Dr. Wang Xingbo Fall ， 2005

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Dr. Wang XingboDr. Wang Xingbo

FallFall ，， 20052005

Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

1.1. Vectors Vectors

2.2. Products of Two VectorsProducts of Two Vectors

3.3. Vector Calculus Vector Calculus

4.4. Fields Fields

5.5. Applications of Gradient, Divergence and Curl Applications of Gradient, Divergence and Curl

Vector Algebra Vector Algebra

Quantities that have both magnitude anQuantities that have both magnitude and direction; the magnitude can stretch od direction; the magnitude can stretch or shrink, and the direction can reverse.r shrink, and the direction can reverse.

In a 3-dimmensional space,In a 3-dimmensional space, a vector a vector X=(xX=(x11, x, x22, x, x33) has three components x) has three components x11,x,x22, x, x33..

Vectors Vectors

Vectors X=(xVectors X=(x11, x, x22, x, x33), Y=(y), Y=(y11, y, y22, y, y33) )

Scalar multiplication:2Scalar multiplication:2X X = (2x= (2x11, 2x, 2x22, 2x, 2x33))

Addition:Addition:X X + + Y Y = (x= (x11+ y+ y11, x, x22+ y+ y22, x, x33+ y+ y33))

The zero vector:The zero vector:0 0 = (0,0,0)= (0,0,0)

The subtraction:The subtraction:X X - - Y Y = (x= (x11- y- y11,x,x22- y- y22,x,x33- y- y33))

Algebraic propertiesAlgebraic properties

Length of Length of XX = (x = (x11, x, x22, x, x33) is calculated by:) is calculated by:

A A unit vector in the direction of unit vector in the direction of XX is is::

Length (magnitude) of a vectorLength (magnitude) of a vector

2 2 21 2 3| | x x x X

2 2 21 2 3

( , , )

ProjProjuuA = A = ((|A| |A| coscos))u u ( |( |uu| = 1)| = 1)

Projection of a VectorProjection of a Vector

1.1. Inner Product ,doc product,scalar proInner Product ,doc product,scalar productduct

2.2. Vector Product,cross product Vector Product,cross product

3.3. Without extension Without extension

Products of Two Vectors Products of Two Vectors

AA=(a=(a11, a, a22, a, a33), ), BB=(b=(b11, b, b22, b, b33))

Inner Product Inner Product

cosA B =| A || B |

a b a b a b 1 1 2 2 3 3A B = +

1.Non-negative law

2.Commutative law:

3. Distributive law:

Properties of Scalar ProductProperties of Scalar Product

A B = A B

( ) A B + C = A B + A C

1. Cross product of two vectors A and B is another vector C that is orthogonal to both A and B

2. C = A×B

3. |C| = |A||B||sin|

Vector Product Vector Product

1. The length of C is the area of the parallelogram spanned by A and B

2. The direction of C is perpendicular to the plane formed by A and B; and the three vectors A, B, and C follow the right-hand rule.

Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

Geometric Meanings of Cross Product Geometric Meanings of Cross Product

B Area

1.A×B = -B ×A,

2.A ×(B + C) = A ×B +A ×C,

3. A||B is the same as A ×B = 0

Properties of Cross Product Properties of Cross Product

1. i1×i1 = 0, i2 ×i2 = 0, i3×i3 = 0,

2. i1×i2 = i3, i2×i3 = i1, i3×i1 = i2

Three Basis Vectors Three Basis Vectors 1 2 3, ,i i i

A = a1i1 + a2i2 + a3i3, B = b1i1 + b2i2 + b3i3

1. (A×B)×C = B(A·C) -A(B·C)

2. A×(B×C) = B(A·C) - C(A·B)

Product of Three Vectors

A = a1i1 + a2i2 + a3i3,

B = b1i1 + b2i2 + b3i3,

C = c1i1 + c2i2 + c3i3 321

1. A·(B×C) = (A×B) ·C = (C×A) ·B

2. (A× B) · (C×D) = (A·C)(B·D) - (A·D)(B·C)

3. (A× B) · (A×C)= B·C - (A·C) (A·B)

4. (A×B) ·(C×D) + (B×C) ·(A×D) + (C×A) ·(B×D) = 0 .

5. A×(B×C) + B×(C×A) + C×(A×B) = 0

6. (A×B) ×(C×D) = C(A·(B×D)) - D(A·(B ×C))= B(A·(C×D))-A(B·(C×D))

Other Useful Formula for Vector Products Other Useful Formula for Vector Products

For any scalar t, a function f(t) is called avector function or a variable vector if thereexists a vector corresponding with f(t).

A(t) = (cos t, sin t, 0) (-∞ < t < ∞)

Vector Calculus Vector Calculus

The Derivatives of a Vector Function The Derivatives of a Vector Function

A(t) = (A1(t),A2(t),A3(t)) = A1(t)i1 + A2(t)i2 + A3(t)i3

( ) ( ) ( )'( ) lim

d t t t tt

31 21 2 3

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

( )( ) ( )

dA tdA t dA td t

dt dt dt dtdA tdA t dA t

dt dt dt

Properties of Vector DerivativeProperties of Vector Derivative

velocity

( )'( ) ( )

d tt t

acceleration 2 ( )"( ) ( )

d tt t

Properties of Vector DerivativeProperties of Vector Derivative

( )d d d

dt dt dt

A BA B

( )d d d

dt dt dt

A BA B B A

( )d d d

dt dt dt

A BA B B A

A(t) = (A1(t),A2(t),A3(t)) = A1(t)i1 + A2(t)i2 + A3(t)i3

The Integral of a Vector Function The Integral of a Vector Function

2 3( ) ( ) ( ( ) , ( ) , ) )t t dt A t dt A t dt A t dt B A (1

Suppose Ω be a subspace, P be any point in Ω,if there exists a function u related with a quantity of specific property U at each point P, namely, Ω is said to be a field of U if

Fields Fields

, ( ),u u u U P P

where symbol means “subordinate to”

1. Temperature in a volume of material is a temperature field since there is a temperature value at each point of the volume.

2. Water Velocity in a tube forms a velocity field because there is a velocity at each point of water in the tube.

3. Gravity around the earth forms a field of gravity

4. There is a magnetic field around the earth because there is a vector of magnetism at each point inside and outside the earth.

Example of fields Example of fields

A real function of vector r in a domain is called a scalar field.

Pressure function p(r) and the temperature function T(r) in a domain D are examples of scalar fields.

Scalar Fields Scalar Fields

A scalar field can be intuitionistically described by level surfaces

1 2 3( , , )x x x c

Directional Derivative Directional Derivative

Directive derivatives and gradient Directive derivatives and gradient

( ) ( )lim

1 2 31 2 3

cos cos cosd

d x x x

Where l is a unit vector

Gradient Gradient

1 2 31 2 3 1 2 3

( , , ) ( , , )x x xx x x x x x

1 2 3i i + i

It can be shown if l is a unit vector

PropertiesProperties1.1. The gradient gives the direction for most raThe gradient gives the direction for most ra

pid increase. pid increase. 2.2. The gradient is a normal to the level surfaceThe gradient is a normal to the level surface

s. s. 3.3. Critical points of f are such that Critical points of f are such that =0 at the=0 at the

se points se points

=constant

Operational rules for gradient Operational rules for gradient

( ) ( )

grad C Cgrad

grad u v gradu gradv

grad uv ugradv vgradu

u vgradu ugradvgrad

v vgrad u gradu

Two important concepts about a Two important concepts about a vector field arevector field are flux,divergence, circul flux,divergence, circul

ation ation and and curlcurl

Vector Fields Vector Fields

A vector field can be intuitionistically described by vector curve tangent at each point to the vector that is produced by the field

dxdx dx

The The FluxFlux is the rate at which some-thi is the rate at which some-thing flows through a surface. ng flows through a surface.

Flux Flux

Let A= A (M) be a vector field, S be an orientated surface, An

be normal component of the vector A over the surface S

dS d A A S

AA((rr)=()=(AA11((xx11, x, x22, x, x

33),),AA22((xx11, x, x22, x, x

33),),AA33((xx11, x, x

22, x, x33)) ))

in Cartesian coordinate systemin Cartesian coordinate system

Flux Flux

1 2 3 2 1 3 3 1 2

A dx dx A dx dx A dx dx

Rate of flux to volume. Rate of flux to volume. In physics called In physics called densitydensity..

DivergenceDivergence

0lim S

divV V

AA((rr)=()=(AA11((xx11, x, x22, x, x

33),),AA22((xx11, x, x22, x, x

33),),AA33((xx11, x, x22, x, x

33)))) In Cartesian coordinate systemIn Cartesian coordinate system

Divergence Divergence

31 2 AA Adiv

Lagrangian Operator Lagrangian Operator

DivergenceDivergence

1 2 3 1 2 3

( , , )x x x x x x

1 2 3i i i

div A A

Operational rules for divergence Operational rules for divergence

1. ( ) ( )

2. ( )

3. ( )

div C Cdiv

div div div

div div

A+ B = A+ B

A A+ A

Circulation is the amount of something through Circulation is the amount of something through a close curve a close curve

Circulation Circulation

A() be a vector field, l be a orientated close curve

AA((rr)=()=(AA11((xx11, x, x22, x, x

33),),AA22((xx11, x, x22, x, x

33),),AA33((xx11, x, x22, x, x

33)))) ll be a orientated close curve be a orientated close curve

Circulation Circulation

1 2 3( + )1 2 3d A dx + A dx A dx l l

AA((rr)=()=(AA11((xx11, x, x22, x, x

33),),AA22((xx11, x, x22, x, x

33),),AA33((xx11, x, x22, x, x

33))))

The Curl of a Vector Field The Curl of a Vector Field

3 32 1 2 1

2 3 3 1 1 2 1 2 3

= ( , , )A AA A A A

rot curlx x x x x x x x x

curl A A

Makes circulation density maximal at Makes circulation density maximal at a point along the curl.a point along the curl.

The Curl of a Vector Field The Curl of a Vector Field

3 32 1 2 11 2 3

2 3 3 1 1 2

( )cos( , ) ( ) cos( , ) ( ) cos( , )

A AA A A Ax x x

x x x x x x

Operational rules for Rotation (Curl)Operational rules for Rotation (Curl)

1. ( ) ( )

2. ( )

3. ( )

5. ( ) 0

6. ( ) 0

rot C Crot

rot rot rot

rot rot

4. rot rot rot

rot grad

div rot

A+ B = A+ B

A A+ A

A B B A - A B

Potential Field Potential Field AA==gradgrad

Tube Field Tube Field divA =0

Harmonic FieldHarmonic FielddivA =0, rotA=0

Several Important Fields Several Important Fields

Summary Summary

Class is Over! Class is Over!

See you Friday Evening!See you Friday Evening!

21:30,1,Dec,2005 21:30,1,Dec,2005

Dr. Wang Xingbo Fall ， 2005

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