Vector calculus - engineering mathematics

1. VECTOR CALCULUS

2. Vector Product
a
b a  b
CROSS PRODUCT
multiplication VECTOR

3. Vector Product (contd.)
a
b

Magnitude : Area of the parallelogram
generated by a and b.

4. Vector Product (contd.)
a
b

Magnitude : 
sin
ab
ah 

sin
b
h 

5. Vector Product (contd.)
a
b

b
a




sin
b
h 
Direction : Perpendicular to
both a and b.

6. Vector Product (contd.)
a
b

b
a



Direction : A rule is required !!

sin
b
h 

7. a b
b
a



Right-Hand Rule

8. The order of vector multiplication
is important.
a
b
a
b




9. Geometrical Interpretation
a

b

b sin

sin
|
b
a
| ab





A = a  b
Area of the parallelogram
formed by a and b

10. Properties
Vector multiplication is not
Commutative.
Vector multiplication is Distributive
c
a
b
a
)
c
b
(
a













Multiplication by a scalar
m
m
m
m )
b
a
(
)
b
(
a
b
)
a
(
)
b
a
(















a
b
b
a








11. Properties(contd.)
0
0
sin
|
b
a
| 



 

ab


and a and b are not
null vectors, then a is
parallel to b.
If 0
b
a 




12. Properties(contd.)
ĵ
î
k̂
;
î
k̂
ĵ
;
k̂
ĵ
î 





Angle between them 0º
0
k̂
k̂
ĵ
ĵ
î
î 





Angle between them =90º
ĵ
k̂
î
;
î
ĵ
k̂
;
k̂
î
ĵ 








j
i
k

13. Vector Product: Components

a

b


)
a
a
(a k̂
3
ĵ
2
î
1

 )
b
b
(b k̂
3
ĵ
2
î
1


)
k̂
3
ĵ
2
î
1
(
k̂
3
)
k̂
3
ĵ
2
î
1
(
ĵ
2
)
k̂
3
ĵ
2
î
1
(
î
1
b
b
b
a
b
b
b
a
b
b
b
a













14. Vector Product: Components
)
k̂
k̂
(
3
3
)
ĵ
k̂
(
2
3
)
î
k̂
(
1
3
)
k̂
ĵ
(
3
2
)
ĵ
ĵ
(
2
2
)
î
ĵ
(
1
2
)
k̂
î
(
3
1
)
ĵ
î
(
2
1
)
î
î
(
1
1


















b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a

15. Vector Product: Components
k̂
)
1
2
2
1
(
ĵ
)
3
1
1
3
(
î
)
2
3
3
2
(
b
a
b
a
b
a
b
a
b
a
b
a
b
a










16. Vector Product: Determinant
3
2
1
3
2
1
k̂
ĵ
î
b
b
b
a
a
a

b
a



17. Examples in Physics
z
x
y
F

r

F
r
τ







The torque
produced by
a force is

18. x
y
z
O
The angular
momentum
of a particle
with respect
to O
Examples in Physics (contd)
v
p


m

r

p
r
L





p
r
L






19. Examples in Physics (contd)
The force acting
on a charged
particle moving
in a magnetic
field,
)
B
v
(
F




 q
v

B

F

F

Positive charge
Negative charge

20. Applications
)
ˆ
ˆ
3
ˆ
2
(
)
ˆ
2
ˆ
ˆ
5
( k
j
i
k
j
i 




A simple cross product
1
3
2
2
1
5
ˆ
ˆ
ˆ



k
j
i
k
j
i
ˆ
)]
1
.
2
(
3
.
5
[
ˆ
)]
1
.
5
(
2
.
2
[
ˆ
]
2
.
3
)
1
.
1
[(










17
9
-5

21. k
j
i ˆ
17
ˆ
9
ˆ
5 



C
H
E
C
K
)
( b
a


 is perpendicular to b
a


&
An Example
)
ˆ
ˆ
3
ˆ
2
(
)
ˆ
2
ˆ
ˆ
5
( k
j
i
k
j
i 




0
17
.
2
9
.
1
5
.
5
)
ˆ
17
ˆ
9
ˆ
5
).(
ˆ
2
ˆ
ˆ
5
(









 k
j
i
k
j
i

22. finding a unit vector
perpendicular to a plane.
Find a unit vector perpendicular
to the plane containing two vectors b
&
a


Applications(contd.)
A vector perpendicular to a
and b is
• Corresponding unit vector
b
a



|
b
a
|
b
a







23. k
j
i
b
a ˆ
17
ˆ
9
ˆ
5 






Cross product
Magnitude 395
289
81
25
|
| 



b
a


Unit vector is )
ˆ
17
ˆ
9
ˆ
5
(
395
1
k
j
i 


An Example
Determine a unit vector perpendicular
to the plane of
and
)
ˆ
2
ˆ
ˆ
5
( k
j
i
a 



)
ˆ
ˆ
3
ˆ
2
( k
j
i
b 




24. SUMMARY
Magnitude and Direction of a vector
remain invariant under transformation
of coordinates.
Product of a vector with a scalar is a
vector quantity
Vector product : directional property,
denotes an area.
Scalar product
a . b = ax bx + ay by + az bz

25. TRIPLE PRODUCTS
Scalar Triple Product
)
c
b
(
a





Vector Triple Product
)
( c
b
a






26. Scalar Triple Product
)]
ˆ
3
ˆ
2
ˆ
1
(
)
ˆ
3
ˆ
2
ˆ
1
[(
)
ˆ
3
ˆ
2
ˆ
1
(
k
c
j
c
i
c
k
b
j
b
i
b
k
a
j
a
i
a








]
ˆ
)
1
2
2
1
(
ˆ
)
3
1
1
3
(
ˆ
)
2
3
3
2
[(
)
ˆ
3
ˆ
2
ˆ
1
(
k
c
b
c
b
j
c
b
c
b
i
c
b
c
b
k
a
j
a
i
a










27. Scalar Triple Product (contd.)
)
(
)
(
)
(
1
2
2
1
3
3
1
1
3
2
2
3
3
2
1
c
b
c
b
a
c
b
c
b
a
c
b
c
b
a






3
2
1
3
2
1
3
2
1
c
c
c
b
b
b
a
a
a


 )
( c
b
a




28. Scalar Triple Product (contd.)
a

b

c


29. 
sin
|
| bc
c
b 



b

c


b sin
|
| c
b


 Area of the base
Scalar Triple Product (contd.)
c
b




30. a

b

c

c
b




a cos  = height

cos
|
|
|
)
(
| c
b
a
c
b
a









Volume of the parallelopiped
Scalar Triple Product (contd.)

31. Properties
Interchanging any two rows reverses
the sign of the determinant, so
)
b
c
(
a
)
c
b
(
a












Interchanging rows twice the original
sign is restored, so
)
b
a
(
c
)
a
c
(
b
)
c
b
(
a


















32. Properties
If any two vectors of the scalar triple
product are equal, the scalar triple
product is zero.
0
)
( 

 c
a
a




33. bac - cab
rule
Vector Triple Product
)
( c
b
a





)
.
(
)
.
( b
a
c
c
a
b









34. SUMMARY
A physical quantity which has
both a magnitude and a direction
is represented by a vector
A geometrical representation
An analytical description: components
Can be resolved into components along
any three directions which are non
planar.

35. SUMMARY
Vector Product
k̂
)
1
2
2
1
(
ĵ
)
3
1
1
3
(
î
)
2
3
3
2
(
b
a
b
a
b
a
b
a
b
a
b
a
b
a









Scalar Product of vectors
a . b = a1 b1 + a2 b2 + a3 b3

36. SUMMARY
Scalar Triple Product : volume of a
parallelepiped.
)
c
b
(
a





Vector Triple Product
Quadruple Product of vectors

37. z
f
k
y
f
j
x
f
i
z
y
x
f
F










 )
,
,
(

z
k
y
j
x
i










F

i.e. gradient of a scalar quantity is a vector quantity,
)
,
,
( z
y
x
f is a scalar quantity
Geometrical Interpretation:
Gradient has magnitude and direction
( , , ). cos
df f x y z dl f dl 
  

38. For fix value of magnitude of dl, df is greatest
when cos  is zero, i.e. we move in the same
direction as f.
o The gradient f points in the direction of
maximum increase of the function f.
o The magnitude f gives the slope (rate of
increase) along this maximal direction.
( , , ). cos
df f x y z dl f dl 
  

39. DIVERGENCE
z
F
y
F
x
F
F z
y
x











.
is a vector quantity.
is a scalar quantity
F

F

.

F

.
 is known as divergence of a vector quantity ( )
Physical Significance
It represents how much the vector spreads out (
diverges) from the point. If divergence of any
vector is positive then it shows Spreading out
and if negative then coming towards that point.
F

z
k
y
j
x
i










x y z
F iF jF kF
  

40. Eg: Divergence of current density:
(Current density : current per unit area)
at a point gives the amount of charge flowing
out per second per unit volume from a small
closed surface surrounding the point.
j
( ) 0
div v  i.e. the flux entering any element of space
is exactly balanced by that leaving it.
Such vectors are known as solenoidal vector
Point works as Source Point works as Sink

41. CURL
is a vector quantity
is a vector quantity known as curl of
Physical Significance
It is a measure of how much the vector curls
around the point.
 
z
y
x kF
jF
iF
z
k
y
j
x
i
F 






















F

F


 F


42. PROBLEMS
 1. If A=3x2y - y3x2, calculate gradient A at a point
(1,-2,-1)
 2. If = x2yi-2xzj+2yzk, calculate divergence and
curl of a vector at (1,2,1).
Ans: 1. 10i-9j
2.(i) 6 (ii) k
A

A


43. SECOND DERIVATIVES
 The gradient, the divergence and the curl are the only
first derivatives we can make with , by applying
twice we can construct five species of second
derivatives.
 The gradient is a vector, so we can take the divergence
and curl of it.
(1) Divergence of gradient : (Laplacian)
(2) Curl of gradient:
o The divergence is a scalar, so we can take its gradient.
(3) Gradient of divergence.
o The curl is a vector, so we can take its divergence and
curl.
(4) Divergence of a Curl.
(5) Curl of curl.


0
)
( 


 A
)
.
( A



0
)
.( 


 A

A
A
A


 2
)
.
(
)
( 








A
A 2
)
.( 


