aut = Urxe + 4tx 02x23, tro Их(Ort) = 1*, Их(3,t) =t to ИСХО)=1 tx, Oext3
G) A 12 meter water t aut -the hole G) A 12 meter water t aut -the hole
Recall that if T: R" R" is a linear transforrmation T(x) = [Tx, where [T is the transformation matrix, then 1. ker(T) null([T] (ker(T) is the kernel of T) 2. T is one-to-one exactly when ker(T) = {0 3. range of T subspace spanned by the columns of [T] col([T) 4. T is onto exactly when T(x) = [Tx = b is consistent for all b in R". 5. Also, T is onto exactly when range of T col([T]) =...
t) (2,t, e') 1. Consider the space curve r and B Tx N (a) Find T N= r°T T' (b) Compute the curvature K(t) of r(t) t) (2,t, e') 1. Consider the space curve r and B Tx N (a) Find T N= r°T T' (b) Compute the curvature K(t) of r(t)
Use the Laplace transform to solve initial value problems 3. tx" + 2(t-1)x' - 2x = 2, x(0) = 0.
Tx N, for the vector-valued Find the vectors T and N, and the unit binomial vector B function r(t) and the given value of t. k, 3 to= 1 + Tx N, for the vector-valued Find the vectors T and N, and the unit binomial vector B function r(t) and the given value of t. k, 3 to= 1 +
3. Solve the initial and boundary value problem aut, c) = a. cut,c), force (0,7), t > 0. u(t,0) = u(t, 7) = 0, (0, 2) = cos , 2 € (0,7).
Expectation of xext ? E[xe^(tx)] = some series? for example E[e^(tx)] = 1 + tx + (t2x2)/2! + (t3x3)/3! + .....
let {X(t), 1 2 0} denote a Brownian motion 8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion (d) Let Using (c) present an argument that let {X(t), 1 2 0} denote a Brownian motion 8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion...
4.1.5 ([Ber14, Ex. 4.2.5]). Fix y € R3 and define T: R3 R3, Tx = 3 xy (cross product of vectors). Find the matrix of T relative to the canonical basis of R3
x is distributed in U (a, b) M x (t) = E ( e ^ tx) = a) ( e ^ (tb) - e ^ (ta))/ ( t (b-a)) for t that is not zero b) 1 for t = 0 Show that it is continuous at zero.