2) Let u(xy,2) = でさす。R。 () Find He(tいり, "g(1,0,り、 Uy(1,0,1). ニ0. (k) Show that oty a22

Q: Compute Div(F) and Curl(F): F--(4y-1) i+yj+(-3y) k

A:

Q: Compute fx + fxy + fyyzz, where f(x, y, z) = z sin(xy)

A: Given f(x,y,z)= z sin(xy)

Q: 2z + 5z + 14 If U(z) = evaluate u %3D Uz and (z – 1)

A: Solution :

Q: 1) Let f(z,y)- . Compute DS(z,y) where u- (4-9)

A:

Q: Evaluate / (x - 2y + z) ds. S: z = 3 - x, 0sxs 3, 0sys 2

A:

Q: Verify that det A = det B + det C, where Uj + vj U2 + v2 Uz + V3 A = A22 аз1 A32 --EE-E: A12 A22 V2…

A:

Q: 2. Calculate the divergence of D = (÷)[10xyzảy + 5x²zãy + (2z3 – 5x²y)ã,] at P(-2,3,5).

A:

Q: { (0,3), (0,1) is linearly independent in R? ylgn O Ins

A: The given vectors are  (0, 3) and (0, 1) To find whether they are linearly dependent or not

Q: Consider thế iñifial a. 2y"- 30y" + 150y' – 234y = 0, b. y"+ 6ay' + (9a² + 1)y = 0, y(0) = -3, y(0)…

A:

Q: Let, T:R' → R³;T(x,y,z)=(2x+ y, y- z,2y+4z) 2. Find the rank and nullity of T.

A:

Q: show that the scalar t that minimizes ||y– tx||

A: In this question, we need to find the t that minimizes ||y-tx||.

Q: Aşağıdaki alıştırmalarda, f., fy ve fiyi bulun. 23. f(x, y, z) = 1 + xy² – 2z2 25. f(x, y, z) = x -…

A:

Q: Verify f U= n1 V = 234 43 snyz %3D +23- 422-24-42- dependend %3D fundlarelly are

A: We know the identity: x3+y3+z3-3xyz=(x+y+z)(x2+y2+z2-xy-yz-zx) --(1) We are…

Q: 2x+ 3y-z =1 4x + y-3z = 11 3x-2y +5z = 21 | %3D

A: Solution:

Q: Find the work done by the force F = x²yi– xy²j along the paths shown from (1, 1) to (4, 2). (4, 2) a…

A:

Q: Prove thut:f focK, 2) Y is a contimans findion then, g=XxX-> YxY 3 3 n K2) = (F(), PU=) Einction is…

A:

Q: 14 Verify the given linear approximation at a = 0. Then deter-

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Q: 4Ux+Uy=U² , U(x,0)=1/1+x²

A: In the given partial differential equation with the given initial condition. 4ux+uy=u2u(x,0)=11+x2

Q: Compute ∇ƒ(-1, 2, 1), where ƒ(x, y, z) = xy/z.

A: Given,f(x,y,z)=xyz     As we know ∇f(x,y,z)=∂f∂x,∂f∂y,∂f∂z

Q: Compute Jx²z ds where C is the line segment from (0,6,–1) to (4,1,5).

A: The objective is to compute ∫Cx2zds where C is line segment from  0,6,-1 to 4,1,5

Q: 2x+ 3y-z =1 4x + y-3z = 11 3x- 2y +5z =21

A:

Q: Let w = -4xy – 5yz – 4xz, x = st, y = et, z = t Compute (3,4) = as (3,4)%3D at

A:

Q: let z- 57 dy .. dy By chain Ruie

A: The objective is to find the second derivative using chain rule.

Q: Show that if AT Ay = y for all y in Rn , then ATA = In .

A: Given that     ATAy=y  ∀y∈Rn. To prove ATA=In

Q: 2- Prove that: Jzt=27 dz %3D 2in. Whot is the princinle nart for f(z) = ze %3D

A: Since you have asked multiple questions in a single request we would be answering only the first…

Q: Prove that F(z) = 81 8z + 1 is chaotic on (-1,1).

A: To prove: Fx=8x4-8x2+1 is chaotic on -1,1. First plot graph of given function with -1,1 as follows:

Q: Compute ƒxz and ƒzz for ƒ(x, y, z) = xyz - x2 z + yz2.

A: Given that f(x,y,z)=xyz-x2z+yz2. We need to find : i)fxz. ii)fzz.

Q: Let f be q functiom that admits contırous second partial denvatives Df (X.Y)=(4x²- 2q²x+ 2q°y ,4y³ +…

A: Given, the second partial derivatives              ∇f(x, y)=4x3-2a2x+2a2y,  4y3+2a2x-2a2ywith a>0

Q: tnd the kes reetor projection of anto r if

A: Given : u→=5i→+2j→ and v→=6i→-3j→

Q: The gradient vector of the function f defined by f(x,y)= In(1+7x² +5y²) is equal to Select one: O…

A:

Q: Compute the discriminant D(x, y) of ƒ(x, y) = x2y2.

A: fx, y=x2y2 Partially differentiating with respect to x, we get ∂f∂x=2xy2 Partially differentiating…

Q: 2. Compute Ja F · ds for F c(t) = (t , In(1+ t²), e*) for 0 <t < 1 VV, where V(r, y, z) = 4x² In(1+…

A: Find your answer below

Q: Assume that u = U (x,y) iu differentiible and x =r ca Ô and y=rsin o eapress 2. entinely in termv 0…

A:

Q: s the linear operator T : R3 --> R3 defined by T (x, y, z) = (2x - 3y + 7z, 5y + z, -z)…

A:

Q: Let F I2). Evaluate JJaw as z² + y? 0

A: In the given question, the concept of the Divergence Theorem is applied. Divergence Theorem Assume S…

Q: Utu = c²uxx , u(0, t) = 0, u(1, t) = 0, u(x,0) = 0 U¿(x,0) = x² – 2x, 00 t> 0 t > 0 0sx<1.

A: Given P.D.E

Q: bot A de groutne conitant. est Cheafuntis uh that f cark3 and f'4)=-s. g(xl=In (f(x)) + A conx)

A:

Q: Q2:show that the force Fexyzi-x4z'k is Conservative. ?

A:

Q: In Exercises 47-50, verify that wy Wyx- 48. w e+ x In y + y Inx

A: To find the second partial derivative of any function z i.e., zxy first differentiate z partially…

Q: If (x* – y*,x – z°) = 0,z = z(x, y), prove: az y3 az +x³z4 ax - %3D ду 3

A:

Q: If z = f(u) + g), where u = = and v = x· y. Prove that y x: Zx - y Zy = y? Zyy – x2 - Zxx using the…

A: We have to prove this

Q: b) Verify uxy = Uyx , if u = In /x² + y² . %3D %3D

A: Consider the given function. u=lnx2+y2 To verify the equation, find the derivative first with…

Q: 4. Find the work done by the force F(r,y,2) = (x,-z,2y) in moving straight from (0,0,0) to (1.1.0),…

A:

Q: Uyy=4Uxx , U(0,y)=U(5,y)=U(x,0)=0 Uy(x,0)=3sin(2pi x)x)-2sin(5pi x), y>0

A:

Q: 4) Let f(r, y, z) = In(r + 2y + 3z). Find Vf = . %3D

A: Given: f(x, y, z)=ln (x+2y+3z)

Q: Evaluate f(x, y, z) ds. Js f(x, y, z) = x2 + y2 + z2 S: x + y = 36, 0 SxS 6, OszS X %3D !3!

A: To evaluate:

Q: Let z = f(x, y) and r =r cos 0, y = r sin 0. Use the chain rule to express: af %3D 11. af dy Te and…

A:

Q: Compute the convolution y(n) = x(n) * h(n) , Where x(n) = {1,1,0,1,1} and h(n) = {1,-2,-3,4} , Using…

A: Given: Given convolution expression is Y(n)=x(n)*h(n) x(n)={1,1,0,1,1} h(n)={1,-2,-3,4}