Page 41 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

32 Fundamentals of Ocean Renewable Energy

respectively (e.g. ˆ i = (1, 0, 0)). Occasionally, instead of boldfaced letters, arrow
notation is used for vectors, especially in handwritten equations, as follows,
u = u = (u, v, w) (2.2)
The inner (dot or scalar) product of two vectors (e.g. u and S = (S x , S y , S z ))
in vector notation is written as
(2.3)
u · S = uS x + vS y + wS z
The cross (outer or vector) product of two vectors (e.g. and R) is a vector
quantity, and can be evaluated as follows
× R = (Ω y R z − Ω z R y ) ˆ i + (Ω z R x − Ω x R z ) ˆ j + (Ω x R y − Ω y R x )k ˆ (2.4)
Alternatively, index or indicial notation is an efficient method of writing
vectors and matrices. Indicial notation is based on indices. For instance, the
velocity vector u = (u, v, w) can be represented by u = (u 1 , u 2 , u 3 ). Indices 1,
2, and 3 correspond to x, y, and z, respectively. If we use a ‘free index’ i which
can take 1, 2, and 3, we can show a vector as

u = u i (2.5)
Lowercase subscript (here i) in the index notation has a range (1, 2, 3).
Similarly, a matrix can be represented by two free indices: i and j.For
example, τ ij (i, j = 1, 2, 3) is a three-by-three matrix or array where τ 11 = τ xx ,
τ 12 = τ xy ,τ 13 = τ xz , τ 21 = τ yx , ..., τ 33 = τ zz .

2.1.1 Einstein Convention
In the index notation, for summations, the rule is that repeated indices are
summed over. For instance, for the inner product, we can write
3

u i S i = u i S i = u · S = u 1 S 1 + u 2 S 2 + u 3 S 3 (2.6)
i=1
Based on this convention, when an index variable appears twice in a single term,
it should be summed over. Here is another example,
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b i = c k u ik = c k u ik (2.7)
k=1
where k is repeated and should be summed over. The expanded version of the
previous equation is

b 1 = c 1 u 11 + c 2 u 12 + c 3 u 13 ⇒ b x = c x u xx + c y u xy + c z u xz
b 2 = c 1 u 21 + c 2 u 22 + c 3 u 23 ⇒ b y = c x u yx + c y u yy + c z u yz (2.8)
b 3 = c 1 u 31 + c 2 u 32 + c 3 u 33 ⇒ b z = c x u zx + c y u zy + c z u zz