9. Using the fact that ||u|| = 2, ||v|| = 3, and u x v = (-1,5, 1), find u · v.

Q: 10. If z = x + f(u), where u = xy, show that dz = x. х дх

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Q: ủ (u x) = 0 for all u, v in R.

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

Q: 9. Use f(x, y) = Vx? + y? to give a linear approximation of v/3.12 + 3.992.

A: Linear approximation to the function

Q: In Exercises 29–31, find the equation of the line tangent to thesurface at the given point P and in…

A:

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: X= R² and T = { $₁,x} V {₂\x>0} where U₁ = {(x, y) | x² + y ² < 2² } Prove (XT) is Topology

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Q: If u =f(x² +2yz, y² + 2zx ).Prove that (v² – zx) + (x² - yz) ди + ди ху

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Q: of af and Let f(x, y, A) = xy + A(5 – x – y). Find· ax' əy` af

A: f(x,y,λ)=xy+5λ-λx-λy .....(1) Differentiating (1) partially with respect to x, ∂f∂x=(1)y+0-λ-0…

Q: 4. Given that T : R² → R* and is defined as T(x, y) = (x + a, y + b) . Prove that

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Q: Given f(x, y, z) = x³yz². Find f.(1,1,1).

A: The equation we have , f(x,y,z) = x3yz2 We have to find out fz, Now we differentiate f(x,y,z) with…

Q: 1. Compute u + v and u-2v where u = -5 and v = T 2

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Given that vec F =x^ 2 y hat i -(z^ 3 -3x) hat j +4y^ 2 hat k find the k- component of ( nabla* vec…

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

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Q: Find ƒxx ( x, y)  and ƒxy( x, y).

A: Given, fx,y=3x+yx-1. The objective is to find fxxx,y and fxyx,y.   fxxx,y is found as follows: First…

Q: 3.2 Evaluate where z = x + iy with y the path on line from z = 2+i to z = 1 + 2i.

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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: Prove or disprove: Vx € Q, Vy e Q, 3z E Q, (x < y) → (x < z< y).

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Q: f(2) — и(х, у) + jo(x, y) и(х, у) — е% сos x — х f (0) = 1

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Q: Given f(2, y, z) =v V-I - 2y + z, find f.(z, y, z) = f(z, y, z) = f(1, y, z) =

A: We will use definition of differentiation. First we will handle derivative of the root of the…

Q: If f(1, y, 2) = In(z³ + y8 + z³) + V(z³ + y³ + z³), then find fr, fy and fa at (1, 1, 1).

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Q: If u = (u, uz) and v = (V1, V2), how do you find u + v?

A: Given: u=(u1,u2)v=(v1,v2)

Q: . If x, y E R with x < y, prove that x < tx + (1 – t)y < y for all t e (0,1).

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: 3. Find (ð v/ôx)x,z if v =x +y²-e²+ sin t and x+y=t

A: Given that  v=x+y2-ez+sint where t is given by x+y=t. So our function v becomes…

Q: (a) Vx E R, 3y E Z such that y > x + 110. (b) Vx E Z, 3y E N such that y > x + 110. (c) 3x E Z such…

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Q: Obtain (i) P (X 1)

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Q: дz (B)If Z = f(x, y), x = s² - t², y = 2st. Prove that s მა 2 (s² + t²) of af əx I t дz at II

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Q: prove. that s U. W V- W 1. U.r V.r

A: Using the vector triple product, it can be written that: a→×b→×c→=a→·c→b→-b→·c→a→. Using the scalar…

Q: 10. If z = x + f(u), where u = xy, show that дz dz х дх = x. ду

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Q: X = R² and = = { $₁x} V { \\x>0} where U4 = {(x,y) | x² + y² <²} Prove (x, t) is Topology

A:

Q: For x = 1, 2, 3 and y = 1, 2, let the joint pmf of X and Y be defined by fX,Y (x, y) = x + y/21…

A: Solution

Q: 1 3. Let f(r, y, z) = Show that Va? + y? + z²° frz + fyy + fz: = 0.

A: answer is in next step

Q: 1. If x, y, z E R, prove that (c) ||x| – |y|| < |x –yl. (d) \xy| < }(x² + y²)

A:

Q: ( b) For n=1,2,K , show that (x² –1)y"*) +2(nx + p)y") +(n² –n)y"-1) =0 where %3D d" y p") = (Hint:…

A: * Given that y = x+1x-1pa dydx = px+1x-1p-1ddxx+1x-1    dydx = px+1x-1p-1x-11-x+1.1x-12   dydx =…

Q: Find Duf at P f(x,y,z)=4x^3y^3z^3;  P(1,−2,1);  u=1/3i+2/3j−

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Q: If F (x, y, z) = xyzî – e² cos xĵ + xy² z³k, what is V · (V×F) (1, 2, 3) 1 2

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Q: Show that u(x,t)= f((x+a*t)+f(x-a*t))/2

A: Given PDE is utt=a2uxx subject to ux,0=fxutx,0=0 We know that the general solution of the PDE is…

Q: Additional Problem Prove the statement: 3X CZ such that Vx E X, 2 | x and 3 | x.

A:

Q: The solution for the IVP: Uxi u(x, 0) =2 x u,(x, 0) -ex. ux, () - 2x + (*-0-(**) O b. u(x,…

A: Consider the given information. utt=uxx                 -∞&lt;x&lt;∞t&gt;0ux,0=2xutx,0=e-x D…

Q: 4. Prove that (À × B) · (B x T) × (T x A) = [ABT)

A: The objective is to prove that

Q: Solve the P.D.E utt = 16 uxz2 s.t u(0,t)=u(5,t)=0 , u:(x,0) = 0, u(x,0)=x² (5 - x) ,h=1,k=0.4

A: See this solution in below

Q: 3) f(x, y, z) Зху %3D z- Vx

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Q: Suppose T(x,y) = (2x, 2y) and U(x,y) = (x, -y). Is T∘U=U∘T?  Justify your answer.

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Q: I ntegrate: Sv V v+2dv HIN T: If u =x+Q, then × = U-a

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Q: If X is a RV with f(x) = 0, when x 0, P(X 2 a) <u/a. %3D

A:

Q: where: %3D Solue ulX, o)=2 X •a u (0,t)=o 1 ノ

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Q: Find Duf at P. 2. f(x, y, z) = 6x°y°z*; P(1, – 2, 1); u = i+ 1 Duf = i

A: We need to find directional derivative at p.

Q: f(L,K) + x [loo-PeL-Pak] max dQ

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 ,…