Lecture Note 05 - Vector Differential Calculus
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Transcript of Lecture Note 05 - Vector Differential Calculus
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8/13/2019 Lecture Note 05 - Vector Differential Calculus
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Lecture Note 5
Basics of Vector Differential Calculus
5.1. Basics of scalar calculus
o Average theorem in calculus (linearization)
Fig. 5.1 Average theorem in calculus (linearization) !et"een t"o #oints
$
($)%&$)($
fflim)x('f
0x
+=
(5%1)
x
f
h)x(f)hx(f
h
)x(f)hfff
x
f
oo
o
=+
+=
+=
o$(
$
($)%$)$(
Lecture Note ' Dr.r. Lili *o +i,o,o- /
5%1
y
f(x)
f(x)
f(x + x)
x
x
x + x
D
(Xo,Yo)
(Xo+h , Yo+k)
$
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For function "ith multi%varia!le
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 ar!itrar2 varia!le
))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 ' Dr.r. Lili *o +i,o,o- /
5%0
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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 Directional ,erivative
Fig. 5.0 Directional ,erivative
X
)x(f)xx(f
x
flim
0x
+=
in $ ,irection
Lecture Note ' Dr.r. Lili *o +i,o,o- /
5%
:
;
C
!
/
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S
)(f)(f
flim
0
+=
in s ,irection
S
)!(f)"(flim
0
=
(5%10)
S
)!(f)"(f
ffD lim
0#
=
=
(5%1)
*
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8/13/2019 Lecture Note 05 - Vector Differential Calculus
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fgrad%#f%#fD
f
fgrad%#f%#
# ===
==
(5%17)
=n fig. 5.0- # is unit ,irectional vector. For an ar!itrar2 & vector- it is o!taine, the
follo"ing ,irectional ,erivative- "hich is in line "ith the ,irection of & vector an, is
e$#resse, as
f%&%&
ffD& =
=
(5%09)
From e
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8/13/2019 Lecture Note 05 - Vector Differential Calculus
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Fig. :h2sical meaning of ,irectional ,erivative over iso%lines
"ill !e ma$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
Lecture Note ' Dr.r. Lili *o +i,o,o- /
5%4
+
rd
*
'
d
r
tangent line
)t(r)t(r
)t(u
(0,0,0)
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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
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[ ]*,,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
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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 =
kyv
xv
jx
v
z
v
iz
v
y
v
vvvzyx
kji
vxv6url
'*
,'
*,
,*'
+
+
=
==
(5%07)
0xgrad6url ==
(5%9)
Lecture Note ' Dr.r. Lili *o +i,o,o- /
5%7
