For the following exercises, find vector u with a magnitude that is given and satisfies the given conditions. 99. v = 〈 7 , − 1 , 3 〉 , ‖ u ‖ = 10 , u and v have the same direction
For the following exercises, find vector u with a magnitude that is given and satisfies the given conditions. 99. v = 〈 7 , − 1 , 3 〉 , ‖ u ‖ = 10 , u and v have the same direction
For the following exercises, find vector
u
with a magnitude that is given and satisfies the given conditions.
99.
v
=
〈
7
,
−
1
,
3
〉
,
‖
u
‖
=
10
,
u
and
v
have the same direction
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Q2: Using the Laplace transform, find the solution for the following equation
y"" +y" = 6et + 6t + 6. Suppose zero initial conditions (y"" (0) = y"(0) = y'(0) = y(0) = 0).
1- Let A = {A1, A2, ...), in which A, A, = 0, when i j.
a) Is A a π-system? If not, which element(s) should be added to A to become a π-system?
b) Prove that σ(A) consists of the finite or countable unions of elements of A; i.c., A E σ(A) if and
only if there exists finite or countable sequence {n} such that A = U₁An (Hint: Let F be such
class; prove that F is a σ-filed containing A.)
c) Let p ≥ 0 be a sequence of non-negative real numbers with Σip₁ = 1. Using p₁'s, how do you
construct a probability measure on σ(A)? (Hint: use extension theorem.)
2- Construct an example for which P(lim sup A,) = 1 and P(lim inf An) = 0.
3. Let
f(z) =
sin (22) + cos (T2)
2(22+1)(z+1)
Compute f(z)dz over each of the contours/closed curves C1, C2, C3 and C4 shown
below.
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