1. Mathematics Review A.1 Vectors A.1.1 Definitions
V = vx ex + vyey + vzez
A.1.2 Products
A Scalar Products
A Vector Product
V x W = ｜v｜｜W｜sin(V,W) nvw
ex ey ez
vx vy vz
wx wy wz =
V．W = ｜v｜ ｜W｜cos(V,W)
= vxwx + vywy + vzwz

2. A.2 Tensors
A.2.1 Definitions
A tensor (2nd order) has nine components, for example, a stress tensor
can 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
[A.2-1]
[A.2-2]
Explanation (Borisenko, p64)

3. The vector product of a tensor t and a vector v, denoted by t．v is a vector defined by
The product between a tensor vv and a vector n is a vector
[A.2-5]

4. The scalar product of two tensors s and t, denoted as s:t, is a scalar defined by
The scalar product of two tensors vw and t is
[A.2-7]
Table A.1-1 Orders of physical quantities and their multiplication signs
Physical quantity Multiplication sign
Scalar Vector Tensor None X ‧ :
Order

5. 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
[A.3-1]
The gradient of a scalar field s, denoted as ▽ s, is a vector defined by
[A.3-2]
A.3.2 Products
The divergence of a vector field v, denoted as ▽‧v is a scalar .
[A.3-5]
Flux is defined as the amount that flows through a unit area per unit time
Flow rate is the volume of fluid which passes through a given surface per unit time

6. Similarly
[A.3-5]
For the operation of
[A.3-7]
For the operation of ▽‧▽s, we have
[A.3-8]
In other words
[A.3-9]
Where the differential operator▽2, called Laplace operator, is defined as
[A.3-10]
For example: Streamline is defined as a line everywhere tangent to the velocity
vector at a given instant and can be described as a scale function of f.
Lines of constant f are streamlines of the flow for inviscid irrotational flow
in the xy plane ▽2f=0

7. The curl of a vector field v, denoted by ▽ x v, is a vector like the vector product
of two vectors.
[A.3-11]
When the flow is irrotational,
Like the tensor product of two vectors, ▽v is a tensor as shown:
[A.3-12]

8. Like the vector product of a vector and a tensor, ▽‧t is a vector.
From Eq. [A.2-2]
[A.3-14]
It can be shown that
[A.3-15]

9. A.4 Divergence Theorem
A.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
[A.4-1]
This equation , called the gauss divergence theorem, is useful for converting from a
surface integral to a volume integral.
A.4.2 Scalars
For a scale s
[A.4-2]
A.4.3 Tensors
For a tensor t or vv
[A.4-3]
[A.4-4]

10. A.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.
A.5.1 Cartesian Coordinates ey y
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. dy ex
P(x,y,z) dz dx ez x z
Fig. A.5-1(a)

11. 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, y, and z.
x = r cosθ [A.5-1] y = r sinθ [A.5-2] z = z [A.5-3]
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:
v = er vr + eθvθ + ezvz
and
Fig. A.5-1(b)
Fig. A.5-1(b)*

12. 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 sinq cosf [A.5-6]
y = r sinq sinf [A.5-7]
z = r cosq [A.5-8]
Fig. A.5-1(c)
The differential increments of a control
unit, as shown in Fig. A.5-1(c)*, in r, θ,
and φ axis are dr, rdθ , and rsinθdφ ,
respectively. A vector v and a tensor τ
can be expressed as follows: θ φ
[A.5-9]
[A.5-10]
Fig. A.5-1(c)*

13. A.5.3 Differential Operators
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
[A.5-13]
Whereas in spherical coordinates,
[A.5-14]

14. The equations for ▽s, ▽‧v, ▽ xv, and ▽2s in rectangular, cylindrical, and
spherical coordinates are given in Tables A.5-1, A.5-2, and A.5-3, respectively. θ φ