Vector Calculus

(1)

Vector Calculus

Vector function is a vector whose components are functions instead of scalars. For instance,

  t x       t x t x t x

x   

₁

,

₂

,

₃



(1)

is a vector function.

Vector function is commonly used for representing quantity on a curve or a surface that must be quantified by both direction and size. It defines what is generally known as vector field. Examples of vector fields are the field of tangent vectors of a curve, and the field of normal vectors of a surface.

The vector function x

in (1) only depends on one independent variable, namely

t

. Nevertheless, it is possible that a vector function depends on more than one independent variable. For instance, in Cartesian coordinate,

 x y z  v  x y z   v x y z   v x y z 

v

v    , , 

₁

, , ,

₂

, , ,

₃

, ,

. (2)

Each component of v

depends on

 x , y , z 

.

Many vector functions that we have to deal with are in Cartesian coordinate. Various combinations of unit vectors, such as

 x ˆ , y ˆ , z ˆ 

_,

 

ⁱ^ˆ^,^ˆ^j^,^k^ˆ ^{, or}



eˆx,eˆy,eˆz



, are sometimes used in place of the vector notation. That is,

z y

x

v e v e

e v k v j v i v z v y v x v v v v

v  

₁

,

₂

,

₃



₁

ˆ 

₂

ˆ 

₃

ˆ 

₁

ˆ 

₂

ˆ 

₃

ˆ 

₁

ˆ 

₂

ˆ 

₃

ˆ

. (3)

In order to describe position of a curve or a surface in Cartesian space, one can define a vector function

r 

where

      t y t z t x   t i y   t j z   t k

x

r   , ,  ˆ  ˆ  ˆ

. (4)

At

t  t

0, the vector function

r 

points from point

 0 , 0 , 0 

to point



x

     

t0 ,y t0 ,z t0



. As

t

changes,

r 

may varies in both direction and size, forming a curve.

Vector

r 

is called positional vector, and is the parametric representation of the curve.

Example: Kreyszig, Example 3, p. 429.

t

does not necessary always have to represent time quantity. It is merely a parameter.

Hereafter, if otherwise stated, we will assume that all quantities are in Cartesian coordinate.

There are several important operations in vector calculus.

1. Derivative of a vector function

r    t

is defined as

(2)

     

^r

     

^t ^r ^t ^r ^t

t t r t t t r

r

t 1 2 3

0 , ,

lim    







 

 



 

. (5)

Notice that

r     t

is still a vector quantity.

If

r    t

is the positional vector, then

r     t

is the tangent vector of the curve represented by

r    t

. This tangent vector is essentially the velocity vector v

which has direction that is tangent to the curve.

Differentiating the velocity vector yield the acceleration r

v a  

. (6)

a

can be separated into tangential and normal components where

v v

a a 



 



 

tan (7)

atan

a

a_norm   (8)

The length of the curve from

t  a

to t b can be found by calculating

  



^b

a

r r dt

l

(9)

We can also define arc length of a curve as

  t  

a^t

r  r  dt

s

(10)

By differentiating (7), we get

2 2

2



 







 







 



 



 dt

dz dt

dy dt

dx dt

r d dt

ds  

. (11)

The line element ds is then

      dx

²

dy

²

dz

²

ds   

. (12)

Some interesting properties of vector differentiation include a.

 

uvu vuv

(13)

(3)

b.



^u^^^v^



^ ^^u^^^^v^^^u^^^v^^ (14) c.



^u^^v^^w^



^ ^û^^^v^^w^^û^^v^^^w^^û^^v^^w^^ (15) 2. Gradient of a scalar function

f  x , y , z 

is defined as

z k j f y i f x f f z k y j x i f grad

f ˆ ˆ ˆ ˆ ˆ ˆ



 



 



 

 

 







 



 



 



 

. (16)

Note that the gradient operation turns a scalar function into a vector function.

Let

f  x , y , z 

be a function that represents curve, and at point P on f , the direction of f is the same as unit vector

b 

. Thus,

f

ds b

df     

(17)

f

points to the direction of the maximum increase of f .

If there is a surface that is characterized by equation

f  x , y , z   c  const

, then f

points in the normal direction of the surface.

3. Laplace of a scalar function

f  x , y , z 

is defined as

2 2 2 2 2 2

2

ˆ ˆ ˆ ˆ ˆ ˆ

z f y

f x

k f z j f y i f x k f j z i y f x

f

f 

 



 



 

 

 







 



 



 

 

 







 



 



 













   

. (18)

Notice that the Laplace operation turns a scalar function into another scalar function.

In physics, there is an important equation called Laplace’s equation which is commonly used to describe potential of a field in vacuum. The equation is

2 0

 f

, (19)

where f represents potential. Laplace’s equation can be applied to electrical or gravitational potential in vacuum.

Example: Kreyszig, Example 3, p. 450-451.

4. Divergence of a vector field

v   x , y , z 

is defined as

(4)

 

z v y v x k v v j v i v z k y j x i v div

v 

 



 



 



 



 







 



 



 





    ˆ ˆ ˆ

₁

ˆ

₂

ˆ

₃

ˆ

¹ ² ³

(20) Notice that the divergence operation turns a vector function into a scalar function.

Notice also that



grad f



div f f  

₂  

. (21)

Divergence operation is an important operation in fluid theory. For example, the continuity equation which describes the mass conservation of a compressible fluid flow is

0





 

 v

t

 

, (22)

where



is the mass density of the fluid, and v

is the fluid velocity.

Gauss’s law of electric field also states that



0







 E

, (23)

where E

is the electric field,



is the charge density, and



₀ is the electrical permittivity in free space.

5. Curl of a vector field

v   x , y , z 

is defined as

 v i v j v k 

z k y j x i v curl

v ˆ ˆ ˆ ˆ ˆ ˆ

3 2

1

 

 



 







 



 



 





   

y k v x j v x v z i v z v y

v

³ ²

ˆ

¹ ³

ˆ

² ¹

  ˆ

 







 



 

 

 







 



 

 

 







 



 

(24)

Notice that the curl operation turns a vector function into another vector function.

Notice also that since

z k j f y i f x

f f ˆ ˆ ˆ



 



 



 

 

(16) Then,

ˆ 0 ˆ

ˆ

² ² ² ²

2

 



 







 



 

 

 







 



 

 

 







 



 





 k

x y

f y x j f z x

f x z i f y z

f z y f f



. (25)

(5)

That is

 grad f   0

curl

. (26)

Also,

 curl v   0

div 

. (27)

Curl operation appears in several electrical theorems such as Faraday’s law:

t E B



 







 



(28)

Ampere’s law:

t J D

H 

 







 



(29)

where E , B

, H ,

J 

, and D

are respectively the electric field, magnetic field, magnetic field intensity, current, and electric displacement field.

6. Line integral of a vector function

Suppose we have a function ^F^

 

^r^ along curve C, then its line integral is defined as

    

   



C C

dt

dt r r d F r d r F

 

  integral

line

. (30)

To evaluate this integral, notice that

F F i ˆ F ˆ j F k ˆ

3 2

1

 

 

and drdxiˆdyˆjdzkˆ

. Thus,

     



C

 

C

  

C

     dt   

dt F dz dt dt

F dy dt dt

F dx dz

F dy F dx F r

d r

F   

₁ ₂ ₃ ₁ ₂ ₃

. (31)

If we want to evaluate from

t  a

to t b then,

  



C

 

a^b

        dt dt F dz dt F dy dt F dx r

d r

F   

₁ ₂ ₃

. (32)

If curve C is a closed curve, e.g. boundary of a circle or rectangular area, one sometimes use



C

 

instead of the normal integration symbol.

In physics, total work done by force is calculated along the path using line integral.

Some properties of line integrals include

(6)

a.



C

k F    r   d r   k 

C

F    r   d r 

(k = constant) (33)

b.



_C

 F      r   G  r    d r   CF    r   d r   CG    r   d r  (34)

c.



CF

 

r ^dr^



C₁F

 

r ^dr^



C₂F

 

r ^dr

(C is subdivided into

C

₁ and

C

2) (35) d. If

F F i ˆ F ˆ j F k ˆ

3 2

1

 

 

is a gradient of some function f , then

  r d r f     B f A F

B

A

  

   

^. ⁽³⁶⁾

That is, the line integral is independent of the integral path between points A and B. Regardless of which path the integral is done between the two points, the result would be the same.

By the same token, consider a close path as a combination of 2 different paths

C

₁ and

C

₂ between points A and B, then

         

2 1

2

1 C

B C A B

C A A

C B B

AF r dr F r dr F r dr F r dr

r d r

F 

 

 

 

 

 

 

 

 

 

 





   



^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^{. (37)}

If

   

2

1 C

B C A B

AF r dr F r dr

 

 

 





^ ^  ^



^ ^ ^ ^{, then}



^F^

 

^r^ ^^d^r^^⁰^.

7. Green’s theorem in a plane is used to transform between double integrals and a line integral which are performed on a plane.

If

F   F

₁

, F

₂

is a vector field on a plane, and both

F

₁ and

F

₂ are continuous, then

 



            

C



R

dy F dx F y dxdy

F x F

2 1 1

2 (38)

where R is the region on the plane that is bounded by curve C. Example: Kreyszig, Examples 2, 3, pp. 488-489.

8. Divergence theorem of Gauss states that

If V is a closed bounded region whose surface is S, and if F



x,y,z



be a vector function that is continuous in V , then



    

S V

dS n F dV

F   



. (39)

(7)

This theory is used to convert triple integrals into double integrals.

9. Stoke’s therem states that

If S is a bounded surface whose boundary is C, and if

F   x , y , z 

be a vector function that is continuous in S, then

    

    

C

 

S

ds s r F dS n

F   



. (40)

This theory is used to convert double integrals into line integrals.