ch3 review of partial differential operators
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3. Review of Partial Differential Operations
1. Partial Derivatives
Given a certain multidimensional function, ),,,( t z y x A , a partial derivative at a specific point defines the local rate of change of that function in a particular direction. For the4-dimensional variable, ),,,( t z y x A , the partial derivatives are expressed as
( ) ( ) x
t z y x At z y x x A x A
t z y x
t z y +=
,,,,,,limconstant,,
0 ,,
= slope of A in the x direction
( ) ( ) y
t z y x At z y y x A y A
t z x y
t z x +=
,,,,,,limconstant,,
0 ,,
= slope of A in the direction
( ) ( ) z
t z y x At z z y x A z A
t y x z
t y x +=
,,,,,,limconstant,,
0 ,,
= slope of A in the ! direction
( ) ( )t
t z y x At t z y x At A
z y xt
z y x +=
,,,,,,limconstant,,0
,,
= the local time rate of change of A
"he subscripts on the brac#ets indicate that those dimensions are held constant.
$otice that the definition of a partial derivative of a multi-variable function is the same as
derivatives of functions of a single variable, but %ith the other variables of the function beingheld constant. &henever ou see the 'bac#%ard-six notation for the derivative, ou shouldthin# about %hat variable ou are operating on, as indicated in the denominator of theexpression, %hile holding the other variables constant.
t is common convention that the directions being held constant are implied and not explicitel%ritten %ith subscripts.
2. Higher order partial derivatives
*
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&e can appl the partial derivative multiple times on a scalar function or vector. Forexample, given a multivariable function, ( ) y x f , , there are four possible second order partialderivatives+
x y f
x f
y y x f
y f
x y f
y f
y x f
x f
x = =
=
=
- - -
"he last t%o partial derivatives, y x
f
,
and x y
f
,
are called 'mixed derivatives. n
important theorem of multi-variable calculus is the mixed derivative theorem . "he proof is be ond the scope of this course and onl the results are stated.
Mixed derivative Theorem: f a function ( ) y x f , is continous and smooth to second order,then the order of operation of the partial derivatives does not matter. n other %ords+
x y f
y x f
=
,,
for a continous and smooth (to second order) function ( ) y x f ,
xample: For the function ( ) ( ) y x xy y x f exp, += , sho% x y
f y x
f
=
,,
!nswer Provided:
( )( ) ( )( ) ( ) ( )( ) y x yx x y y x x xy x
y x xy y x y
f x y x
f exp*expexp ++=+
=
+
=
=
( )( ) ( )( )( ) ( ) ( )( ) y x yx x y y x xy y y
y x xy x y x
f
y x y
f exp*expexp ++=+
=
+
=
=
&e can see that the order of operation of the partial derivative on a continous and smooth scalarfunction does not matter.
3. Del operator:
"he del operator is a linear combination of spatial partial derivatives. n rectangularcoordinates, it is expressed as
zk
y j
xi ++=
///
(*)
$otice the second e ualit above is missing the vector arro%. is al%a s a vector operatorand thus it is common convention to 1ust leave off the vector arro%.
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"he anal sis of the del operator on various ob1ects such as scalar functions or vectors can be rather complex. n rectangular coordinates, ho%ever, the rules %e learned about in chapter on 'multipl ing vectors appl to the del operator as %ell. t is important to notice ho%ever thatthe order is extremel important in the use of e uation (*). "he del operator acts on all ob1ects tothe right of it. "t is #r$i#ial to note that the del operator is not #omm$tative when applied to
s#alars or ve#tors% &o$ onl' appl' del operators on what is to the right in the term andnever on the o()e#ts to the left.
*. +radient Operator
ppl ing the gradient operator, , on a scalar function ( ) z y x ,, = , simpl re uiresscalar multiplication. "he gradient of ields the follo%ing+
k z
j y
i x z
k y
j x
i 222222+
+=
+
+= ( )
$otice that e uation ( ) is a linear combination of vector components and basis vectors. nother %ords the gradient of a scalar ields a vector. &o$ will (e tested on the appli#ationleading to e,$ation -2 as well as the fa#t that the res$lt of is a ve#tor. 3ince thegradient of a scalar function is a vector, it obe s all the rules that %e learned about in chapter .
xample: For scalar function xyz = sho% that
( ) ( ) x x
xample Given a velocit vector
///
k w jviuu ++= and the gradient of a scalar function, as
defined in e uation ( ), expand out u in rectangular coordinates+
!nswer provided: 5sing e uation (*0) from chapter ,
zw
yv
xuk
z j y
i x
k w jviuu+
+=
++
++= 222
///
6$otice the result is a scalar as re uired for the dot product of t%o vectors.66$otice that there %ere no parantheses given for the application of the operation above. &etoo# the dot product of the vector u %ith the vector . &e could have 1ust as %ell ta#en thedot product of the vector u %ith the operator and then applied that on the scalar function+
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( ) zw
yv
xu
zw
yv
xuk
z j y
i x
k w jviuu+
+=
++
=+
+++= 222
///
.
n other %ords, ( ) = uu . "his e ualit is onl relavent %hen %e are operating on ascalar .
n this course, %e %ill onl ta#e gradients of s#alar f$n#tions . t is possible to ta#e gradientsof vectors but ou obtain a 8 element matrix called the Dyadic product 1 of the vector field, xu .
For example, given the vector
///
k w jviuu ++= , the gradient of u is
=+
+
=
zw
zv
zu
yw
yv
yu
xw
xv
xu
k zu
j yu
i xu
u///
9ou can see %h %e %ant to avoid operations li#e this.
+radient properties: magnit$de
: uation ( ) is a vector since it has a magnitude and direction. For a function( ) z y x f f ,,= , the magnitude of f is simpl found using the rules of chapter .
+
+
==
z f
y f
x f
f f f (7)
* "he operation is also related to the transpose of the ;acobian
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+radient properties: dire#tion
"he direction of f is a bit more complicated. From the previous chapter %e can see that
the direction of f can be expressed b the unit vector, f f
, but %e also can interpret the
direction of f in a more geometric or ph sical %a . First %e need to use the differential of f , %hich is labeled df . differential is an infintesimal (meaning reall small) change in thevalue of the multivariable function f and has components+
dz z
f dy
y f
dx x f
df +
+=
f %e define the vector line element,
///
k dz jdyidxd ++= , then %e can see b inspection that thedifferential ta#es the simple form
d f df =
$o% let us appl the geometric definition of the dot product d f +
( ) cos d f d f df == %here is the coplanar angle bet%een the vector f andd .
f d is perpendicular f then o80= and 0=df . n other %ords, d is along lines ofconstant f %hen it is perpendicular to f . lternativel , %e find that df is a maximum%hen d is parallel to f . "his means that df is maximum %hen d is in the samedirection as f (and also perpendiculal to contours of constant f ). "his also means that f must al%a s be in the direction that leads to the greatest df . "he direction of f is alsocalled the asecendant of f. Figure *, on page >, sho%s ou a picture relating the direction of
f to lines of constant f .
/: The #hange of a ,$antit' in the dire#tion of the velo#it' field -!dve#tion
&e can find the change of a scalar, ( ) z y x f ,, , in an arbitrar direction,/
7
/
,
/
*
/
k u juiuu ++= %here*7* =++ uuu , b ta#ing the dot product of
/
u %ith f . "he results is+
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f udt df
=/
(4)
"o derive e uation (4), parametri!e the spatial curve, ( ) ( ) ( )///
k t z jt yit x ++= %ith respect to the variable t +
( ) ** ut x
tu xt x o =
+=
( ) ut y
tu yt y o =
+=
( ) 77 ut z
tu z t z o =
+=
"hen, using the #hain r$le , %e obtain e uation (4)+
/
u f t
z z f
t y
y f
t x
x f
dt df
=
+
+
=
Finding variations in a specific direction often occurs %hen %e tr to find that variationof a ph sical uantit in the direction of the flow field , u . &e usuall discuss the rate of changeof the scalar uantit , ( ) z y x f ,, due to variations in f along the flo% field, u . "his isrepresented mathematicall as+
f udt df
=
"he term on the right side of the e ualit is called the advective term and is one of t%ocontributions to the total or material derivative that %e %ill learn more about later in thesemester. @ften %e are interested in determining if there is an variation in the direction of flo%.
f one obtainAs the result+
0= f u
&e sa that the function, f , is spatiall constant along the flo% field, u . For example, if ourscalar uantit is a time-independent pressure field, ( ) z y x p ,, , then the e uation 0= pu ,tells us that isobars are constant along the flo% field %hich also means that isobar contours areever %here parallel to the velocit vector field.
&hat %e have done here is to create a curve that is parameteri!ed along the direction of differentiation. "hisho%ever, is exactl %hat %e %ant to find the change in the function f along a specific direction. f ou find these
points confusing, ou ma %ant to revie% the definition of a curve from a calc textboo#.
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0$mmar':* Find the gradient of the multivariable function ( ) z y x f ,, , f , is a vector %ith magnitude
+
+
==
z f
y f
x f f f f and unit vector f
f
.
- "he direction of the gradient of f is al%a s in the direction of the greatest increase in f and perpendicular to the contours of constant f. "he direction of f is also called the ascendant of
f . 7 - &e can find the spatial rate of change of a function in a specific direction/
u b ta#ing the
dot product of/
u %ith f , f u /
. f 0/
= f u then f is constant along/
u .
xample: Find the gradient of the function ( ), y x y x f =
graph of the surface ( ), y x y x f = is sho%n in figure *.
"he vector field of the gradient is sho%n in figure .
-1-0.8
-0.6-0.4
-0.20
0.20.4
0.60.8
1
-1
-0.5
0
0.5
1-2
-1.8
-1.6-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
distance (x)
graph of the function f(x,y)=-x2-y2
distance (y)
f ( x
, y )
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
distance (x)
Overhead view of the function f(x,y)=-x 2-y2
d i s
t a n c e
( y
)
Figure * Graph of the function ( ), y x y x f = in 7-B and from an overhead vie%.
C
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-1 -0.8 -0 .6 -0.4 -0 .2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
distance (x)
d i s
t a n c e
( y )
Overhead view of the vector field, grad(f(x,y)=grad(-x2-y2)
Figure @verhead vie% of the vector field, ( ) y x f = . $otice that the arro%s are alldirected to%ard the maximum increase in the function, f.
. Divergen#e of a e#tor $antit'
"here are t%o possible %a s to appl the del operator to a vector. "he first, called thedivergence, results in a scalar function. "he second operation, the curl, results in a vector field.
For a vector field,
///
k w jviuu ++= , the divergence of u is defined as
z
w
y
v
x
ukw jviu
zk
y j xiu
+
+
=++
+
+
= //////
(?)
n index notation, e uation (?) is expressed as
D
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ii uu =
6 4oti#e that the divergen#e of a ve#tor is a s#alar ,$antit'5 )$st li6e the dot prod$#t.
Eh sicall , the divergence is a measure of the addition or removal of a vector uantit .magine a sin# full of %ater. f %e examine the flo% of the %ater near the drain of the sin# %e
%ill notice it is directed radiall in%ard indicating a net loss of the fluid. "his %ould result in anegative divergence. f %e attached a hose to the drain, so %e are adding %ater to the s steminstead of removing it, then the flo% %ould be radiall out%ard, indicating a net outflo% and a
positive divergence. f the divergence is !ero, 0= u , then there is no net inflo% or outflo%. fluid field %here 0= u is called solenoidal or divergenceless .
Figure 7 (left) ector field near a sin# drain indicating a negative divergence 0 u
xample: For the flo% field/
,/
,
j yi xu += as seen in figure 4
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
distance (x)
d i s
t a n c e
( y
)
vector flow field u=(-x2,y2)
Figure 4 ector flo% field/
,/
, j yi xu +=
3in# drain - ;et or hose outflo% -
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a) Find u at x=0, =0.?+ b) Find u at x=0.?, =0+
. The 7$rl of a e#tor $antit'
"he other %a to appl the del operator on a vector field is the curl . For a vector field,///
k w jviuu ++= , the curl of u is defined as
///
///
k yu
xv j
xw
zui
zv
yw
wvu z y x
k ji
u + + ==(>)
"he curl is a measure of the rotational properties of a vector field about a point. For a velocitfield, u , the curl is a measure of the rotation of a fluid parcel about its center of mass and iscalled the vorticity . "he vorticit is usuall denoted b the vector omega, . @ne %a to
imagine the vorticit is to place a small compass arro% in a fluid and to see ho% the arro%rotates about its center as it travels throughout the medium. f the vorticit of a fluid is !ero, it iscalled irrotational .
n oceanographic and atmospheric applications, one has particular interest in the vertical
vorticit component, ( )
== yu
xv
k u /
. "his is simpl a measure of the hori!ontal shear of
the fluid medium.
xample: Find the vorticit of the velocit field+
//
j xi yu += sho%n in figure ?+
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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
distance (x)
d i s t a n c e
( y )
velocity field (-y,x)
Figure ? Bepiction of the velocit field -
//
j xi yu +=
8. 9apla#ian of a s#alar f$n#tion:
n certain circumstances, it is possible to relate a velocit field to a scalar function called
the velocit potential, k z j
yi
xu 222
+
+
== . &hen %e examine the divergence of the
velocit field, %e obtain a ne% operation on the scalar called the aplacian.
( ) z y xu
+
+
==
"he aplacian, a scalar operation, is defined generall as
z y x +
+ (C)
n index notation, it ta#es the form
( )iii == - $otice that i is a dumm or repeating index .
t consists of the divergence of the gradient and thus is a measure of the spatial rate of change of
the gradient on a scalar function.n alculus, %e learned for *-B functions that, b setting the first derivative of a function
e ual to !ero %e can find the extrema of the curve. &e can resolve if the extrema points arelocal maximums or minimums b observing the sign of the second derivative. f the secondderivative is less than 0, the local extrema is a maximum. f the second derivative is greater than0, the local extrema is a minimum. f the second derivative is e ual to !ero, then %e have aturning point .
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For multivariable functions, there are similar results. f the gradient of a scalar function,, is 0, then %e have a local extrema in the surface. &e can then use the aplacian to measure
the concavit of the surface and %hether the local extrema is a maximum 0 or a saddle point 0= .
xample: Hecall from our second example, that %e too# the gradient of the surface,( ), y x y x f = . &e can see from figure *, that %e have an extrema at the point x=0, y=0. t
is easil verified b finding %here the gradient of the surface is !ero, 0,,//
== j yi x + "his isclearl at x=0, y=0. &e can also see from figure *, that the extrema is a maximum. "his can beconfirmed b ta#ing the aplacian of + 04
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*22
,
,,//
=+=++== r y xr j yi xr j yi xrrr
7. For the vector, ( )//
,
/
, exp k xyz j xyi yxu += find u
From e uation (?)
zw
yv
xukw jviu zk y j xiu ++=++
++=
//////
For ( )//
,/, exp k xyz j yi xu +=
, %e obtain
( )( ) ( ) ( ) ( ) xyz xy xyz xy xy xy
xyz z
xy y
yx x z
w yv
xu
u
==
+
=
+
+
=
expexp
exp
&hich part of the functionAs domain is the divergence positive, negative or 0.
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0> u for xJ0, K0 and xK0, J0 ( the second and fourth uadrant of the x, plane for everfinite value of !)
0
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From e uation (C), z y x +
+ and since %e have no ! or vertical dependence %e
onl need to consider,
( ) ( ) ( ) ,*
,,
,
,
,
*,,
,
,
,
*,,
,
,
,
,, * +
++=+
+
=
y x y
y x x
y x y xr h
First examine ( ) ( ) ( )
+=+
+
,
7,,
,
*,,
,
*,,
,
,
+ y x x y x x
y x x
3o ( ) ( ) ( ) ( )( ) ,
?,,
,,
,
?,,,
,
7,,
,
7,,
,
*,,
,
, ,7
y x
y x y x x y x y x x
x y x
x +=+++=
+
=+
&here %e simplified in the last step b finding a least common denominator. &e can seeimmediatel b s mmetr of the problem that %e can obtain the second partial %ith respect to y
b s%apping the xAs and As above+
( )( ) ,
?,,
,,
,
*,,
,
,
,
y x
x y y x y +
=+
dding the t%o terms together %e obtain our solution+
( ) ( ) ( ) 7
,
?,,
,,
,
?,,
,,
,
?,,
,,, *,,*
r y x
y x
y x
x y
y x
y x
r h =
+
+=+
++
=
4ote: &e can see that evaluating
r *
in artesian coordinates is tedious. f one use polarcoordinates, ho%ever, it is rather simple. oo# up the aplacian in polar coordinates and see
ho% simple it is to evaluate
r *
.
!ppendix !: 3ome useful vector calculus identities are (memori!e the first five)+
". The gradient prod$#t r$le of two s#alar f$n#tions:( ) g f f g fg +=
"". The divergen#e prod$#t r$le with a ve#tor and a s#alar:( ) uuu +=
""". The divergen#e of the gradient of a s#alar The 9apla#ian: =
" . The #$rl of the gradient of a s#alar:
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( ) ( )0,0,00 == $otice the solution is the vector !ero, 0 , of %hich each component is !ero. t is commonnotation to impl the vector s mbol of the vector !ero since the curl is al%a s a vector result.
. The divergen#e of the #$rl of a ve#tor:0= u
$otice that this is 1ust the scalar number 0 since the divergence al%a s results in a scalarfunction or number.
". The #ross;prod$#t prod$#t r$le with a ve#tor and a s#alar:( ) = uuu
"". The divergen#e of the #ross prod$#t:( ) ( ) ( )!aa!!a =
""". The #$rl of the #ross prod$#t of a ve#tor:( ) ( ) ( ) ( ) ( )a!!a!aa!!a +=
"