂z
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient


P
 

 × Q 
R

Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Divergence of a Curl


div curl ~F = 


= 
∂
∂x
∂
∂y
∂
∂z
∂
∂x
∂
∂y
∂
∂z
 


P
 
 

 · 
 × Q 
R

  ∂
∂
∂yR − ∂zQ

  ∂
 ·  ∂ z P − ∂∂x R 
∂
∂
∂xQ − ∂yP
∂
∂x
∂
∂y
∂
∂z
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient

Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Divergence of a Curl

div curl ~F =
=
=
∂
∂x
 ∂
 ∂y
∂
∂z

∂
 ∂∂x
 ∂y
∂
∂z
∂2
∂ x∂ y
 


P
 
 

 · 
 × Q 
R

  ∂
∂
∂yR − ∂zQ

  ∂
 ·  ∂ z P − ∂∂x R 
∂
∂
∂xQ − ∂yP
R−
∂
∂x
∂
∂y
∂
∂z

∂2
∂2
∂2
∂2
∂2
Q+
P−
R+
Q−
P
∂ x∂ z
∂ y∂ z
∂ y∂ x
∂ z∂ x
∂ z∂ y
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science
Curl and Grad
Div and Grad
Div and Curl
The Divergence of a Curl

div curl ~F =
=
=
∂
∂x
 ∂
 ∂y
∂
∂z

∂
 ∂∂x
 ∂y
∂
∂z
∂2
∂ x∂ y
= 0
 


P
 
 

 · 
 × Q 
R

  ∂
∂
∂yR − ∂zQ

  ∂
 ·  ∂ z P − ∂∂x R 
∂
∂
∂xQ − ∂yP
R−
∂
∂x
∂
∂y
∂
∂z

∂2
∂2
∂2
∂2
∂2
Q+
P−
R+
Q−
P
∂ x∂ z
∂ y∂ z
∂ y∂ x
∂ z∂ x
∂ z∂ y
Bernd Schröder
The Relationship Between Divergence, Curl and the Gradient
Louisiana Tech University, College of Engineering and Science