Page 44 - Aeronautical Engineer Data Book

Fundamental dimensions and units 31

d dB dA
33 (A 2 B)= A 2 33 + 33 2 B
dt dt dt
d
= – 33 (B 2 A)
dt
Gradient
The gradient (grad) of a scalar field +(x, y, z) is
� ∂ ∂ ∂ �
grad + = 6+ = i 33 + j 33 + k 3 3
+
∂ z
∂y
∂x
∂ + ∂ + ∂ +
= 3 3 i + 3 3 j 3 3 k
∂x ∂
y ∂
z
Divergence
The divergence (div) of a vector V = V(x, y, z)
(x, y, z) i + V (x, y, z) j + V (x, y, z)k
= V x y z
∂V x ∂ V y ∂ V z
div V = 6 · V 3 3 + 3 3 + 3 3
∂x ∂y ∂z
Curl
Curl (rotation) is:
i j k
∂
∂
∂
curl V = 62 V = 33 33 33
∂x ∂y ∂z
V x V y V z
� ∂ V z ∂ V y � � ∂ V x ∂ V z �
= 3 3 – 3 3 i + 3 3 – 3 3 j
∂z
∂z
∂y
∂x
� ∂ V y ∂ V x �
+ 3 3 – 3 3 k
∂y
∂x
2.8.8 Differentiation
Rules for differentiation: y, u and v are
functions of x; a, b, c and n are constants.
d du dv
33 (au ± bv) = a 33 ± b 33
dx dx dx
d (uv) dv du
33 = u 33 + v 33
dx dx dx
� �
d u 1 du u dv
33 33 = 33 33 – 33 33
2
dx v v dx v dx