05 - lecture note 5 - vector differential calculus

8/13/2019 05 - Lecture Note 5 - Vector Differential Calculus

1/10


2/10

For function !ith multi$variale :

z

f

y

fk

x

fh)z,y,x(f)z,ky,hx(f oooooo

++

=+++

z

f

y

fk

x

fh)z,y,x(fd

+

+

=

(5$0)

or

f f ff dx dy dz

x y z

= + +

(5$)

5.0. ulti$variate scalar calculus

o Chain rule

Chain rule for multi$variate scalar function against aritrar2 variale:

))v,u(z),v,u(y),v,u(x(fW=

v

z

z

w

v

y

y

w

v

x

x

w

v

wu

z

z

w

u

y

y

w

u

x

x

w

u

w

+

+

=

+

+

=

(5$3)

Chain rule for multi$variate scalar function against s"ace an, time:

))t(z),t(y),t(x(fW=

+

+

=

t

z

z

w

t

y

y

w

t

x

x

w

t

w

(5$5)

z

w)t('z

y

w)t(y

x

w)t(x

t

w

+

+

=

(5$4)

o Variation against s"ace (s"atial Variation)

zz

wy

y

wx

x

ww

+

+

=

Lecture Note & 'r.r. Lili *o +i,o,o- /

5$0


3/10

dzz

wdy

y

wdx

x

wdw

+

+

=

w w ww dx dy dz

x y z

= + +

(5$)

*#am"le : variation of flu# against s"ace an, timeis e#"resse, as :

dtt

vdz

z

vdy

y

vdx

x

vdv

+

+

+

=

(5$6)

t

v

z

vv

y

vv

x

vv

Dt

Dvzyx

+

+

+

=

(5$7)

5.. 8ra,ientof scalar fiel,(s"atial variation)

kz

fj

y

fi

x

f

z

f,

y

f,

x

ffgrad

+

+

=

=

(5$19)

=

z

f,

y

f,

x

ff

kz

jy

ixz

,y

,x

+

+

=

=

(5$11)

o 'irectional ,erivative

Fig. 5.0: 'irectional ,erivative

X

)x(f)xx(f

x

flim

0x

+=

: in # ,irection


5$

;

C

/


4/10

S

)(f)(f

flim

0

+=

: in s ,irection

S

)!(f)"(flim

0

=

(5$10)

S

)!(f)"(f

ffD lim

0#

=

=

(5$1)

*


5/10

fgrad%#f%#fD

f

fgrad%#f%#

# ===

==

(5$17)

=n fig. 5.0- # is unit ,irectional vector. For an aritrar2 & vector- it is otaine, the

follo!ing ,irectional ,erivative- !hich is in line !ith the ,irection of & vector an, is

e#"resse, as

f%&%&

ffD& =

=

(5$09)

From e


6/10

Fig. : h2sical meaning of ,irectional ,erivative over iso$lines

!ill e ma#imum if coinci,e !ith rd or # or &

o 8ra,ient as normal vector of intersection "lane

Fig. 5.3: 8ra,ient as a normal vector


5$4

+

rd

*

'

d

r

: tangent line

)t(r)t(r

)t(u

(0,0,0)


7/10

Fig. 5.5: Normal vector on iso$lines

rove :

/calar function is generall2 e#"resses as :

)z,y,x( or ))t(z),t(y),t(x(

(5$0)

arametric e


8/10

[ ]*,,3),0,( =

2%1)',0,'( =

0) 8iven : (x,y,z) x*/ -y*/ xy + *z + 1. Com"ute ,erivative of ,irecte,

to!ar, vector [ ]43,4152,4151a= at "oint (', 0, ')%

Ans!er :

[ ]*,,3),0,( =

[ ] [ ]*,,3%43,4152,415141

%fD

# == a

'0%*41

'**==

) 8ive unit vector normal of "lane ( x, y , z) *x + y + 3z + '0

Ans!er :

[ ]3,,*)0z3yx*( =+++=

[ ]3,,*.=

[ ]3,,*1

.%

.

+ ==

Character of gra,ient in general:

o 8ra,ient of scalar fiel, has similar ,irection !ith its normal-

o 8ra,ient increases to!ar, increasing value of corres"on,ing fiel,-

o >rans"ort "hanomena !ill ,irect to the ,ecreasing value of corres"on,ing fiel,-

o /calar value (mo,ulus) of gra,ient e


9/10

5.3. 'ivergenceof vector fiel,

[ ]* v,v,vv =

' * div ( v ) v v v % vx y z

= + + =

(5$0)

[ ]* v,v,v%z

,y

,x

=

Com"are ?

fz

,y

,x

ffgrad

==

La"lace o"erator(*)

( ) ( )

===

z,

y,

x%

z,

y,

x%%div *

*

*

*

*

*

*

zyxz,

y,

x%

z,

y,

x

+

+

=

=

(5$06)

5.5. Curlof vector fiel,

[ ]* v,v,vv =


5$7


10/10

ky

v

x

v

jxv

zv

iz

v

y

v

vvv

zyx

kji

vxv6url

'*

,'

*,

,*'

+

+

===

(5$07)

0xgrad6url ==

(5$9)


5$19

05 - lecture note 5 - vector differential calculus

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