Mathematics Review A.1 Vectors A.1.1 Definitions
-
Upload
beatrix-harris -
Category
Documents
-
view
225 -
download
0
description
Transcript of Mathematics Review A.1 Vectors A.1.1 Definitions
1
Mathematics ReviewA.1 VectorsA.1.1 Definitions
A.1.2 ProductsA.1.2.1 Scalar Products A.1.2.2 Vector Product
V.W = | v | | W | cos(V,W) = vxwx + vywy + vzwz
V x W = | v || W | sin(V,W) nvw
V = vx ex + vyey + vzez
ex ey ez
vx vy vz
wx wy wz
=
2
Mathematics Review
A.2 TensorsA.2.1 Definitions A tensor (2nd order) has nine components, for example, a stress tensorcan be expressed in rectangular coordinates listed in the following:
A.2.2 Product The tensor product of two vectors v and w, denoted as vw, is a tensor defined by
xx xy xz
yx yy yz
zx zy zz
τ
x x x x y x z
y x y z y x y y y z
z x z y z zz
v v w v w v wv w w w v w v w v w
v w v w v wv
vw
[A.2-1]
[A.2-2]
3
Mathematics ReviewThe vector product of a tensor and a vector v, denoted by . v is a vector defined by
xxx xy xz
yx yy yz y
zx zy zz z
x xx y xy z xz x yx y yy z yz
x zx y zy z zz
x y
z
vv v
v
v v v v v v
v v v
τ
e e
e
[A.2-3]
xx x x y x z
y x y y y z y
z x z y z z z
x x x x y y x z z y x x y y y y z z
z x x z y y z z z
x y z x x y y z z
x y
z
x y z
nv v v v v vv v v v v v nv v v v v v n
v v n v v n v v n v v n v v n v v n
v v n v v n v v n
v v v v n v n v n
vv n
e e
e
e e e
v v n
[A.2-5]
The product between a tensor vv and a vector n is a vector
4
The scalar product of two tensors and , denoted as , is a scalar defined by
: :xx xy xz xx xy xz
yx yy yz yx yy yz
zx zy zz zx zy zz
xx xx xy yx xz zx yx xy yy yy yz zy
zx xz zy yz zz zz
σ τ[A.2-6]
:x x x y x z xx xy xz
y x y y y z yx yy yz
z x z y z z zx zy zz
x x xx x y yx x z zx y x xy y y yy y z zy
z x xz z y yz z z zz
v w v w v wv w v w v wv w v w v w
v w v w v w v w v w v w
v w v w v w
vw [A.2-7]
The scalar product of two tensors vw and is
Physical quantity Multiplication sign Scalar Vector Tensor None X : ‧Order 0 1 2 0 -1 -2 -4
Table A.1-1 Orders of physical quantities and their multiplication signs
5
Mathematics Review
x y zx y z
e e e
x y zs s ssx y z
e e e
A.3 Differential Operators
A.3.1 Definitions
The vector differential operation , called “del”, has components similar to those of a vector. However, unlike a vector, it cannot stand alone and must operate on a scalar, vector, or tensor function. In rectangular coordinates it is defined by
The gradient of a scalar field s, denoted as ▽s, is a vector defined by
[A.3-1]
[A.3-2]
A.3.2 Products
The divergence of a vector field v, denoted as ▽‧v is a scalar .
[A.3-5]
x y z x x y y z z
x y z
v v v vx y z
v v vx y z
e e e e e e
6
Mathematics Review
x y z x x y y z z
x y z
x y zx y z
a av av avx y z
av av avx y z
v v v a a aa v v vx y z x y z
v e e e e e eSimilarly
[A.3-5]
For the operation of [A.3-7] a a a v v v
For the operation of s, we have ▽‧▽
[A.3-8]
[A.3-9]
2
2
2
2
2
2
)eee()eee(zs
ys
xs
zs
ys
xs
zyxs zyxzyx
In other words ss 2Where the differential operator▽2, called Laplace operator, is defined as
2
2
2
2
2
22
zyx
[A.3-10]
7
Mathematics Review
x y z
x y z
x y x z y xx y z
e e e
x x xv v v
v v v v v ve e ey z z x x y
v
x y z
x y zx y z
x y z
v v vx x x x
v v vv v vy y y y
v v vz z z z
v
The curl of a vector field v, denoted by ▽x v, is a vector like the vector product of two vectors.
[A.3-11]
[A.3-12]
Like the tensor product of two vectors, ▽v is a tensor as shown:
8
[A.3-13]
Like the vector product of a vector and a tensor, ▽‧ is a vector.xx xy xz
yx yy yz
zx zy zz
yx xy yy zyxx zx
yzxz zz
x y
z
x y z
x y z x y z
x y z
τ
e e
e
x x x y x z
y x y y y z
z x z y z z
x x x y x z x
y x y y y z y
z x z y z z z
v v v v v vv v v v v v
x y zv v v v v v
v v v v v vx y z
v v v v v vx y z
v v v v v vx y z
vv
e
e
e
[A.3-14]
From Eq. [A.2-2]
[A.3-15]It can be shown that vv v v v v
9
A.4 Divergence TheoremA.4.1 Vectors
Let Ω be a closed region in space surrounded by a surface A and n the outward-directed unit vector normal to the surface. For a vector v
Ad dA
v v n [A.4-1]
This equation , called the gauss divergence theorem, is useful for converting from a surface integral to a volume integral.
A.4.1 Scalars
A.4.1 Tensors
For a vector s
For a tensor or vv
Asd s dA
n
A
d dA n
A
d dA vv vv n
[A.4-2]
[A.4-3]
[A.4-4]
10
Mathematics ReviewA.5 Curvilinear Coordinates
For many problems in transport phenomena, the curvilinear coordinates such as cylindrical and spherical coordinates are more natural than rectangular coordinates.A point P in space, as shown in Fig. A.5.1, can be represented by P(x,y,z) in rectangular coordinates, P(r, θ,z) in cylindrical coordinates, or P(r, θ,ψ) in spherical coordinates.
Fig. A.5-1(a)
A.5.1 Cartesian Coordinates
For Cartesian coordinates, as shown in A.5-1(a), the differential increments of a control unit in x, y and z axis are dx, dy , and dz, respectively.
x
y
z
dx
dy
dzP(x,y,z) ex
ey
ez
11
Mathematics Review
Fig. A.5-1(b)
A.5.1 Cylindrical Coordinates
For cylindrical coordinates, as shown in A.5-1(b), the variables r, θ, and z are related to x, θ, and z.
x = r cosθ [A.5-1] y = r sinθ [A.5-2] z = z [A.5-3]
Fig. A.5-1(b)*
v = er vr eθvθ + ezvz
rr r rz
r z
zr z zz
τ
and
The differential increments of a control unit, as shown in Fig. A.5-1(b)*, in r, q, and z axis are dr, rdq , and dz, respectively. A vector v and a tensor τcan be expressed as follows:
12
Mathematics Review
Fig. A.5-1(c)
A.5.2 Spherical Coordinates
For spherical coordinates, as shown in A.5-1(c), the variables r, θ, and ψ are related to x, y, and z as follows
x= r sin cos-y = r sin cos[A.5-7]z = r cos-
[A.5-9]
rr r rz
r z
zr z zz
τ [A.5-10]
The differential increments of a control unit, as shown in Fig. A.5-1(c)*, in r, θ, and ψ axis are dr, rdθ , and rsinθdψ, respectively. A vector v and a tensor τcan be expressed as follows:
vvvrr eeev
Fig. A.5-1(c)*
13
Mathematics ReviewA.5.3 Differential Operators
100
rz
r z
zr z zz
r zr
rr r
r
r rr z
τ e
e e e
100
r
r
r
r r
rr r
r
r rr
τ e
e e e
1r zs s ssr r z
e e e
1 1sinr
s s ssr r r
e e e
Vectors, tensors, and their products in curvilinear coordinates are similar in form to those in curvilinear coordinates. For example, if v = er in cylindrical coordinates, the operation of τ . er can be expressed in [A.5-11], and it can be expressed in [A.5-12] when in spherical coordinates
[A.5-11] [A.5-12]
In curvilinear coordinates, assumes different forms depending on the orders of ▽the physical quantities and the multiplication sign involved. For example, in cylindrical coordinates
Whereas in spherical coordinates,
[A.5-13]
[A.5-14]
14
Mathematics ReviewThe equations for s, ▽ ▽‧v, x▽ v, and ▽2s in rectangular, cylindrical, and spherical coordinates are given in Tables A.5-1, A.5-2, and A.5-3, respectively.