Page 47 - Aeronautical Engineer Data Book

34 Aeronautical Engineer’s Data Book
Partial derivatives Let f(x, y) be a function of
the two variables x and y. The partial deriva­
tive of f with respect to x, keeping y constant is:
∂ f f (x + h, y) – f (x, y)
3 = lim 333
3
∂x h→0 h
Similarly the partial derivative of f with respect
to y, keeping x constant, is
∂
f f (x, y + k) – f (x, y)
3 3 = lim 333
∂y k→0 k
Chain rule for partial derivatives To change
variables from (x, y) to (u, v) where u = u(x, y),
v = v(x, y), both x = x(u, v) and y(u, v) exist and
f(x, y) = f [x(u, v), y(u, v)] = F(u, v).
∂ F ∂x ∂ f ∂y ∂ f ∂ F ∂ x ∂f ∂ y ∂f
3 3 = 33
3 + 3 3 3, 3 3 =
3 333 +
3 333
3
3
∂ u ∂u ∂x ∂u ∂y ∂ v ∂v ∂v ∂v ∂y
∂ f ∂ u ∂ F ∂ v ∂ F ∂ f ∂ u ∂ F ∂ v ∂ F
33 = 3 3 3 + 3 3 3 3, 33 = 3 3 3 + 3 3 3 3
3
3

∂
∂
∂x ∂ x u
∂ x ∂ v ∂y ∂ y u
∂ y ∂ v
2.8.9 Integration
f(x) F(x) = ∫f(x)dx
x a+1
x a 3 3, a ≠ –1
a + 1
x –1 ln | x |
e kx
e kx 3 3
k
a x
a x 3 3, a > 0, a ≠ 1
ln a
ln x x ln x – x
sin x –cos x
cos x sin x
tan x ln | sec x |
cot x ln | sin x |
sec x ln | sec x + tan x |
1
1
= ln | tan (x + π)|
33
3 3
2
2