Download Formula sheet for engineering mathematics and more Cheat Sheet Mathematics in PDF only on Docsity! 2 4 -1
> 1 is a linear combination of | —1 ] and 1 }, since
—2 2 -2
2 4 —1
1})={-1)+2) 1
—2 2 —2
To find the length (or magnitude, norm, modulus, etc) of any vector
a
(5). ‘we use:
length = Va? + b?
If we have a 3 dimensonal vector it works exactly the same, to find the
a
length of | b |, we use:
c
length = Va? +b? +c?
Often it is useful to use vectors of length 1. Such a vector is called a unit
vector. If v is a (non-zero) vector, we denote the unit vector in the
direction of v by Vv.
If we know the length of v, it is easy to find V, we just take:
x v
v=
lv|
For example, let v = ({). Then |v| = V3? + 4? = 25 = 5.
3
Sov=¥=iv= (:).
5
Dot product: Parallel vectors: Orthogonal vectors: Find the angle between two vectors: Cross Product: Log functions have the following properties:
> The domain of the log function is (0,00), and the range is all of R
> As x approaches 0, log, (x) takes a large negative value.
> As x takes a large positive value, so does
log, (2) + log, (y) = log, (2) log, (#) — log, (9) = log, (=)
log, (2) = kloga (2) log, (a) =1
_ log, (z)
08a) = Tog, (a)
if y = a*, then x = log, (y).
where P is the measured sound pressure.
Why not just use P?
dB SPL corresponds better to how people experience sound.
OdB corresponds to the threshold of human hearing
30dB corresponds to background office noise
60dB corresponds to a people talking
100dB will start to hurt your ears...
Negative dB values correspond to pressure levels too low to hear.
2 0
If A= 72 then AT =] 1 2].
02 1 304
_(0 2 r_{0 4
WB=({ _ then B =(° |)
A square matrix A is called symmetric if AT = A.
3 -1 2
For example, {| —1 5 3] is symmetric.
2 3 0
A square matrix A is called skew-symmetric if AT = —A.
For example, a
1 0 ) is skew-symmetric.
AA1+=A'A=I,
Differentiation:
Function f(x) | Derivative f’(z)
constant 0
x” ner!
e* e”
In(x) x
sin(x) cos(z)
cos(z) — sin(z)
tan(z) sec?(z)
sec(x) sec(x) tan(x)
cesc(a — csc(z) cot (x)
cot(z) — csc? (x)
Sum Rule: If f(x) = g(x) + h(x), then f’(x) = g'(x) + h(x).
Constant Multiple Rule: If f(2) = c(g(x)) (where c is a constant),
then f!(2) = e(g/(2)).
Product Rule: If f(x) = g(x)h(x), then f’(x) = g/(x)h(x) + g(x)h'(x).
. . _ 9{2) tony — o(x)h(w) — g(x)h'(z)
Quotient Rule: If f(z) = Aa)’ then f’(x) = — (h(a)y2
Chain Rule. If f(x) = g(h(x)), then f’(z) = h’(x)g'(h(z)).
Function f (a) Integtal [ f(a) dx
0 c
x” ware" +e
e* ev +C
2 In(z) +c
cos(z) sin(z) +c
sin(z) —cos(xz) + ¢
tan(x) In|sec(x)| +c
Tae sin *(£)+c
ig qtan'(%) +e
Sum rule:
[t@+9@) a= | He) a+ | (a) ae
