2. Let ū = (-1, 2, –1) and i = (-2, 0, 3). Find ū · ū.

Q: Find g(u(х, у, г), в(х, у, г), и(х, у, 2)) if g(x, y, z) = z sin xy, u(x, y, z) = x²z", v(x, y, 2) =…

A: Given:  (1) gx, y, z=zsinxy (2) ux, y, z=x2z3 (3) vx,y,z=πxyz (4) wx,y,z=xyz To find : gux, y,z, vx,…

Q: ủ (u x) = 0 for all u, v in R.

A: We will prove whether this is true or false as following.

Q: Find x, y e lº such that || x || ∞ = 1 = || y || = || x+y 2 || x #y.

A: As per the question we have to find two sequences x, y from the norm-space l∞ such that : ||x||∞ = 1…

Q: Prove that Cov (x, y) = E (xy) – E (x) E (y).

A: Cov(x,y) is covariance of x and y. E(x) is expected value of x E(y) is expected value of y.

Q: = + E (0 < x < L,0 < t <ox) L Ot u (0, t) %3D 0, и(L,t) %3 0, и(х,0) — А, и (т,0) %3D0.

A: Given - ∂2u∂x2 = ∂2u∂t2 + πL∂u∂t , u0, t = 0 , uL, t = 0 , ux, 0 = A , ∂∂tux, 0 = 0 , 0 &lt; x &lt;…

Q: こ= (t)} () て= () %3D - (^),J %3D てんn ZトXC

A: →∫eax+bdx = eax+ba + C, where 'C' is the constant of integration.→∫xndx = xn+1n+1 + C

Q: c) Sinh3u = Sinh'u + 3Cosh'u.Sinhu = 3Sinhu + 4Sinh'u d) Sinh'u- Sinh'v Cosh'u-Cosh'v

A: We have to show that the following results

Q: where C is rE by the lines X

A:

Q: Let u=3i|-2j and v=-4i+5i find ||u||

A: Norm of a vector is given by- u2=u→.u→

Q: solve the following P.D.E: azu ди + u at u(0, t) = u(1, t) = 0 %3D || (1, 0< x < 1/2 0,1/2 < x < 1…

A:

Q: The functions y1, Y2, . .. , Yn are linearly independent, if and only if W (y1, 32, ..., Yn)(to) #0…

A: If two functions are linearly dependent then the Wronskian is zero for every value of t.

Q: Let F = cos(5t) i+ 3tj+ 5t° k and G = ° cos(5t) i +²j+5e* k Find the cross product Fx G =

A:

Q: ôu ôu ôt ôx? u (0, 1) — и(1,1) - 0,1> 0, : 0 in (0,1) × (0,+∞), u(х,0) — и, (х),0 <x<1,

A: Given that the heat equation ∂u∂t-∂2u∂t2=0withu(0,t)=u(1,t)=0with ICu(x,0)=u0(x)

Q: Find a linear transformation T:

A: Consider the provided information, T11=23   and T1-1=05 Find a linear transformation T. We can put…

Q: Obtain characteristics y² Uxx + Uyy = O

A: Given equation is  y2Uxx+Uyy=0 Now comparing with  AUxx+BUxy+CUyy=0 We have A=y2, B=0, C=1

Q: Two functions u and r of x and y are said to satisfy the Cauchy- Riemann equations if u, = v, and -…

A: Given that the functions u and v are said to satisfy Cauchy Riemann equations if ux=vy   and uy=-vx…

Q: Show that: (a) Cov(X,Y) = E[XY] – E[X]E[Y]. (b) If X and Y are independent, then fY|X(y|x) = fY(y)

A: Solution:a.Show that COV(X,Y)=E(XY)–E(X)E(Y)

Q: (a) r(u) = e" i+e"j. u € [0, 1]. r(u) = (1- u)i, ue [0. 2].

A:

Q: Prove that f-(U B: -U(B.), %3D i=1 i=1 and n N (B.). 3=D1 i=1

A:

Q: Condition JW=3.6, j0=0.4

A:

Q: δu/δt = δ2u / δx2    Given that u (0,t ) = 0                          u (x, 0) = 10sin 2x -6sin 4x

A:

Q: Find g(u(x, y), v(x, y)) if g(x, y) = y sin(xy), u(x, y) = x²y', and v(x, y) = Txy. %3D

A: We have given g(x, y)=ysin(x2y), u(x, y)=x2y3 and v(x, y)=πxy. To find gu(x, y), v(x, y), first…

Q: 7. Which of the following sets of functions is linearly dependent on R0,2? (a) {6, z, sinz} (b) {1,…

A:

Q: (IV). (10') Let u : 3 Find the coordinates of x with and x = respect to uj and u2.

A:

Q: 4. Classify and reduce the following equations to a canonical form: (a) yu -x*u, = 0, x>0, y>0. (b)…

A: To classify and reduced in canonical form of the given equation is..

Q: Use Cramer's rule to solve for y without solving for x, z, and w. 3х + y + z + = -6 W 2x + 2y z + =…

A:

Q: Use the Chain Rule to find az/əs and ôz/Ət. z = tan(u/v), u = 8s + 3t, V = 3s – 8t

A: z=tanuv,   u=8s+3t,   v=3s−8tFormula→∂z∂t=∂z∂u⋅∂u∂t+∂z∂v⋅∂v∂t

Q: 1 Assume that 1 < a < b_and equation is solvable. + x = Show that this * – b'

A:

Q: Q1. If w = f(u, v), u = cosx and v = sinx, show that: d?w v2 fuu + u² fov – 2uv fuv - ufu - vf dx2

A:

Q: 2- Sketch the signal x(t) = u(t +1) + 2u(t) –u(t –1) –u(t - 2) -u(t-3)

A: Given signal is x(t)=u(t+1)+2u(t)-u(t-1)-u(t-2)-u(t-3) For signal u(t+1), graph is For signal…

Q: Let z = e*y, where x = Vuv and y = . Find in terms of u and v.

A: Change the z in the terms of u and v and the differentiate.

Q: b) = f ) then prove that x ux + y uy + z uz = 0 u

A:

Q: 2. -4 let u= dx=

A: Indefinite Integration using substitution

Q: dt Determine dw if t = (6r* +3r*) -(*+2r-5)'. -s)'. and w =

A:

Q: find if w= fcusu)ou=X+y Wxy xy Express the of u,v, fu , fv V= an Swer interns

A: Solve the following

Q: If f(z) = 6tan(z) , then f'(r) = (A) In(6) 6tan(2) (B) sec2(z) 6tan(2) (C) In(6) sec2() 6tan(z) (D)…

A:

Q: Determine whether the work done along the path C is positive, negative, or zero. Ozero negative…

A: To determine whether the work done along the path C is positive , negative or zero .

Q: üx (u x i) = 0 for all u, v in R.

A: we have to identify the given statement is true or false

Q: c) Sinh3u= Sinh’u+ 3Cosh’u.Sinhu = 3 Sinhu + 4Sinh’u d) Sinh'u- Sinh'v = Cosh'u– Cosh'v

A: We have to prove the following results

Q: find th e angle between u and v in th e given exercise. U = [0.9, 2.1, 1.2], V = [-4.5, 2.6, -0.8]

A: Given: u = [ 0.9 , 2.1 ,  1.2 ]  and v = [ - 4.5, 2.6 , - 0.8 ]    To find: Angle between u and v.

Q: Let u=2i+3j-2k and v=-1i-4j +k Find u . v, ||u|| and u . v

A: Given that u and v vector and we find its dot product (u.v) and mode of u(||u||). Here i attached…

Q: Find those constants a, b, c, d such that f(z) = ax² + bxy + y² + i(x² + cxy + dy?)

A: The given function is : fz=ax2+bxy+y2+i(x2+cxy+dy2)

Q: x+y Show that: B(x+1, y) = B(x, y)

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 3 and ajj = i. j. Th

A:

Q: Find f'(3), where f(t) = u(t) · v(t), u(3) = (1, 2, -2), u'(3) = (4, 0, 7), and v(t) = (t, t², t³).

A: Introduction: The outside-inside rule, composite function rule, and function of a function rule are…

Q: Let R=[0,1]x[0,1]. xy3 dA= R.

A: Let's find.

Q: u(0,t) = u(3,t)=0, t≥0 x 0<x<1 u(x,0) = 2-x 1<x<3

A:

Q: Reduce Z= ( 1+i) (1+i) / (1-í), to the form a + ib -1-i .a 1-i .b 1+i .C -1+i .d others .e

A:

Q: 6. Let f(u, v) = u sin(v) – v cos(u) and u(r,y) = r² – ry² v(r, y) = y – r Find af af and both by…

A: We need to find the derivatives using direct substitution and by chain rule . Now , we have ,…