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Understanding Fundamental Identities: Gradient, Divergence, Curl & Laplacian, Lecture notes of Vector Analysis

Sharda UniversityVector Analysis

The fundamental identities of vector analysis, focusing on the gradient, divergence, curl, and laplacian. It covers the identities of these operators, their relationships, and applications to maxwell's equations. Students of physics, engineering, and mathematics will benefit from understanding these concepts.

Typology: Lecture notes

2021/2022

Uploaded on 09/27/2022

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LECTURE 15
Identities of Vector Analysis
1. Differential Operator Notation
Let
∇
denote the formal symbol
∂
∂x,∂
∂y,∂
∂z
thought of as a 3-dimensional vector. Of course, the components of
∇
really don’t make any sense until
they act of a function. But if we permit ourselves this notational absurdity, we can better understand the
notation used for the gradient, divergence and curl:
∇
f
=
∂
∂x,∂
∂y,∂
∂z
f
=
∂f
∂x,∂f
∂y,∂f
∂z
∇·
F
=
∂
∂x,∂
∂y,∂
∂z
·
(
F
x
,F
y
,F
z
)=
∂F
x
∂x
+
∂F
y
∂y
+
∂F
z
∂z
∇×
F
=
∂
∂x,∂
∂y,∂
∂z
×
(
F
x
,F
y
,F
z
)
=
∂F
z
∂y
−
∂F
y
∂z ,∂F
x
∂z
−
∂F
x
∂z ,∂F
y
∂x
−
∂F
x
∂y
Along these same lines we now introduce a second order differential operator, the
Laplacian
that is defined
by
∇
2
f
≡
(
∇·∇
)
f
=
∇·
(
∇
f
)=
∂
∂x,∂
∂y,∂
∂z
·
∂f
∂x,∂f
∂y,∂f
∂z
=
∂
2
f
∂x
2
+
∂
2
f
∂y
2
+
∂
2
f
∂z
2
Theorem
15.1
.
(Fundamental Identities of Vector Analysis). Let
f
and
g
be real-valued functions on
R
3
and let
F
and
G
be vector fields on
R
3
. Then
1.
∇
(
f
+
g
)=
∇
f
+
∇
g
2.
∇
(
cf
)=
c
(
∇
f
)
for any constant
c
3.
∇
(
fg
)=
g
(
∇
f
)+
f
(
∇
g
)
4.
∇
(
f/g
)=(
g
∇
f
−
f
∇
g
)
/g
2
5.
∇·
(
F
+
G
)=
∇·
F
+
∇·
G
6.
∇×
(
F
+
G
)=
∇×
F
+
∇×
G
7.
∇·
(
f
F
)=
f
(
∇·
F
)+
∇
f
·
F
8.
∇·
(
F
×
G
)=
G
·
(
∇×
F
)
−
F
·
(
∇×
G
)
9.
∇·
(
∇×
F
)=0
10.
∇×
(
f
F
)=
f
(
∇×
F
)+
∇
f
×
F
11.
∇×
(
∇
f
)=0
12.
∇
2
(
fg
)=
f
∇
2
g
+2(
∇
f
·∇
g
)+
g
∇
2
f
13.
∇·
(
∇
f
×∇
g
)=0
14.
∇·
(
f
∇
g
−
g
∇
f
)=
f
∇
2
g
−
g
∇
2
f
15.
∇×
(
∇×
F
)=
∇
(
∇·
F
)
−∇
2
F
1
pf3

Partial preview of the text

Download Understanding Fundamental Identities: Gradient, Divergence, Curl & Laplacian and more Lecture notes Vector Analysis in PDF only on Docsity!

LECTURE 15

Identities of Vector Analysis

  1. Differential Operator Notation

Let ∇ denote the formal symbol

∂x

∂y

∂z

thought of as a 3-dimensional vector. Of course,the components of ∇ really don’t make any sense until

they act of a function. But if we permit ourselves this notational absurdity,we can better understand the

notation used for the gradient,divergence and curl:

∇f =

∂x

∂y

∂z

f =

∂f ∂x

∂f ∂y

∂f ∂z

∇ · F =

∂x ,

∂y ,

∂z

· (Fx, Fy , Fz ) =

∂Fx

∂x +^

∂Fy

∂y +^

∂Fz

∂z

∇ × F =

∂x

∂y

∂z

× (Fx, Fy , Fz )

∂Fz

∂y

−^ ∂Fy

∂z

, ∂Fx

∂z

−^ ∂Fx

∂z

, ∂Fy

∂x

−^ ∂Fx

∂y

Along these same lines we now introduce a second order differential operator,the Laplacian that is defined

by

∇^2 f ≡ (∇ · ∇)f = ∇ · (∇f) =

∂x

∂y

∂z

·

∂f

∂x

∂f

∂y

∂f

∂z

∂^2 f

∂x^2

∂^2 f

∂y^2

∂^2 f

∂z^2

Theorem 15.1. (Fundamental Identities of Vector Analysis). Let f and g be real-valued functions on R^3

and let F and G be vector fields on R^3. Then

1. ∇(f + g) = ∇f + ∇g

2. ∇(cf) = c(∇f) for any constant c

3. ∇(fg) = g(∇f) + f(∇g)

4. ∇ (f/g) = (g∇f − f∇g) /g^2

5. ∇ · (F + G) = ∇ · F + ∇ · G

6. ∇ × (F + G) = ∇ × F + ∇ × G

7. ∇ · (fF) = f(∇ · F) + ∇f · F

8. ∇ · (F × G) = G · (∇ × F) − F · (∇ × G)

9. ∇ · (∇ × F) = 0

10. ∇ × (fF) = f (∇ × F) + ∇f × F

11. ∇ × (∇f) = 0

12. ∇^2 (fg) = f

∇^2 g

+ 2 (∇f · ∇g) + g

∇^2 f

13. ∇ · (∇f × ∇g) = 0

14. ∇ · (f∇g − g∇f) = f

∇^2 g

− g

∇^2 f

15. ∇ × (∇ × F) = ∇ (∇ · F) − ∇^2 F

1

  1. APPLICATION: MAXWELL’S EQUATIONS 2

Since we shall use Identity 15 below,let me give a brief indication as to why it should be true. Both sides

of this equation are vector fields (in the end); we shall look only at the x component

(∇ × (∇ × F))x =

(

∇ (∇ · F) − ∇^2 F

) x

Now the x-component of the right hand side is

(RHS)x =

(

∇ (∇ · F) − ∇^2 F

) x

= ∇x (∇ · F) − ∇^2 Fx

∂x

( ∂Fx ∂x

∂Fy ∂y

∂Fz ∂z

) −

( ∂^2 ∂x^2

∂^2

∂y^2

∂^2

∂z^2

) Fx

(^2) Fx ∂x^2

(^2) Fy ∂x∂y

+ ∂Fz

∂x∂z

(^2) Fx ∂x^2

(^2) Fx ∂y^2

(^2) Fx ∂z^2

∂^2 Fy ∂x∂y

∂Fz ∂x∂z

∂^2 Fx ∂y^2

∂^2 Fx ∂z^2

The x-component of the left hand side is

(LHS)x = (∇ × (∇ × F))x

[( ∂ ∂x

∂y

∂z

) ×

( ∂Fz ∂y

∂Fy ∂z

∂Fx ∂z

∂Fz ∂x

∂Fy ∂x

∂Fx ∂y

)]

x

∂y

( ∂Fy ∂x

∂Fx ∂y

) −

∂z

( ∂Fx ∂z

∂Fz ∂x

)

∂^2 Fy ∂y∂x −^

∂^2 Fx ∂y^2 −^

∂^2 Fx

∂z^2 +^

∂^2 Fx ∂z∂x

∂^2 Fy ∂x∂y

∂Fz ∂x∂z

∂^2 Fx ∂y^2

∂^2 Fx ∂z^2

= (RHS)x

So weve now confirmed the x-component of Identity 15.

  1. Application: Maxwell’s Equations

As an example of the utility of the identities listed in the preceding section,let us consider the equations

governing the behavior of electric and magnetic fields. These are Maxwell’s equation:

∇ · E = 4 πε^1 o ρ(x) (Gauss’ Law)

∇ · B = 0 (Gauss’ Law for Magnetic Field)

∇ × E = − ∂ ∂tB (Faraday’s Law)

∇ × B = μo εo∂ ∂tE + μo j(x) (Ampere’s Law)

Here E = E(x, t) is the electric field strength at the point x at time t, B = B(x, t) is the magnetic field

strength at the point x at time t, ρ(x) is the charge density at the point x. εo is a constant called the electric

permitivity of the vacuum,it is determined experimentally by measuring the force of attraction b etween electric charges

F = q 1 q 2

4 πεo ‖r‖^3

r

and is equal to

εo = 8. 85 × 10 −^12 (colomb)^2 (sec)^2 (kg)−^1 (meter)−^3

j(x) is the density of electrical current at the point x,and μo is another experimentally determined constant. It is called the magnetic permeability of the vacuum and its value is

μo = 1. 26 × 10 −^6 (coulomb)−^2 (kg)(meter)