Dr. Wang Xingbo Fall ， 2005

Dr. Wang XingboDr. Wang Xingbo

FallFall ，， 20052005

Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

1.1. Dual spaceDual space2.2. Wedge ProductWedge Product3.3. ““d” operationd” operation4.4. 1-form1-form5.5. 2-form2-form 6.6. Surface Integral Surface Integral


Differential FormsDifferential Forms

Let V be a real vector space over R; a real linear Let V be a real vector space over R; a real linear transformation on V is such a transformation thtransformation on V is such a transformation that returns a real-value when it applies on the eleat returns a real-value when it applies on the element v of V, namely ment v of V, namely


Dual spaceDual space

, ( ) v vV R

We also call such transformation a linear map or We also call such transformation a linear map or linear functional. The dual space of V, denoted linear functional. The dual space of V, denoted by V*, consists of the set of all linear maps from by V*, consists of the set of all linear maps from V to R, and satisfies V to R, and satisfies



( )( ) ( ) ( )( )( ) ( ( ))a a

v v v

v v

, *, a vV V, R

The basis of V is The basis of V is

Then the basis of V* is Then the basis of V* is

A element in V* isA element in V* is



{ }ix

{ }idx

* * 1 * 2 * 31 2 3( , , )i

ia dx a dx a dx a dx *v v*PT ,


Geometric meaningGeometric meaning

(a) (b)

A wedge product is denoted by symbol ∧, A wedge product is denoted by symbol ∧, also called exterior product. A wedge also called exterior product. A wedge product of two quantities product of two quantities and and is the is the directed area swept out from directed area swept out from to to . Thus . Thus the wedge product satisfies the the wedge product satisfies the following rulesfollowing rules


Wedge productWedge product

0 0 0

1

)(

Let be two forms and f be a real Let be two forms and f be a real function. The rules for “d” operation are function. The rules for “d” operation are as follows:as follows:


““d” Operation d” Operation

,

kk

fdf dxx

( )d d d

( )d fd df d

( ) ( )d f df fd

( )d d d 2( ) 0d d d

Complement to vector analysisComplement to vector analysis many complex formulas and many complex formulas and

operations in vector operations in vector analysis can be converted analysis can be converted to a simple form by to a simple form by differential forms differential forms


Differential FormDifferential Form

A differential 1-formA differential 1-form


1-form1-form

( , ) ( , )F x y dx G x y dyA very important example

dyyfdx

xfdf

In general, a 1-form on an open set of RIn general, a 1-form on an open set of Rnn can be the can be the following formfollowing form


1-form1-form( , , ) ( , , ) ( , , )F x y z dx G x y z dy H x y z dz

1 2( , ,..., )n ii x x x dx

A differential form is very similar to a veA differential form is very similar to a vector field ctor field


1-form1-form

1i i idx i ie e

A CA C22 function exists with function exists with


Geometric and Physical Interprets of 1-form Geometric and Physical Interprets of 1-form

( , )f x y

( , ) ( , )df F x y dx G x y dy We will say that it is exact Most differential forms are not exact unless they satisfy

xG

yxf

xyf

yF

22

If F and G satisfy the above condition, we will call the differential form closed.

Exactness is a very important Exactness is a very important concept. The case occurs concept. The case occurs frequently in differential frequently in differential equations. Given an equationequations. Given an equation::


Exactness Exactness

),( yxFdxdy

0Fdx dy If the differential on the left is exact 0df Fdx dy then the curves give solutions to this equation

( , )f x y c

When is it exact ?

Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

Related with Line Integrals Related with Line Integrals

( , ) ( , )F x y dx G x y dy

It is a closed form on all of R2 with C1 coefficients, then it is exact. Exactness is its path independence

21 CCGdyFdxGdyFdx

Y B C1 C2 A O X

If ω is exact and C1 and C2 are two parameterized curves with the same endpoints (or more accurately the same starting point and ending point), then:


Related with Line IntegralsRelated with Line Integrals

1 2c c

The work done by a force along a displacement


Application in Physics

= kkFF i k= kdss i

j 1 1 2 2 3 3= -( ) ( ) = -( )kk jW F ds F ds F ds F ds = -F s i i + +

If the force and the displacement vary with the position on the path C 1 1 1 1 3 3

c cW d F ds F ds F ds F s

Thus, if F is a conservative force and C is a close path 0

cW d F s

Given a vector field


Physical MeaningsPhysical Meanings

1 2F FF i + j

A function called the potential energy , such that ( , )P x y

P FThe force is called conservative if it has a potential energy function

F is conservative precisely when is exact in terms of differential forms

1 2F dx F dy

Given a vector field


Physical MeaningsPhysical Meanings

1 2F FF i + j

1 2

1 1

c c

2 1( ( ) ( ))W d P P c

F r r r


1-form & vectors 1-form & vectors

Table 5.1 Comparative relationships between a 1-form and a force

1-form A vector (force)

1 2( , ) ( , )F F x y dx F x y dy 1 2F F 1 2F e + e

Exact if there is an f such that df F Conservative if there a potential function P such that P F

Path-independent integral if F exact Path-independent work if F is conservative

A typical 2-form is as follows


2-forms and Curl of A Vector Field 2-forms and Curl of A Vector Field

2 ( , , ) ( , , ) ( , , )F x y z dx dy G x y z dy dz H x y z dz dx 3 1 2dx dy e e e

1 2 3dy dz e e e

2 3 1dz dx e e e

1 2 3F G H V e e e

1 2 3( - ) ( - ) ( - )y z z x x yH G F H G FV = e + e + e



1 ( , , ) ( , , ) ( , , )F x y z dx G x y z dy H x y z dz 1 ( ) ( ) ( )x y y z z xd G F dx dy H G dy dz F H dz dx

3 1 2

( ) ( ) ( )

( ) ( ) ( )x y y z z x

x y y z z x

G F dx dy H G dy dz F H dz dx

G F H G F H

e e e

1d V


““d” of a 1-form and the Curld” of a 1-form and the Curl

A C1 1-form is called exact if there is a C2 function f (called a potential) such that then is called closed if

Fdx Gdy Hdz

df 0d

, = , =y z x y z xH = G G F F H

If is a closed form on R3 with C1

coefficients, thenωis exact. Fdx Gdy Hdz



A 1 -fo rm A v e c to r f ie ld

1 ( , , ) ( , , ) ( , , )F x y z d x G x y z d y H x y z d z 1 2 3F G H V e e e

1

( ) ( ) ( )x y y z z x

dG F d x d y H G d y d z F H d z d x

1 2 3( - ) ( - ) ( - )y z z x x yH G F H G F

V =

e + e + e

1 0d 1 is e x a c t 0 V V is c o n se rv a tiv e


2-form and Divergence of a Vector Field 2-form and Divergence of a Vector Field

converting the above 2-form to the vector field F G H V i j k

Then the coefficient of dx dy dz

x y zV F G H

2 Fdy dz Gdz dx Hdx dy 2 ( )x y zd F G H dx dy dz

A 3-form is simply an expression



( , , )f x y z dx dy dz

( ) ( )x y zd Fdy dz Gdz dx Hdx dy F G H dx dy dz

converting the above 2-form to the vector field F G H V i j k

Then the coefficient of dx dy dz

x y zV F G H



2 - fo rm A v e c to r f ie ld

2 F d y d z G d z d x H d x d y 1 2 3F G H V e e e

2 ( )x y zd F G H d x d y d z x y zF G H V

Proposition


““d” of a 2-form and Divergence d” of a 2-form and Divergence

2 0d

( ) 0f ( ) 0 V

Let S be a smooth parameterized surface


Surface Integrals Surface Integrals

Dvuvuhzvugyvufx

),(),(),(),(

dvvxdu

uxdx

dvvydu

uydy

)()( dvvydu

uydv

vxdu

uxdydx

dvduvuyxdydx

uy

vx

vy

ux

),(),()(

The integral of a 2-form on S is given by


Surface IntegralsSurface Integrals

In practice, the integral of a 2-form can be calculated by first converting it to the form

and then evaluating

( , ) ( , ) ( , )[ ]( , ) ( , ) ( , )

S

D

Fdx dy Gdy dz Hdz dx

x y y z z xF G H dudvu v u v u v

( , )f u v du dv

D dudvvuf ),(

Green TheoremGreen Theorem


Forms & Integral Forms & Integral

CS

d

Stokes TheoremStokes Theorem

SV

d SV

dSd nFVF

dsdSCS

FnF

See You!See You!


Class is OverClass is Over

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Dr. Wang Xingbo Fall ， 2005

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