guide to calculus 3

7/18/2019 Guide to Calculus 3

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1

A guide to calculus 3

Chapter 12, functions of several variables

12.1-functions of two variables

Distance= √ ( x−a)2+( y−b)2+( z−c)2

12.2- graphs of functions of two variablesf is the set of all points (x,y,x) such that z=f(x,y)

12.- contour !iagra"s- #every year $ go to %olora!o to hi&e'

12.- linear functions

$f a plane has slope m in the x !irection, slope n in the y !irection an! passesthrough the point ( x0, y0 , z0) , the the e*uation is

z= z0+m ( x− x0 )+n( y− y0 )

+his is the graph of the linear function

f ( x , y )= z0+m ( x− x0 )+n ( y− y0 )

linear function can be recognize! fro" its table by the following features

each row an! each colu"n is linear

all the rows have the sa"e slope

all the colu"ns have the sa"e slope (although the slope of the rowsan! the slope of the colu"ns are generally !ierent)

12./-functions of three variables level surface, or level set of a function of three variables, f(x,y,z), is asurface of the for" f(x,y,z)=c, where c is a constant. +he function f can berepresente! by the fa"ily of level surfaces obtaine! by allowing c to vary

0lliptical paraboloi!

z= x

2

a2+

y2

b2

yperbolic paraboloi!

z=− x

2

a2 +

y2

b2

0llipsoi! x

2

a2+

y2

a2+

z2

a2=1

yperboloi! of one sheet

x2

a2+

y2

a2−

z2

a2=1



2

yperboloi! of two sheets

x2

a2+

y2

a2−

z2

a2=−1

%one x

2

a2+ y

2

a2− z

2

a2=0

lane xa+by+cz=d

%ylin!rical surface x2+ y

2=a2

arabolic cylin!er y=a x2

12.3 li"its an! continuity

+he function f has a limit 4 at the point (a,b), writtenlim

( x , y)→(a , b)f ( x , y )= L

$f f(x,y) is as close to 4 as we please whenever the !istance fro" the point(x,y) to the point (a,b)is su5ciently s"all, but not zero.

function f is continuous at the point (a,b) if lim( x , y)→(a , b) f ( x , y )=f (a , b)

function is continuous on a region 6 in the by-plane if it is continuous ateach point in 6.

Chapter 13: a fundamental tool: vectors

1.1 !isplace"ent vectors +he displacement vector fro" one point to another is an arrow with its tailat the 7rst point an! its tip at the secon!. +he magnitude (or length) of the!isplace"ent vector is the !istance between the points, an! is represente!by the length of the arrow. +he direction of the !isplace"ent vector is the

!irection of the arrow.

!!ing an! subtracting vectors8!uh !uh !uh

1.2 vectors in general- nothing to it

1. the !ot pro!uct +he !ot pro!uct is a scalar pro!uct

Geometric denition v ∙ w=‖w‖‖v‖cosθ

Algebraic denitionv ∙ w=v1 w1+v2 w2+v3 w3

+he equation of the plane with nor"al vector n=a i+b j+c k an!

containing the point P

0=( x0

, y0

, z0 ) is

a ( x− x0 )+b ( y− y

0 )+c ( z− z0 )=0



4ettingd=a x0+b y 0+c z0 (a constant), we can write the e*uation of the

plane in the for"ax+by+cz=d

ro9ection of v onto the line in the !irection of vector u

v¿=(v ∙ u )u

‖u‖2

+he wor&, :, !one by a force F acting on an ob9ect through a

!isplace"ent d is given by

W = F ∙ d

1. the cross pro!uct

Geometric denition v × w=(‖v‖‖w‖sinθ ) n

Algebraic denition

v × w=| i j k

v1 v2 v3

w1 w2 w3|=(v2 w3−v3 w2 ) i ; ( v

3w

1−v

1w

3 ) j+ (v1w

2−v

2w

1 ) k

Area of parallelogram with e!ges v∧w Area=‖v × w‖

Volume of a parallelepiped with e!ges a , b , c volu"e= |(b × c ) ∙ a|

Chapter 1, di!erentiating functions of several variables

1.1 +he partial !erivative

"artial derivatives at the point #a,b$

f x ( a ,b )= Rate of can!eof f wit re"#ect ¿ x at (a , b )=lim →0

f ( a+ ,b )−f (a , b)



f y (a , b )= Rateof can!e of f witre"#ect ¿ y at (a , b )=lim →0

f ( a , b+ )−f (a , b)

$f we let a an! b vary, we have the partial derivative functions f x ( x , y )

an! f y( x , y)

1.2 co"puting partial !erivatives algebraically- 9ust ta&e the !erivative withrespect to x, or y, or potato

1. local linearity an! the !ierential

%angent plane to the surface z=f ( x , y ) at the point #a,b$

ssu"ing f is !ierentiable at (a,b), the e*uation of the tangent plane is

z=f ( a ,b )+ f x (a , b ) ( x−a )+f y(a , b)( y−b)

%angent "lane Appro&imation to f#&,'$ for #&,'$ near the point #a,b$rovi!e! f is !ierentiable at (a,b) we can approxi"ate f(x,y)

f ( x , y )$ f ( a ,b )+ f x (a , b ) ( x−a )+f y(a , b)( y−b)

+he !ierential of a function z=f(x,y) +he !ierential, !f (or !z), at a point (a,b) is the linear function of !x an! !y

given by the for"ula df =f x (a , b )dx+ f y ( a ,b ) dy

< a general pointdf =f

x

dx+ f y

dy

1. gra!ients an! !irectional !erivatives in the plane(irectional derivative of f at #a,b$ in the direction of a unit vector

u

a+u1,

lim →0

f (¿b+u2)−f (a , b)

f u (a , b , c )=rate of can!e of f ∈directionof u at (a , b )=¿

%he gradient vector of a di!erential function at the point #a,b$

!radf ( a ,b )=f x (a ,b ) i+ f y (a , b) j

%he directional derivative and the gradient

$f f is !ierentiable at (a,b) an! u is a unit vector then



/

f u (a , b )=f x ( a ,b )u1+ f y (a , b ) u2=!rad f (a ,b) ∙u

Geometric properties of the gradient vector in the plane +he !irection of gra! f(a,b) is

erpen!icular to the contour of f through (a,b)

$n the !irection of the "axi"u" rate of increase of f +he "agnitu!e of the gra!ient vector is

+he "axi"u" rate of change of f at that point4arge when the contours are close together an! s"all when they are

far apart

1./ gra!ients an! !irectional !erivatives in space

f u (a , b , c )=f x (a , b , c )u1+ f y (a , b , c )u2+ f z(a , b , c)u3

!rad f ( a , b , c)=f x

( a , b , c ) i+ f y

( a , b , c ) j+f z

(a , b , c )k

"roperties of the gradient Vector in )pace

$f f is !ierentiable at (a,b,c) an! u is a unit vector, then

f u (a , b , c )=!rad f ( a , b , c ) ∙ u

$f , in a!!ition, !rad f ( a , b , c) %0 , then

• !rad f (a , b , c ) is in the !irection of the greatest rate of increase of f

• !rad f ( a , b , c )

is perpen!icular to the level surface of f at (a,b,c)

• ‖!rad f ( a , b , c )‖ is the "axi"u" rate of change of f at (a,b,c)

%angent plane to a level surface

$f f ( a , b , c ) is !ierentiable at (a,b,c), then an e*uation for the tangent

plane to the level surface of f at the point (a,b,c) is

f x ( a , b , c) ( x−a )+ f y (a , b , c ) ( y−b )+ f z (a , b , c ) ( z−c )=0

1.3 the chain rule

$f f, g, an! h are !ierentiable an! if z=f ( x , y ) , an! x=!(t ) , an!

y= (t ) , then

dz

dt =

&z

&x

dx

dt +

&z

&y

dy

dt



3

+o 7n! the rate of change of one variable with respect to another in a chain of co"pose! !ierentiable functions

• Draw a !iagra" expressing the relationship between the variables, an!

label each lin& in the !iagra" with the !erivative relating the variables atits en!s

• or each path between the two variables, "ultiply together the!erivatives fro" each step along the path

• !! the contributions fro" each path

$f f, g, h are !ierentiable an! if z=f ( x , y ) , with x=! (u , v ) an!

y= (u , v ) , then

&z

&u=

&z

&x

&x

&u+

&z

&y

&y

&u

&z

&v

=&z

&x

&x

&v

+&z

&y

&y

&v

1.> secon! or!er partial !erivatives

%he second order partial derivatives of z=f ( x , y )

& 2

z

& x2=f xx=( f x ) x

& 2

z&y&x

=f xy=( f x ) y

& 2 z

&x&y=f yx=( f y ) x

& 2

z

& y2=f yy=( f y ) y

*qualit' of mi&ed partial derivatives

$f f xy (a , b )=f yx(a , b)

%a'lor pol'nomial of degree 1 appro&imating f(x,y) for # x,y) near#+,+$

f ( x , y ) $ L ( x , y )=f (0,0 )+ f x (0,0 ) x+f y (0,0 ) y

%a'lor pol'nomial of degree 2 appro&imating f#&,'$ for #&,'$ near#+,+$

$f f has continuous secon! or!er partial !erivatives, then

f ( x , y ) $ ' ( x , y )=f (0,0)+ f x (0,0 ) x+f y (0,0 ) y+f xx (0,0)

2 x

2+ f xy (0,0 ) xy+f yy(0,0)

2 y

2



>

%a'lor pol'nomials appro&imating f#&,'$ for #&,'$ near #a,b$?i"ply substitute in (x-a) for x an! (y-b) for y

Chapter 1: -ptimi.ation: local and global e&trema

1/.1 local extre"a

• f has a local ma&imum at the point P0 if

f ( P0)( f ( P ) for all

points near P0

• f has a local minimum at the point P0 if

f ( P0)) f ( P )

for all

points near P0

points where the gra!ient is either @ or un!e7ne! are calle! critical points of the function

)econd derivative test for functions of t/o variables

?uppose ( x0, y0) is a point where!rad f ( x0

, y0 )=0

. 4et

*=f xx ( x0 , y 0) f yy ( x0 , y0 )−( f xy( x0 , y0))2

if *>0 an! f xx ( x0 , y0)>0 then f has a local "ini"u" at ( x0 , y0 )

if *>0 an!f xx ( x0, y0 )<0 then f has a local "axi"u" at ( x0 , y0 )

if *<0 then f has a sa!!le point at ( x0 , y0 )

if *=0, anyting can happen f can have a local "axi"u", or a local

"ini"u", or a sa!!le point, or none of these at ( x0 , y0 ) +

1/.2 opti"ization

• f has a global ma&imum at the point P0 if

f ( P0)( f ( P )

for all

points in 6

• f has a global minimum at the point P0 if

f ( P0)) f ( P )

for all

points in 6

0or an unconstrained -ptimi.ation problem

7n! the critical points

investigate whether the critical points give global "axi"a or "ini"a

closed region is one which contains its boun!ary



A

bounded region is one which !oes not stretch to in7nity in any !irection

*&treme value theorem for multivariable functions$f f is continuous function on a close! an! boun!e! region 6, then f has a

global "axi"u" at so"e point ( x0 , y0 ) in 6 an! a global "ini"u" at so"e

point ( x1 , y 1 ) in 6.

1/. %onstraine! opti"ization 4aBrange "ultipliers

?uppose P0 is appoint satisfying the constraint ! ( x , y )=c

• f has a local ma&imum at P0 subect to the constraint if

f ( P0 ) ( f ( P ) for all points near

P0 satisfying the constraint

• f has a global ma&imum at P

0 subect to the constraint if

f ( P0 ) ( f ( P ) for all points satisfying the constraint.

4ocal an! global "ini"a are !e7ne! si"ilarly.

-ptimi.ing f subect to the constraint gc$f a s"ooth function, f, has a "axi"u" or "ini"u" sub9ect to a s"ooth

constraint !=c at a point if P0 , then either

P0 satis7es the e*uations

!rad f = !rad ! an! !=c

Cr P0 is an en!point of the constraint, or

P

(¿¿ 0)=0

!rad !¿. +o 7n!

P0

co"pare values of f at these points. +he nu"ber is calle! the aGrange

multiplier

)trateg' for -ptimi.ing f#&,'$ subect to the constraint ! ( x , y )) c

• 7n! all points in the interior ! ( x , y )) c where !rad f is zero or

un!e7ne!

• se 4aBrange "ultipliers to 7n! the local extre"a of f on the boun!ary! ( x , y )) c

• 0valuate f at the points foun! in the previous two steps an! co"pare

the values



E

+he value of is the rate of change of the opti"u" value of f as c

increases (where ! ( x , y )=c ) if the opti"u" value of f is written as

f ( x0 (c ) , y 0 (c )) then

ddc

f ( x0 (c ) , y0 (c ) )=¿

Chapter 14: integrating functions of several variables

13.1 the !e7nite integral of a function of two variables

?uppose the function f is continuous on 6, the rectangle a ) x ) b , c ) x ) d . $f

(uij , v ij) is any point in the ij-th subrectangle, we !e7ne the denite

integral of f over 6

∫ R

fdA= lim- x,- y→0

∑i , j

f (uij , v ij)- x - y

?uch an! integral is calle! a double integral

$f x,y,z represent length an! f is positive, then

vo.ume under !ra#of f above re!ion R=∫ R

fdA

area ( R )=∫ R

❑

dA

avera!e va.ue of f on te re!ionR= 1

area of R∫ R

fdA

13.2 iterate! integrals

5riting a double integral as an iterated integral

$f 6 is the rectangle a ) x ) b , c ) x ) d an! f is a continuous function of 6, the

integral of f over 6 exists an! is e*ual to the iterated integral

∫ R

fdA=∫ y=c

y=d

(∫ x=a

x=b

f ( x , y )dx )dy

+his expression can be re written as ∫c

d

∫a

b

f ( x , y )dx dy



1@

Fnote, or!er of the integral !oes not "atter, can be !one !x!y or !y!x

imits on 6terated 6ntegral

+he li"its on the outer integral "ust be constants

+he li"its on the inner integral can involve only the variable in the

outer integral. or exa"ple, if the inner integral is with respect to x, itsli"its can be functions of y.

13. triple integrals

%riple integral as an iterated integral

∫W

fd/ =∫ #

0

(∫c

d

(∫a

b

f ( x , y , z )dx )dy)dz

:here y an! z are treate! as constraints in the inner"ost (!x) integral, an! zis treate! as a constant in the "i!!le (!y) integral. Cther or!ers ofintegration are possible.

imits on triple integrals

• +he li"its for the outer integral are constants

• +he li"its for the "i!!le integral can involve only one variable (that in

the outer integral)

• +he li"its for the inner integral can involve two variables (those on the

two outer integrals)

13. !ouble integrals in polar coor!inates

:hen co"puting integrals in polar coor!inates, use

x=r cosθ , y=r sinθ , x2+ y2=r2 . ut dA=rdrdθ∨dA=r d θ d r +

13./ integrals in cylin!rical an! spherical coor!inates

7elation bet/een Cartesian and c'lindrical coordinates

0ach point in -space is represente! using 0) r<1 ,0) θ )2 2 ,−1< z<1 ,

x=r cosθ y=r sinθ z= z

s with polar coor!inates x2+ y

2=r2

:hen co"puting integrals in cylin!rical coor!inates d/ =rdrdθdz

7elationship bet/een Cartesian and spherical coordinates

0ach point in -space is represente! using 0) 3<1 ,0)ϕ ) 2 ,0) θ )22

x= 3 sinϕ cosθ y= 3 sinϕ sinθ z= 3 cosϕ



11

lso 32= x

2+ y2+ z

2

:hen co"puting integrals in spherical coor!inates d/ = 32sin ϕd3 dϕ dθ

Chapter 18: parameteri.ation and vector elds

1>.1 para"eterize! curves

"arametric equations of a line through the point ( x0, y0 , z0) an! parallel

to the vector a i+b j+c k are

x= x0=at y= y0+bt z= z0+ct

"arametric equation of a line in vector form

+he line through the point with position vectorr0= x0 i+ y0 j+ z0 k

in the

!irection of the vector v=a i+b j+c k has para"etric e*uation

r ( t )=r0+t v

1>.2 "otion, velocity, an! acceleration

+he velocit' vector of a "oving ob9ect is a vector v such that

• +he "agnitu!e of v is the spee! of the ob9ect

• +he !irection of

v is the !irection of "otion

+hus the spee! of the ob9ect ‖v‖ an! the velocity vector is tangent to the

ob9ectGs path

+he velocit' vector v (t ) , of a "oving ob9ect with position vector r (t ) at

ti"e t is

- r

-t =¿ lim

- t →0

r (t +- t )−r (t )- t

v ( t )= lim- t → 0

¿

:henever the li"it exist. :e use the notation v=d r

dt =r

4 (t ) (velocity is

!erivative of position)



12

+he acceleration vector of an ob9ect "oving with velocity

v (t )= lim- t →0

v ( t +- t )−v (t )-t

$f the li"it exists. :e use the notation a=d v

dt

=d2r

d t 2

9niform circular motion: for a particle whose "otion is !escribe! by

r (t )= R cos (wt ) i+ R sin (wt ) j

• Hotion in a circle of ra!ius 6 with perio! 22 /|w|

• Ielocity, is tangent to the circle an! spee! is constant ‖v‖=|w| R

• cceleration points towar! the center of the circle with ‖a‖=‖v‖2/ R

otion in a straight line: for a particle whose "otion is !escribe! by

r ( t )=r0+f ( t )v0

• Hotion is along a straight line through the point with position vector

parallel to velocity

• Ielocity an! acceleration are parallel to the line

$f the curve % is given para"etrically for a ) t ) b by s"ooth fuctions an! if

the velocity vector is not @ for a<t <b , then

Len!tof 5 =∫a

b

‖v‖dt

1>. vector 7el!s

a vector eld in 2-space is a function F ( x , y) whose value at a point (x,y)

is a 2-!i"ensional vector. ?i"ilarly, a vector 7el! in -space is a function

F ( x , y , z ) whose values are -!i"ensional

1>. the Jow of a vector 7el!

;o/ line of a vector 7el! v = F (r ) is a path r (t ) whose velocity vector

e*uals v + +hus

r4 (t )=v= F (r (t ))

+he ;o/ of a vector 7el! is the fa"ily of all of its Jow lines



1

1>./ para"eterize! surfaces

"arametric equations for a plane +he plane through the point with position vector

r (" ,t )=r0+ " v1+t v2

Chapter 1<: ine 6ntegrals

1A.1 the i!ea of a line integral

curve is sai! to be oriented if we have chosen a !irection of travel on it

+he line integral of a vector 7el! F along an oriente! curve is

∫5

F ∙ d r= lim

‖- r i‖→0

∑i=0

n−1

F (ri) ∙ - ri

work done by force F (r ) a.on!curve 5 = lim‖- ri‖→0

∑i

F (ri ) ∙ - r i=∫5

F ∙ d r

$f % is an oriente! close! curve, the line integral of a vector 7el! F aroun!

% is calle! the circulation of F aroun! %

1A.2 co"puting line integrals over para"eterize! curves

$f r (t ) , for a ) t ) b is a s"ooth parar"eterization of an oriente! curve %

an! F is a vector 7el! which is continuous on %, then

∫5

F ∙ d r=∫a

b

F (r (t ))∙ r4 (t ) dt

$n wor!s +o co"pute the line integral of F over %, ta&e the !ot pro!uct of

F evaluate! on % with the velocity vector r4

(t ) , of the para"eterization of %,

then integrate along the curve

1A. gra!ient 7el!s an! path-in!epen!ent 7el!s

0undamental theorem of calculus for line integrals



1

?uppose % is a piecewise s"ooth oriente! path with starting point an!en!ing point K. $f f is a function whose gra!ient is continuous on the path %,then

∫c

!rad f ∙ d r=f ( ' )−f ( P )

vector 7el! F is sai! to be path independent, or conservative, if for

any two points an! , the line integral ∫5

F ∙ d r has the sa"e value along

any piecewise s"ooth path % fro" to K lying in the !o"ain of F

$f F is a continuous gra!ient vector 7el!, then F is path in!epen!ent

"ath independent elds are gradient elds

$f F is a continuous path in!epen!ent vector 7el! on an open region 6, the

F =!rad f for so"e f !e7ne! on 6

continuous vector 7el! F !e7ne! on an open region is path in!epen!ent

if an! only if F is a gra!ient vector 7el!

$ a vector 7el! F is of the for" F =!rad f for so"e scalar function f,

then f is calle! a potential function for the vector 7el! F

1A. path !epen!ent vector 7el!s an! greenGs theore"

vector 7el! is path-in!epen!ent if an! only if ∫5

F ∙ d r=0 for every close!

curve %.

$f F ( x , y )= F 1i+ F 2 j is a gra!ient vector 7el! with continuous partial

!erivatives, then& F 2

&x −

& F 1

&y =0



1/

:e call

& F 2

&x −

& F 1

&y the 2-!i"ensional or scalar curl of the vector 7el! F

Green=s theorem?uppose % is a piecewise s"ooth si"ple close! curve that is the boun!ary ofa region in the plane an! oriente! so that the region is on the left as we "ove

aroun! the curve. ?uppose F = F

1i+ F

2 j

is a s"ooth vector 7el!

on an open region containing 6 an! %. +hen

∫5

F ∙ d r=∫ R( & F 2

&x −

& F 1

&y )dx dy

%he Curl test for vector elds in 3>space

?uppose F is a vector 7el! on -space with continuous partial !erivatives,

such that

• +he !o"ain of F has the property that every close! curve in it can

be contracte! to a point in a s"ooth way, staying at all ti"es withinthe !o"ain.

• cur. F =0

+hen F is path-in!epen!ent, so F is a gra!ient 7el! an! has a potential

function

Chapter 1? ;u& integrals

1E.1 the i!ea of Jux integral

t each point on a s"ooth surface there are two unit nor"al, one in each!irection. Choosing an orientation "eans pic&ing one of these nor"al atevery point of the surface in a continuous way. +he nor"al vector in the

!irection of the orientation is !enote! by n . or a close! surface (that is,

the boun!ary of a soli! region), we usually choose the outwar! orientation.

+he area vector of a Jat, oriente! surface is a vector A such that

• +he "agnitu!e of A is the area of the surface

• +he !irection of A is the !irection of the orientation vector n

$f v is constant an! A is the area vector of a Jat surface, then

f.ux trou! "urface=v ∙ A



13

+he ;u& integral of the vector 7el! F through the oriente! surface ? is

∫6

F ∙ d A= lim‖- r i‖→0

∑ F ∙ - A

$f ? is a close! surface oriente! outwar!, we !escribe the Jux through ? asthe Jux out of ?.

Rate f.uid f.ow" trou! "urface6= F.ux of v trou! 6=∫6

v ∙ d A

1E.2 Jux integrals for graphs, cylin!ers, an! spheres

area vector of #ara..e.o!ram= A=v ×w

%he ;u& of F through a surface given b' a graph of z=f ( x , y )

?uppose the surface ? is the part of the graph of z= f ( x , y ) above a region

6 in the xy-plane, an! suppose ? is oriente! upwar!. +he Jux of F through

? is

∫6

F ∙ d A=∫ R

F ( x , y , f ( x , y ) )∙ (−f x i−f y j +k ) dxdy

%he ;u& of a vector eld through a c'linder +he Jux of F through the cylin!rical surface ?, of ra!ius 6 an! oriente!

away fro" the z-axis, is given by

∫6

F ∙ d A=∫7

F ( R , θ , z ) ∙ (cosθi+sin θ j ) Rdzdθ

:here + is the θz−re!ion correspon!ing to ?.

%he ;u& of a Vector 0ield through a )phere

+he Jux of F through the spherical surface ?, of ra!ius 6 an! oriente!

away fro" the origin, is given by

F ∙ d A=∫6

F ∙ r

‖r‖dA=¿

∫6

¿



1>

∫7

F ( R , θ ,ϕ ) ∙(sin ( ϕ )cos (θ ) i+sin (ϕ ) sin (θ )θ j+cos (ϕ ) k ) R2sin (θ ) d dθϕ

:here + is the θϕ−re!ion correspon!ing to ?

1E. Jux integrals over para"eterize! surfaces

%he ;u& of a vector eld through a parameteri.ed surface

+he Jux of a s"ooth vector 7el! F through a s"ooth oriente! surface ?

para"eterize! by r=r (" ,t ) , where (s,t) varies in a para"eter region 6, is

given by

r (" , t ) ∙( & r

&" ×

& r

&t )d"dt

¿ F ¿

F ∙ d A=∫ R

¿

∫6

¿

:e choose the para"eterization so that & r /&"×& r /&t is never zero an!

points in the !irection of n everywhere.

%he area of a parameteri.ed surface

+he area of a surface ? which is para"eterize! byr=r(

" ,t ) , where (s,t)

varies in a para"eter region 6, is given

∫6

dA=∫ R‖& r

&" ×

& r

&"‖d" dt

Chapter t/ent': calculus of vector elds

[email protected] the !ivergence of a vector 7el!

Geometric denition of divergence

+he divergence or ;u& densit' of a s"ooth vector 7el! F , written ¿ F

, is a scalar value! function !e7ne! by

¿ F ( x , y , z)= limvo.ume→0

∫6

F ∙ d A

/o.ume of 6



1A

ere ? is a sphere centere! at (x,y,z), oriente! outwar!, that contracts !ownto (x,y,z) in the li"it

Cartesian Coordinate denition of divergence

$f F = F 1 i+ F 2 j+ F 3 k

, then

¿ F =& F 1

&x + & F 2

&y + & F 3

&z

[email protected] +he !ivergence theore"

%he divergence theorem$f : is a soli! region whose boun!ary ? is a piecewise s"ooth surface, an! if

F is a s"ooth vector 7el! on an open region containing : an! ?, then

∫6

F ∙ d A=∫W

¿ F d/

:here ? is given the outwar! orientation

2@. the curl of a vector 7el!

+he circulation densit' of a s"ooth vector 7el! F at (x,y,z) aroun! the

!irection of the unit vector n is !e7ne! to be

∘n F ( x , y , z )= lim Area→ 0

5ircu.ationaround 5 Areain"ide 5

= lim Area→ 0

∫5

F ∙ d r

Area in"ide 5

rovi!e! the li"it exists.

Geometric denition of curl

+he curl of a s"ooth vector 7el! F , written cur. F , is the vector 7el!

with the following properties

• +he !irection of cur. F ( x , y , z ) is the !irection n for which

∘n F ( x , y , z ) is the greatest

• +he "agnitu!e of cur. F ( x , y , z ) is the circulation !ensity of F

aroun! that !irection$f the circulation !ensity is zero aroun! every !irection, then we !e7ne thecurl to be @

Cartesian Coordinate denition of curl



1E

$f F = F 1 i+ F 2 j+ F 3 k

, then

cur. F =( & F 3

&y −

& F 2

&z ) i+( & F 1

&z −

& F 3

&x ) j+( & F 2

&x −

& F 1

&y )k

:hile not a true cross pro!uct, the cur. F can be represente! using a crosspro!uct "atrix

cur. F =| i j k

&

&x

&

&y

&

&z

F 1 F 2 F 3|=( & F 3

&y −

& F 2

&z )i+(& F 1

&z −

& F 3

&x ) j+( & F 2

&x −

& F 1

&y )k

sing greenGs theore" in %artesian coor!inates we can show thatcur. F ∙ n=∘n F

2@. sto&esG theore"

)to@es= theorem$f ? is a s"ooth oriente! surface with piecewise s"ooth, oriente! boun!ary %,

an! if F is a s"ooth vector 7el! on an open region containing ? an! %,

then

∫5

F ∙ d r=∫6

cur. F ∙ d A

+he orientation of % is !eter"ine! fro" the orientation of ? accor!ing to theright han! rule.

2@./ the three fun!a"ental theore"s

0undamental theorem of calculus for ine integrals

∫5

!rad f ∙ d r=f ( ' )−f ( P )

)to@es= %heorem

∫5

F ∙ d r=∫6

cur. F ∙ d A

(ivergence %heorem

∫6

F ∙ d A=∫W

¿ F d/



2@

%he curl test for vector elds in 3>space

?uppose F is a s"ooth vector 7el! in -space such that

• +he !o"ain of F has the property that every close! curve in it can

be contracte! to a point in a s"ooth way, staying at all ti"es withinthe !o"ain

• cur. F =0

+hen F is path-in!epen!ent, an! thus is a gra!ient 7el!

%he divergence test for vector elds in 3>space

?uppose F is a s"ooth vector 7el! in -space such that

• +he !o"ain of F has the property that every close! surface in it is

the boun!ary of a soli! region co"pletely containe! in the !o"ain

• ¿ F =0

+he F is a curl 7el!

Lou have now learne! all that is worthy of being learne! MNoe Oeeson

guide to calculus 3

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