Upload
aquila-townsend
View
26
Download
0
Tags:
Embed Size (px)
DESCRIPTION
Mathematical & Mechanical Method in Mechanical Engineering. Dr. Wang Xingbo Fall , 2005. Mathematical & Mechanical Method in Mechanical Engineering. Vector Algebra. Vectors Products of Two Vectors Vector Calculus Fields Applications of Gradient, Divergence and Curl. - PowerPoint PPT Presentation
Citation preview
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
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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..
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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))
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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::
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Length (magnitude) of a vectorLength (magnitude) of a vector
2 2 21 2 3| | x x x X
1 2 3
2 2 21 2 3
( , , )
| |
x x x
x x x
X
X
ProjProjuuA = A = ((|A| |A| coscos))u u ( |( |uu| = 1)| = 1)
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Products of Two Vectors Products of Two Vectors
AA=(a=(a11, a, a22, a, a33), ), BB=(b=(b11, b, b22, b, b33))
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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:
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Properties of Scalar ProductProperties of Scalar Product
A B = A B
( ) A B + C = A B + A C
0 A B
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|
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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
A× B
B Area
A
1.A×B = -B ×A,
2.A ×(B + C) = A ×B +A ×C,
3. A||B is the same as A ×B = 0
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Three Basis Vectors Three Basis Vectors 1 2 3, ,i i i
A = a1i1 + a2i2 + a3i3, B = b1i1 + b2i2 + b3i3
2 3
1 2 3
1 2 3
a a a
b b b
i i i
A B1
1. (A×B)×C = B(A·C) -A(B·C)
2. A×(B×C) = B(A·C) - C(A·B)
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Product of Three Vectors
A = a1i1 + a2i2 + a3i3,
B = b1i1 + b2i2 + b3i3,
C = c1i1 + c2i2 + c3i3 321
321
321
)(
ccc
bbb
aaa
CBA
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))
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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 < ∞)
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Vector Calculus Vector Calculus
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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
0
( ) ( ) ( )'( ) lim
t
d t t t tt
dt t
A A A
A
31 2
31 21 2 3
( )( ) ( )( )( , , )
( )( ) ( )
dA tdA t dA td t
dt dt dt dtdA tdA t dA t
dt dt dt
A
i i i
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Properties of Vector DerivativeProperties of Vector Derivative
velocity
( )'( ) ( )
d tt t
dt v
rr
acceleration 2 ( )"( ) ( )
d tt t
dt
rr a
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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.
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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.
Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Directive derivatives and gradient Directive derivatives and gradient
0
( ) ( )lim
d
d
r l r
l
1 2 31 2 3
cos cos cosd
d x x x
l
Where l is a unit vector
Gradient Gradient
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Directive derivatives and gradient Directive derivatives and gradient
1 2 31 2 3 1 2 3
( , , ) ( , , )x x xx x x x x x
1 2 3i i + i
d
d
ll
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
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Directive derivatives and gradient Directive derivatives and gradient
▽
=constant
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Operational rules for gradient Operational rules for gradient
2
0
( ) ( )
( )
( )
( )
( ) '
gradC
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
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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
31 2
1 2 3
dxdx dx
A A A
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.
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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
n
S 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
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Flux Flux
1 2 3 2 1 3 3 1 2
S
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..
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
DivergenceDivergence
0lim S
V
d
divV V
A S
A
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
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Divergence Divergence
1 2 3
31 2 AA Adiv
x x x
A
Lagrangian Operator Lagrangian Operator
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
DivergenceDivergence
1 2 3 1 2 3
( , , )x x x x x x
1 2 3i i i
div A A
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Operational rules for divergence Operational rules for divergence
1. ( ) ( )
2. ( )
3. ( )
div C Cdiv
div div div
div div
A A
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
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Circulation Circulation
A() be a vector field, l be a orientated close curve
d l
A l
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
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Circulation Circulation
1 2 3( + )1 2 3d A dx + A dx A dx l l
A l =
AA((rr)=()=(AA11((xx11, x, x22, x, x
33),),AA22((xx11, x, x22, x, x
33),),AA33((xx11, x, x22, x, x
33))))
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
The Curl of a Vector Field The Curl of a Vector Field
1 2 3
3 32 1 2 1
2 3 3 1 1 2 1 2 3
1 2 3
= ( , , )A AA A A A
rot curlx x x x x x x x x
A A A
i i i
A A
curl A A
Makes circulation density maximal at Makes circulation density maximal at a point along the curl.a point along the curl.
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
The Curl of a Vector Field The Curl of a Vector Field
0
3 32 1 2 11 2 3
2 3 3 1 1 2
lim
( )cos( , ) ( ) cos( , ) ( ) cos( , )
nS S
A AA A A Ax x x
x x x x x x
n n n
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
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 A
A+ B = A+ B
A A+ A
A B B A - A B
A
Potential Field Potential Field AA==gradgrad
Tube Field Tube Field divA =0
Harmonic FieldHarmonic FielddivA =0, rotA=0
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Several Important Fields Several Important Fields
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
Summary Summary
Class is Over! Class is Over!
See you Friday Evening!See you Friday Evening!
Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering
21:30,1,Dec,2005 21:30,1,Dec,2005