Flashcards
VC Chapter 1
Cards in this deck (13)
- 1N-Dimensional Vector?is an element of Rn
{(x1, ..., x2) : x1 , ... , xn in R}
notated as - 2Addition Scalar Multiplication Dot Productx + y = (x1 + y1, .... , xn + yn)
Lx = (Lx1, .. , Lxn) for all L in R
x.y = x1y1 + ... + xnyn - 3Zero Vectorx + 0 = x = 0 + x
0x = 0
0.x = 0
for all vectors x - 4Length of vector x||x|| = sqrt (x.x) = sqrt{x1^2 + .... + xn^2}
- 5What is Br(a) = {x : ||x-a|| < r}A subset of Rn
Open ball of radius r centred at a - 6Interior Point int(S)S c Rn and a in S. If there exists Br(a) c S for some r>0 then a is said to be an interior point.
- 7Exterior Point ext(S)If b in S(compliment) and be is an interior point of S(c), then b is said to be an exterior point of S.
- 8Boundary PointIf c in Rn is neither, then c is a boundary point.
- 9int(S) = int (S(c)) = int(S) U ext (S) U ds = dS(c) =ext(S(c))
ext(S)
Rn
dS - 10Open Subset Closed SubsetS is said to be an open subset of Rn if S n dS = 0.
If a set is open then S = int(S)
If dS c S then S is closed.
S is closed iff S(c) is open. - 11Neighbourhood of aIf S is open and a in S then we say that S is a neighbourhood of a
And S \ {a} is a punctured neighbourhood of a - 12Continuousf is continuous at a if f(a) is defined and:
lim(x->a) f(x) = f(a) - 13Composition Rule for Vectorf: Rn -> Rm g: Rm -> Rp be any vector map satisfying
(i) f is cts at a in Rn
(ii) g is cts at f(a)
then g o f : Rn -> Rp is cts at a.
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- BSc Maths (Vector Calculus) Flashcards on VC Chapter 1, created by joe.clay11 on 20/07/2013.
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