MATH 2209 Calculus Notes Compilation

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MATH 2209: Calculus


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Callum Biggs 20498066 MATH 2209: Calculus pg. 1

Table of ContentsParametric Functions ............................................................................................................................................................ 4

Definition ........................................................................................................................................................ 4

Stating Parametric Functions ...............................................................................................................................................4

Eliminating the Parameter ...................................................................................................................................................4

Drawing Parameter Functions .............................................................................................................................................4

Tangents to Parametric Curves ........................................................................................................................ 4

Petal (Lemniscate) ........................................................................................................................................... 5Vector Functions .................................................................................................................................................................... 5Vector Function Definition...................................................................................................................................................5

Limits .............................................................................................................................................................. 5

General Limit Definition ......................................................................................................................................................5Value of Vector Function Limit ............................................................................................................................................5

Theorem........................................................................................................................................................................................................ 5Corollary: One Sided Limits ......................................................................................................................................................................... 5

Process to Find Vector Valued Limit.....................................................................................................................................6

Continuity ....................................................................................................................................................... 6

Continuity Conditions ..........................................................................................................................................................6Continuous on I ............................................................................................................................................................................................. 6

Properties of Continuous Functions .....................................................................................................................................6Differentiability ............................................................................................................................................... 7

Differentiable at to (point) ............................................................................................................................................................................ 7One-sided Differentiability ........................................................................................................................................................................... 7Checking Differentiability............................................................................................................................................................................ 7

Smooth Parameterisation ....................................................................................................................................................7

Vector Integration ........................................................................................................................................... 7Indefinite Integrals: ...................................................................................................................................................................................... 7

Arc Length ....................................................................................................................................................... 8

General Arc Length ..............................................................................................................................................................8Polar Arc Length ........................................................................................................................................................................................... 8

General Formula ..................................................................................................................................................................8

Parameterisation w.r.t Arc Length .......................................................................................................................................9Parameterisation Process ............................................................................................................................................................................. 9

Curvature ...................................................................................................................................................... 10

Tangential .........................................................................................................................................................................10

Normal ..............................................................................................................................................................................10

Curvature ..........................................................................................................................................................................10

Multivariate Functions ...................................................................................................................................................... 10Functions of Several Variables ....................................................................................................................... 10

Definition ..................................................................................................................................................................................................... 10Level Set ...................................................................................................................................................................................................... 10

Multivariate limits ......................................................................................................................................... 11

Finding Limits at Origin ......................................................................................................................................................11

Sets .............................................................................................................................................................. 11Open and Closed Balls .......................................................................................................................................................11

Open Ball ..................................................................................................................................................................................................... 11Closed Ball ................................................................................................................................................................................................... 11Sphere ......................................................................................................................................................................................................... 11

Sets ...................................................................................................................................................................................11Subsets:....................................................................................................................................................................................................... 12

Existence of Maximums .....................................................................................................................................................12Bounded Sets .............................................................................................................................................................................................. 12

Partial Derivatives ......................................................................................................................................... 12

Clairault's Theorem ...........................................................................................................................................................13

Tangent Planes .............................................................................................................................................. 13

Tangent Plane Process................................................................................................................................................................................ 13Approximation Formula .....................................................................................................................................................13Approximation Formula Process ................................................................................................................................................................ 14

Continuity ..................................................................................................................................................... 14Single Variable Continuity: ......................................................................................................................................................................... 14


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Differentiability ............................................................................................................................................. 14

Differentiability on Multivariate ........................................................................................................................................14Checking Differentiability ........................................................................................................................................................................... 15

Chain Rule ..................................................................................................................................................... 15

Chain Rule Form 1 .............................................................................................................................................................15

Chain Rule Form 2 .............................................................................................................................................................15Chain Rule on F(x,y,z) .................................................................................................................................................................................. 15

Gradients ...................................................................................................................................................... 16

Directional Derivative ........................................................................................................................................................16

Tangent Planes with parameterised functions ...................................................................................................................16

Angles ...............................................................................................................................................................................16

Mean Value Theorem .................................................................................................................................... 16Definitions ................................................................................................................................................................................................... 16

Mean Value Theorem ........................................................................................................................................................17Corollary 1: Constant Function ..........................................................................................................................................17

EVT ....................................................................................................................................................................................17

Second Derivative Test .................................................................................................................................. 17

Extremum..........................................................................................................................................................................17

Second Derivative Test ......................................................................................................................................................17

Lagrange Multiples ........................................................................................................................................ 18

Orthogonal Surfaces ..........................................................................................................................................................18Lagrange Theorem.............................................................................................................................................................18

Method of Lagrange Multiples...........................................................................................................................................18

Taylors Theorem ........................................................................................................................................... 19

0th

Degree Approximation .................................................................................................................................................191

stDegree Approximation ..................................................................................................................................................19

2nd

Degree Approximation .................................................................................................................................................19Finding the Second Polynomial .................................................................................................................................................................. 19

Vector Functions ........................................................................................................................................... 20

Jacobian Matrices ......................................................................................................................................... 20The Jacobian ............................................................................................................................................................................................... 20

Jacobian Chain Rule ...........................................................................................................................................................20

Inverse Function Theorem .................................................................................................................................................21Invertible Transform ................................................................................................................................................................................... 21

Multiple Integrals ................................................................................................................................................................ 21Properties of Multiple Integrals ..................................................................................................................... 21

Double Integral Properties.................................................................................................................................................21

Cartesian ...................................................................................................................................................... 21

Cartesian Double Integral ..................................................................................................................................................21Type I Planar Region .................................................................................................................................................................................. 22Type II Planar Region................................................................................................................................................................................. 22

Cartesian Triple Integral ....................................................................................................................................................22Triple Cartesian Integral Method ............................................................................................................................................................... 22Type I Solid Region..................................................................................................................................................................................... 23Type II Solid Region.................................................................................................................................................................................... 23

Type III Solid Region................................................................................................................................................................................... 23Polar ............................................................................................................................................................. 24

General Theory ..................................................................................................................................................................24Type I Polar Region .................................................................................................................................................................................... 24Type II Polar Region ................................................................................................................................................................................... 24

Spherical Polar .............................................................................................................................................. 24Cartesian Spherical Polar Conversion ..................................................................................................................................................... 24

General Rule ......................................................................................................................................................................25

Cylindrical ..................................................................................................................................................... 25

General Theory ..................................................................................................................................................................25

Applications .................................................................................................................................................. 25

Surface Area ......................................................................................................................................................................25

Volume ..............................................................................................................................................................................26Determining Volume ................................................................................................................................................................................... 26

2D Mass and Moments ......................................................................................................................................................26Mass............................................................................................................................................................................................................ 26Moment of Lamina .................................................................................................................................................................................... 26


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Moment of Inertia...................................................................................................................................................................................... 26

3D Mass and Moments ......................................................................................................................................................27Mass............................................................................................................................................................................................................ 27Moments..................................................................................................................................................................................................... 27Moment of Inertia...................................................................................................................................................................................... 27

Charge Density ..................................................................................................................................................................282D Charge Density ...................................................................................................................................................................................... 283D Charge Density ...................................................................................................................................................................................... 28

Transformations ............................................................................................................................................ 28

Theorem for R

2

..................................................................................................................................................................28Transforms to find Integrals ....................................................................................................................................................................... 28

Theorem for R3

..................................................................................................................................................................29Special Cases ............................................................................................................................................................................................... 29


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Parametric FunctionsDefinitionIf variables x1, x2,...,xn are defined by the same variable t,{x1=f1(t), x2=f2(t),...,xn=fn(t)}, then as tvaries the point (x1,x2,...,xn) traces out a curve C in The vector form of C is () =11 +22+ . . . + Consider t to be time, then r(t) is the position vector at time t, and as time elapses, the point movesalong the curve CThis means that the point can double back

2: =, = ; = + 3: =, = , = ; = + + Stating Parametric FunctionsFor functions not in vector notation

1=

1

2 = 1 =()

Eliminating the Parameter

A direction relationship between x1, x2,...,xn can be obtained via simultaneous equation manipulationof the defining functions of x1, x2,...,xn ,{x1=f1(t), x2=f2(t),...,xn=fn(t)}.

However eliminating the parameter is not always possible

The ability to eliminate the parameter depends upon the nature of the defining functions

Drawing Parameter FunctionsEliminate the parameter and plot the resultant function

Create a table of the defining functions and their values at specific points; this can be used to trace outthe curve.Arrows are drawn along the curve C to indicate increasing t

Tangents to Parametric CurvesRemember by the chain rule

=

, if 0 = 0 0 Horizontal Tangent 0 = 0 Vertical Tangent = 0 = 0 Need more informationAlways must find both

and

before determining the above tangents


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Petal (Lemniscate) = Acos or sin , = 2,3,4 , Note that 1x axis intercept = A

Since cos < 0 for 4

< < 34

and54

< < 74

, these angles ranges are not plotted, as we only consider > 0in this unit.

Do not plot:

0, s. t . a < a < , , +

For n=1 > 0, > 0,s . t. , + , +

Continuous on IA vector function is continuous on an interval I if it is continuous at every point along I, (end points must be

right and left hand continuous)

Properties of Continuous Functions

If f and g are continuous on = then the following combinations are continuous1. Sum and Difference: 2. Dot Product: 3. Constant Multiples: 4. Quotient:/, given that 05. Powers: , given that it is defined on an open ball containing , where r and s are integers

iff lim

o

=

o

lim + = lim =

Continuous at to (Continuous at a point)

Continuous at a (Right-Hand Continuous)

Remember: Right hand is positive, its right

Continuous at b (Left-Hand Continuous)

Remember: Left hand is negative, its bad

1. Define(, ), and the limit, (0 , 0)2. Set = 0 and determine value function approaches as 0 3. Set y= 0 and determine value function approaches as 0 4. If the limit is at the origin, (0,0), then let = , and set(, ) (Not 0) and determine this5. Can create lines that travel through this point, and test with these lines


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DifferentiabilityDefine: A vector function : I , defined on some interval I(a, b), with an interior point ptoICompare in components

Differentiable at to (point)

lim o o =

=

One-sided Differentiability

= lim 0 + Checking Differentiability(See Method for Multivariate differentiability)

Smooth Parameterisation

A curve C is smooth if it has a parameterisation : I with;1. 2. So to check if a curve is a smooth parameterisation,

1. Find if continuous2. Check if = 0 I

Vector IntegrationConsider component functions

= + + = + + Indefinite Integrals:Indefinite integrals of vector functions will include + C, where C is a vector


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Arc LengthGeneral Arc Length 2 = 2 + 2

2 = 2 + 2

=

2

+

2

= 2 + 2 = 0 Polar Arc LengthFind , and treat as parameter = = cos

=

cos

sin

= sin + cos 2 + 2 = cos sin 2 + sin + cos 2 = 2 + 2 = 2 + 2

General Formula

= 2 + 2 Polar3

= 2 + 2 + 2 2 = 2 + 2 = 1 + 2

Use first formula primarily = General

ds

dx

dy


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Parameterisation w.r.t Arc LengthA vector function can be parameterised in terms of the arc length of the curve it plots out.

Let: , 3 be continuously differentiable, with C being the curve parameterised by(t). C istraversed only as t runs from a to b. Note integral ranges from a to t

S = =

2

+ 2

+ 2

S() = 0, S() = =Arc length of C

Suppose > 0, S = > 0. This means that S is strictly increasing, and hence musthave an inverse function S-1

S: [, ] [0, ]1 = : 0, , For S

0, L

,

(

) is the unique

s.t

S = 0 So the parameterisation of C w.r.t arc length isC: S = S , S0,

Where: S = 1

=1

S = S S= S S S = 1 S[0, ]This means that speed is one unit

This applies to Parameterisation Process

4. Determine S = 0

1. State C:

=

1

,

2

,

,

,

0

2. Define

3. Define 5. Determine inverse function of S(t) above, this only requires switching the variables and solving,

this defines t in terms of s6. Then sub in the defined inverse function, for t, in the r(t) function

i.e. replacing t with a function in terms of S


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CurvatureT Tangential VectorN Normal VectorTangential

C is a smooth curve in

2 or

3 parameterised by

(

),

(

) continously, and

> 0

I

T = T = |Normal

C is a smooth curve in 2 or 3 parameterised by (), () continously, and > 0 IN = TT = 1 dT

N = 2

2

= Curvature

= T

(t) =

3

If 2D vectors assign component = 0, note that cross product will only result in component, thisproduct will be =

Multivariate FunctionsFunctions of Several VariablesDefinition

Before: , , , Now: , , , Let D be a subset of , then: D is a rate that assigns for each = 1, 2, , D, a realnumber =1, 2 , ,

Range of: {: D}, written as: D Graph of: {,: D} +1

Level Set

Level sets are the sets at which a function takes on a specific value.Given , the level set of corresponding to is{: = }


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Multivariate limitsLet r(t) be a -values vector function, defined on some open interval containing oexcept possiblyat to itself. The limit as t approaches to is a vector,

lim exists iff, satisfying > 0, > 0, s.t . 0 < < < i.e.

> 0,

> 0, s. t.

;

and

(

,

+

)

Finding Limits at OriginFind the one sided limits of different variables, moving along the axis. This allows for the othervariables to be set to zero.

As lim + = = lim , 0 = (0, )

SetsOpen and Closed BallsIf and > 0, then aOpen Ball:Closed ball at centred at and radius > 0 is described by the set

,

=

:

0 is described by the set(Difference is ), = : For n=1:; = , + For n=2:

;

=

,

:

2 +

2

2

Closed Disc

For n=3:; = , , : 2

+ 2

+ 2

2

Sphere:Sphere centred at and radius > 0 is described by the set, = : = For n=1:; = , + For n=2:; = , : 2 + 2 = 2}For n=3:; = , , : 2 + 2 + 2 = 2SetsLet D

, a point

is then a

i. Interior point of D if it is centred at some open ball contained in D i.e. < 0 . . (, ) will be a ball with a radius s.t. it is containedii. Boundary point of D if every open ball centred at contains a point of D and a point not of DNote: An element of D is either an interior or boundary point, it CANNOT be both

1. Define(, ), and the limit, (0 , 0)2. State that function must have same value from different approaches3. Set

=

0and determine value function approaches as

0

4. Set = 0 and determine value function approaches as 05. Let = , and set(, ) (Not 0) and determine this


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Notation:

Do= The set of all interior points of DD= The set of all boundary points of DSubsets:

Open: A subset D is called open if every point of D is an interior point of D, Do=DClosed: A subset K

is called closed if it contains its boundary points, i.e.

K

K

, = , = , , = , = A subset is Neither open nor closed if it contains boundary points only for portions.A set cannot beboth open and closed, with the exception of,

1. Every open ball , in is an open set2. For , with < the subsets, , , , , , (, ) are all open sets

,

,

,

,

,

, (

,

) are all closed sets

3. Open Annulus,In 2, 0 < < , , ; 2 < 2 + 2, 2

Existence of MaximumsLet D be a closed and bounded subset of .Let: D be continuous on D.Then : is a bounded subset of and an absolute maximum and minimum of exists on D, D . .

,

D

Bounded SetsA subset of is called bounded if it is contained in some closed ball1, 2 , , Partial DerivativesLet be a defined on some open ball centre .The partial derivative

() of f w.r.t. of is the derivative of taking all other variables to beconstant.

This derivative doesnt exist if the defining limit doesnt this limit is the same as for normal derivative Higher Order Derivatives

Note: The order that the derivatives occur in affects the derivative scripts2 = = 22 = =


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Clairault's TheoremLet: D where D is an open subsetof

If2 is continous on D then2

=

2

on D

This could possibly be used to imply continuity

Tangent PlanesGiven a surface =(, ) (graph of), with , continuous, the normal to the surface at the point, ,, is = , , , , 1 Therefore the eqn of the plane has the formula, , , , = 0

,

+

,

=

Tangent Plane Process

Approximation FormulaThe tangent plane approximates the surface near , , =, , surface = , + , tangent plane , , + , + , near ,

Letting

=

,

=

+ , + , + , + , ExampleFind the approximation to 91.852 + 8.12let, = 92 + 22,8 = 10 = 18

292 + 2 = 1810 = 95 = 2292 + 2 = 810 = 45

So

1.85,8.1

10 +

9

5

0.15

+

4

5

0.1

= 9.81

= , , , , 1 = , ,

1. Determine the partial derivatives, in terms of, , , then determine at the given point2. Find normal vector, via two methods

a. If given =(, ) then use the formulab. If given as(, , ), then use the formula

3. State that

4. Solve , , , , = 0


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Approximation Formula Process

ContinuityPerform normal checks for continuity, apply the formula

Example:, , = 2 + 2 + 1 is continuous on , , : 1Single Variable Continuity:A function defined on some open interval containing is differentiable

lim existThis can be rewritten that f will be differentiable at

if

s.t.

lim = 0Then = DifferentiabilityIf is a real valued function defined on some open ball centre in , we say is differentiable at if a linear transformation of: s.t.

lim

= 0

= Differentiability on MultivariateLet: D where D is an open subset of

1. If is differentiable at then the partial derivatives exists and = 1 , 2 ,

3. If all partial derivatives are continuous on D then is differentiable at all points of DNotes:

is continuously differentiable on D

diff on D

continuous on cont at

= , + , , , + , + ,

+

,

+

,

+

,

+

,

+ , + , + , + ,

1. State =(, ), rearranging might be needed2. For a known point (, , ) the tangent plane is3. State that therefore for points near (, ) (Variables other than that being approximated)4. Determine(, )5. Determine() and()6. Determine and , consider that = , = , where and are the values

being approximated, and and are the known points7. Dont forget to use 8. Determine


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Checking Differentiability

Chain RuleChain Rule Form 1

Let: D be differentiable on open subset D Let: I be differentiable on open interval ISuppose = 1, 2, , and D IThen the function = is differentiable on = =

=

1

1

+

2

2

+ +

I =

= (1,2 , ) 12

Chain Rule Form 2When multiple independent variables, that is, variables which the intermediable variables vary with

Intermediate variables :

1,

2,

,

Independent variables:

1,

2,

,

This means that =(1, 2, , )Note from the equation , this gives the way to choose what variable to differentiation to = 1 1 + 2 2 + + Example: =, = 2 + 3 + 1, = 3 + 2, =

=

+

Chain Rule on F(x,y,z)

=

To find

Full Method1. Treat as ()2. Treat y as a constant3. Implicitly differentiate4. i.e.

Alternate Quick Method:

1. Find all partial derivatives2. Check all partial derivatives for continuity


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GradientsLet be a real-valued function defined on some open ball B centre If all partial derivatives exists at

= 1 , 2 , , Notes: is always given in component formsDirectional DerivativeThe directional derivative of in the direction of vector is =

|| Must remember to make into a unit vectorTangent Planes with parameterised functions

Let: D be differentiable on D,Considering the level set = D: = , 0, Let be any curve on with parameterisation , () where is differentiable on open interval and = , , So therefore = Differentiating with chain rule = 0 = 0

= , , AnglesFastest Increase atDirection=() Rate =Fastest Decrease atDirection=() Rate= Mean Value Theorem

DefinitionsClosed Line Segment: For , the closed line segment joining and , = + : 0 1Open Line Segment:For , the open line segment joining and (difference is < instead of)

(, ) = + : 0 < < 1Connected:Open subset D of is connected if every pair of points in D can be joined by polygonal path lying entirely in Di.e. a continuous curve consisting of a finite no. of closed line segmentsMaxima and Minima:: defined in a subset has a level max/min at for x in some open ball in D, centre a


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Mean Value TheoremSame as old MVT, except uses gradients

1. Let: D be differentiable on the open subset D 2. Let, D with and , D3. Then there exists , s.t

=

Corollary 1: Constant FunctionLet: D be differentiable in the open connected set Let = 0Then is constant on D, =Proof

Determine that is differentiable and continuous hence connected on Determine that = 0 for all Hence

= 0

=

(

)

EVTIf: is cont on the closed bounded subset D of then attains an absolute max at some point inD and an absolute min at some point in D

1. Check interior points for local max and minima2. Check end points

Second Derivative TestExtremumAn extreme point is one in which all partial derivatives = 0, or at least one partial derivative is

undefined

Proof

Local extremum at and all partial derivatives exist, then they all equal zeroLet =, 2, 3, , , where = (1, 2, , 3) has a local extreme at then g has a local extreme at1Since 1 = 1 () if they exist, by fermats theorem we get 1 = 0Then

1 = 0Second Derivative TestLet: 2 , = , = 0 = 2

1. , > 0, > 0 local min2. , > 0, < 0 local max3. , < 0 saddle point


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Lagrange MultiplesOrthogonal SurfacesLet: D be continuously differentiable on the open subset 3 which contain sthe smooth curve withsmooth parameterisation = , , If PO is a point in C wherea local max/min relative to its values has on C, then is orthogonal to C at PoProofThe composite function =, , is differentiable w/ derivative 0 (Fermats theorem)Hence = () so at ( ) Since ( ) is tangential to C, to CLagrange TheoremLet, be continuously differentiable on open set 3 containing the level surface : , , = 0

If has a local maximum at on relative its values on and if 0 then = ( )Proof

has a relative local extremum at relative to its value son any smooth curve on passing thru So is at Since is a non-zero vector at there must exist a scalar so that = ( )Method of Lagrange MultiplesPrinciple remains the same regardless of number of variables , ,

= = 0

1. Define and = 0 = 2. Define and 3. Prove that

= 0 is bounded

a. Show that each , , is bounded between two values4. Show that is continuous on = 0, hence by EVT has a global maximum and minimum on = 05. Prove 0, using = 06. State: Using Lagrange Multiples:

2 Variables:

1. Use = , via substitution of to define =() or =()2. Substitute this function into = 0 and solve for or 3. Use function

=

(

) or

=

(

) to define the remaining variable

3 Variables:

1. Define each , , in terms of()2. Substitute these definitions into = 0 and solve for 3. Then solve for , ,


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Taylors TheoremLet B be an open ball in containing Left: B and letB with Then the tailor polynomial of of degree n is denoted by , 0

thDegree Approximation

has continuous and bounded first order partial derivatives on B

0, = + , Where , = . for some , , const. 1

stDegree Approximation has continuous and bounded first and second order partial derivatives on B1, = + . + 1,

Where

1, = 12! =1=1 , 1, < const. 22

ndDegree Approximation has continuous and bounded first, second and third order partial derivatives on BNOTE: R is negative

1,

=

+

.

+

1

,

+

1

2!

(

)

=1

=1

2

,

Where2, < const. 3Finding the Second Polynomial

,

=

,

+

,

+

1

2 ,

2 + 2

,

+

,

2

1. Determine, 2. (, )3. (, ) (, ) (, ) (, )4. (, ) (, ) (, ) (, )


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Vector Functions : curves: e. g. Temperature: e. g.m n matrixLet Vector function: is a rule which assigns to each D a unique vector = 1,2, , Then functions

: D

= 1,2,

are called the coordinate functions of

Polar coordinates map, = cos , sin in = , , 002 an mxn matrix gives a map = C = C 12 =

C111 + C121 + + C1nC212 + C222 + + C2n

Cm1 1 + Cm22 + + Cmn lim = lim1 , lim2 , , lim

Jacobian MatricesGiven: D with D open, and D, the mxn matrix of at is called the Jacobian matrix

=

11 12 1 21 22 1

1

2

= ()The JacobianThe determinant of a Jacobian matrix of at is called the Jacobian of at = Jacobian Matrix

= 1,2, , (1, 2, , )Jacobian Chain Rule

Let U be an open subset of

and let V be an open subset of

Let

: U

and

: U

with

V

If is diffable on U and is diffable on V then is diffable on U (and is diffable on v) = =


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Inverse Function Theorem

The inverse function theorem uses Jacobian matrices to determine the derivative of x and y, in terms

of u and v. And avoids having to solve complex simultaneous equations, to obtain the inverse

functions

=

1 = 1

For (, ), where = (, ), = (, ) = 1 , = 1 = 1Invertible TransformA transform is invertible over the domain of, s.t. the det > 0When this occurs there is said to be a open disc on which T is invertible and continuous

Multiple IntegralsOrder of integration is such that variables created in earlier integrals are eliminated by subsequent onesThe point to integrate from must correlate to what you are integrating with respect to

Properties of Multiple Integrals

Double Integral Propertiesa) If, are integrable over 2 and k, l then

( k + lg ) = k + L b) If

,

,

,

,

then

c) If , , , Area Area

where: = Aread) If = 12 with 1 2 having no int. point, then if all integrals exist

= 1 + 2 CartesianCartesian Double IntegralLet: , where 2, be a bounded functionSuppose is bounded, i.e. for some rectangle RDefine : , = , , if,

0, if ,

is integrable over

if

is integrable over

, = ,


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Type I Planar Region 2 is a region of the form: = , : and 1 2 1, 2 are continuous on [, ] 1 2, [, ]

Then, if: is continuous and integrable over D , = ,

2(

)

1() This applies when the area is two functions, bounded between 2 values, or poles.Type II Planar Region 2 is a region of the form:

= , : and 1 2 1, 2 are continuous on [, ]

2

1

,

[

,

]

Then, if: is continuous and integrable over D , = , 2()

1() This applies to an area that is two vertical functions inverse functions in a sense, that is bounded by two y valuesGenerally: You can often think of an area as a type I or II, but you must decide

Cartesian Triple IntegralIf

:

is continuous on the rectangular box

=

,

,

,

then

is integrable over

(, , ) = , , 3 is a bounded subset, i.e. for some rectangular box :Define : by , = , , , if, ,

0, if , , is integrable over if is integrable over Triple Cartesian Integral Method

All regions are pretty much the same

2. Will always be integrating for a variable between two functions, and then after, willintegrate for D

3. To integrate between D, consider what you are integrating with respect to, and yourboundary regions

1. Boundary regions are always given as numerical values (unless complicated?)4. The parameters of each integral and the parameters of the variable that is being integrated

with respect to5. This means that the two integrations for D, are between the boundary regions indicated

i.e. if the region is bounded for x=3, x=1, then when integrating for dx, integrate between 1 and 3

7 8 9 10 11 12

200

400

600

800

1000


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Type I Solid RegionMoving area in x,y along z, hence start with defining functions and dz = , , and 1, 2,

1, 2 are continuous on 1, 2, , (, ) Bounded region in , plane, moved about

Then, if

:

is continuous and integrable over D

= , 2(,)1(,)

Where subsequent integration occurs for boundaries

Type II Solid RegionMoving area in y,z along x = , , and 1, 2,

1,

2 are continuous on

1

,

2

,

,

(

,

)

Bounded region in , plane, moved aboutThen, if: is continuous and integrable over D = ,

2( ,)1( ,)

Where subsequent integration occurs for boundaries

Type III Solid RegionMoving area in x,z along y

= , , and 1, 2, 1, 2 are continuous on 1, 2, , (, ) Bounded region in , plane, moved about

Then, if: is continuous and integrable over D = ,

2(,)1( ,)

Where subsequent integration occurs for boundariesFor product of functions

() =


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PolarSome integrals are simplified if they are performed with polar coordinates, rather than CartesianPolar forms involve rectangles in a range of angle and radius r. These polar rectangles could belikened to rainbows.

Polar only occurs with double integrals = , : , RectangleGeneral Theory

If is continuous on the polar rectangle D, given by and , then, = ( cos , sin )

,

[ ,] Once again, the order of integration must be such that, any variable introduced by the first integral, will be

eliminated by the second.

In this particular case, r is part of the first integral

Type I Polar Region

2 is a region of the form:

= , : , 1() 2() 1, 2 are continuous on [, ] 2 1, [, ]


2()1()

The area is for steady radius, and defined angles (looks like a rainbow)

Type II Polar Region 2 is a region of the form: = , : and 1 2 1, 2 are continuous on [, ] 1 2, [, ]


2()

1(

)

This applies to an area that is two vertical functions inverse functions in a sense, that is bounded by two y valuesSpherical PolarSpherical coordinates are a 3 dimensional coordinates in which points are denoted by (, , )p = Distance p from the origin = Angle between and axis of Q, the projection of point P onto the , plane = Angle between OP and axis, it should be noted that 0 Cartesian Spherical Polar Conversion

=

sin

cos

= sin sin = cos This conversion is needed to make integration simpler


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General RuleA solid region , inner spherical polar coordinates = , , : , , 1, 2,

+ 2 0

1,

2 continuous and 0

1

2

, , = [ sin cos , sin sin , cos . 2 sin dp2 ,

1 , d

d In which = 2 sin Dont forget CylindricalThe cylindrical co-ords of Q at a point p in

3 are (

,

,

) where

,

the polar co-ords is polar

coordinates of the projection onto the

,

plane, and

is the cartesian co-ord of P

Note that = cos , = sin General TheorySuppose is a Type I Solid region, = , , and 1, 2, Suppose is a Type II Polar region, = , : 1 2Then the integral simply becomes the first step of a type I solid, and the second and third step, are fora type II polar. This is really just finding the D

Then if is bounded = cos , sin ,

2(

,

)

1(,)2

1() It does help the integration if any , can be converted into values, i.e.2 + 2 = 2Dont forget

ApplicationsSurface AreaFinding the surface area of a 2D object does not mean it has to lie flat on the

,

plane, all that it requires is that

the surface can be modelled by a function(, )Let be a bounded region in 2Let: be continuously diffable with, 0, , The area of the surface is =, , ,

Area = 2, +2, + 1 State that, 0, ,


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VolumeThrough the use of triple integrals

Volume = 1 Determining Volume

2D Mass and MomentsHave a laminar occupying a region D in the , planeSuppose laminar at point, is given by (, ), (: )

Then p=mass/area

Mass = , Moment of LaminaAbout the X-Axis

Distance from x axis is y, therefore

= , About the Y-Axis

Distance from x axis is y, therefore = , Centre of Mass

The centre of mass occurs at (, ), where;, = , Moment of InertiaAbout the X-Axis2 is the difference to moment = 2, About the Y-Axis2 is the difference to moment

=

2

,

Moment of Inertia about the Origino = +

1. Determine

, the set of conditions for the volume

2. To create this, it is ideal to consider the restrictions placed on , , a. Consider the restrictions on each variable , , b. The restrictions for each variable can in the form of function of the remaining two variables,e.g. for1, 2(, )

3. Once has been created, must integrate in terms ofa. When integrating with respect to each variable, the parameters of this integration are the

restrictions on the variable determine above4. If only 2 conditions were given, it is much easier to use cylindrical method, so all that is required

is converting , to ,


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3D Mass and MomentsFor a bounded solid by 3, with a density function (, , )Mass

m = , , Moments

Moments are about planes, not axis. Integral process is the same as for mass, but the initial integrationinvolves the variable not in plane of the moment.

Moment of Body about , plane = , , Moment of Body about , plane = , , Moment of Body about , plane = , , Centre of Mass

The centre of mass occurs at (, ), where;, , = , , , , , Moment of Inertia

About the X-Axis

2 is the difference to moment = (2 + 2), , About the Y-Axis2 is the difference to moment = (2 + 2), , About the Z-Axis

2 is the difference to moment

= (2 + 2), , Moment of Inertia about the Origino = + +


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Charge Density

2D Charge Density: is distributed over a bounded region 2, the total charge is given by = , 3D Charge Density

= TransformationsIf S in the , domain exists such that = , = , , Then transformation T from S in the -, plane to = () in the - plane

,

=

,

=

,

,

,

Theorem for R2Let: be continuously diffable transforms in 2 one-to-one onto region in 2Let the Jacobian of is non-zero on SIf, = , , , and: is cont on , then for suitable regions and , = , , ,

, ,

,

=

,

,

, = , , , . (, ) Notes:

1. Must determine , as functions of, 2. Integration is du then dv3. Notice area of integration changes from R to S4. Need additional conditions on and

The above theorem holds if and are Type I or II planar regions5. Theorem basically substitute

,

in terms of

,

and change

to

(

,

)

6.

S is called the pulltrack/pre-image of by = 1

Transforms to find Integrals (, ) = , = (, )

1. Write down transforms = (, ) = (, )2. Write down the restrictions on R in terms of and , i.e. the inequalities3. Substitute (, ), (, ), and solve. This determines 4. Drawing out the region found can help5. Determine the Jacobian of the transformation, and its absolute value6. Replace, with (, ) and (, )7. Integrate over


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Theorem for R3 , , = , ,

Special CasesIf, = 1, , Area

=

|

(

,

|

If = , is constant thenArea = Area()This is what will occur if is a linear transformation

MATH 2209 Calculus Notes Compilation

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