Let f(x, y, z) = x2y + y2z. Use the Chain Rule to calculate af /as and af /at (in terms of sand t), where x=s+1, y=st, z=2s-1

Q: Let f € L'(0,1). Assume that f(t)t2"dt = 0 for all n f = 0 a.e. 1,2,3,.... Prove that %3D

A:

Q: Use the Chain Rule to calculate f(r(t)). dt f(x, y) = x³ - 5xy, r(t) = (cos(4t), sin(5t)), t = 0…

A:

Q: Let w = f(r.0) where r and 0 are functions of s and t, then according to the chain rule is equal to…

A:

Q: Let f(x, y,z) = x² sin y +y e, where x t +3,y = net-1 and z = In 2t. %3D at t = 1 is: dt O -12n O…

A:

Q: Consider the functions z =2e* In y, x In (u cos v), and y= u sin v. dz (a) Express and du dz as…

A: Since you have posted a question with multiple sub-parts, we will solve first two subparts for you.…

Q: Let F(x, y, z) = 6x?i – cos(xy)(i + 5). Calulate the divergence: div F(x, y, z) =

A:

Q: Let x = s e and y =t es. Compute the Jacobian

A: Given: x=set and y=tes.

Q: f(x, y, z) =(x²+y²+z) 2

A:

Q: Use the Chain Rule to find az/as and az/at. z = e" cos(0), r = st, 0 = V s8 + t8 az as az at

A:

Q: Let g(r, y) = ye-, r(t) = In(t² + 1), and y(t) = vt. Find g(r(t), y(t)).

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

Q: Let L(z, y) f(1, y) at (1, 1) . Then f-(1,1) = = y-2x +3 be the local linear approximation to O a. 3…

A: Topic:- partial differentiation

Q: Consider the functions z = 4 e* In y, x= In (u cos v), and y =u sin v. dz and du dz (a) Express as…

A: Given, z=4exlny,  x=lnucosv,  y=usinv Part (a): z=4exlny,  x=lnucosv,…

Q: By using the multivariable chain rule, compute each of the following deriva- ves. a) dz/dt and…

A: b Given: f(x,y)=exy...(1) x=s-t .....(ii)y=2st .....(iii) By 1 ∂f∂x=yexy, ∂f∂y=xexy By ii and iii…

Q: Find a function v(x, y) such that, given u = 2x(1– y), f(z) = u + jv is analytic in z.

A:

Q: Use the Chain Rule to find ôz/ðs and ôz/ôt. z = tan(u/v), u = 9s + 8t, v = 8s – 9t as az at

A:

Q: find laplace of f(x,y,z)=2x3y2z2

A:

Q: Let z = g (x, y) = x³ + xy³ + y4 - 6x - 2y + 4, where x = r cos (0) and y дz Use the chain rule to…

A: We will use chain rule to find the derivative

Q: 3) Let F = (y+ev=,8x+ cos(y*)) Use a theorem to compute ſ F •dT where c is the boundary of the…

A: Given that  F→=y+ex,8x+cosy3 To compute ∫cF→⋅dr→ where c is the boundary of the planar region…

Q: Let u = cos(r + y), x = 2st +t and y = t – 2s. Use the chain rule theorem to compute du for du and…

A:

Q: Let f(x, y) xy, x = 1- Vt, and y = 1+ Vt. Find df using the chain rule. dt

A:

Q: Compute d/dt f (r(t)) at the given value of t for f (x, y, z) = xz − y^2, r(t) =〈t, t^3, 1 − t〉at t…

A: Given: The function fx,y,z=xz-y2and rt=t,t2,1-t, To find: ddtfrt at t=-2,

Q: 17. Integrate f(x, y, z) = (x + y + z)/(x² + y² + z²) over the path r(t) = ti + tj + tk, 0 < a < t<…

A:

Q: Evaluate O F.dT where +(3y %3D y + 1 and C is the triangle with vertices: (0,0), (6,0), (3, 5).

A:

Q: Let w(z, y, z) = 1 + y² + z?, where z = 3', y = Int, and z = = et. Find I using the Chain Rule.

A:

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

A: Given data: *z=tan(uv) *u=8s+3t and v=3s-8t The partial derivative of z with respect to s is given…

Q: Use the Chain Rule to find dz/ðs and əz/ət. z = tan(u/v), u = 5s + 8t, v = 8s – 5t %3D az %3D as az…

A: We have to use chain rule to find the partial derivatives of z.

Q: Let f(x, y) = 8x + 2y2, r(t) = (cos(t), sin(t)). Use the Chain Rule to calculate f(r(t)). !! dt…

A:

Q: se the chain rule to compute and at (u, v) = (1, I) where f(x, y, z) = x²y + y²z + z²x and r(u, v) =…

A: Given: fx,y,z=x2y+y2z+z2xr→u,v=eucosv,eusinv,u2u,v=1,π4 To find: ∂f∘r→∂u=?∂f∘r→∂v=? Concept used:…

Q: 2. Use the Chain Rule to find for z = e¤/y where x = st and y = s2 +t

A:

Q: Use the Laplace transform to solve Utt – Uxr = 100, x > 0, t > 0. u (x, 0) = u4 (x, 0) = u (0, t) =…

A:

Q: The signal (t) = tan(t) satisfies the Dirichlet conditions. O True O False

A: False

Q: Express dw/ar and aw/as as functions of r and s using chain rule w = In(x² + y² + z²), x = re sin r,…

A:

Q: If w = x²y - y²z³,x = rCost,y = rSint, dw z = re' , (r = 2)and(t = 0)then dt

A: Given equations - w = x2y-y2z3       ...............(1)x = rcost        ............(2)y = rsint…

Q: Let z = f(x, y) where g = e° cos t and y = e sin t. Use the Chain Rule to show: az, a z = e

A:

Q: Suppose that w = f(x,y, z), x = h(s), y= g(s, t), z= k(s,t) and all the conditions required for the…

A: Given that w=fx,y,z, x=hs, y=gs,t and z=ks,t. We have to compute ∂w∂t by using chain rule. Since, by…

Q: Find the divergence of F. F(x, y, z) = 8xi + 8y²j + 8z²k

A: Given,           Fx,y,z=8x2i+8y2j+8z2k

Q: 2. Let z = sin(xy) + x siny,x=u² + w² and y = uw. 2a. Compute Vz= when u = 0 and w = 1.

A:

Q: Consider f(t) = (cos t, sin t, t²) and g(x,y,z) = (x² + y² + z², 3x + 4y –- 12z) and h = g(f(t)) =…

A:

Q: Show that F(x, y, z) = (y² – e* sin z)i + (2xy + z)j + (y – e* cos z)k is conservative and hence…

A:

Q: If f(z) = sin(z 2) = U(x, y) + iV (x, y), determine U and V . Verify that the C-R conditions are met…

A:

Q: Let F(x, y, z) = 4z²ei+ (y³ + tan(z))j+ (4x²z – 3y²)k. Use the Divergence Theorem to evaluate f, F .…

A: Given,       F(x, y, z)=4z2xi+43y3+tan(z)j+(4x2z-3y2)k use the divergence theorem to evaluate…

Q: Let z = f(x, y) where g = e° cos t and y = e° sin t. Use the Chain Rule to show: z = e

A: Let z = f(x, y) where x = es cos t and y = es  sin t. We will be using the chain rule to show,…

Q: Use the Chain Rule to find ôz/ds and ôz/ðt. z = tan(u/v), u = 5s + 9t, V = 9s – 5t az as az at

A:

Q: Use the Chain Rule to find ôz/ðs and dz/dt. z = tan(u/v), u = 3s + 6t, v = 6s – 3t dz %3D as Ot

A:

Q: Let c(t) Use the chain rule to compute (foc)'(t). (cos(t), sin(t), t²) and f(x, y, z) = x² + y² +…

A: The objective is to find the derivative.

Q: (a) Let F z sin y, where r(s,t) = st and y(s, t) = st. Use the chain rule to find %3D %3! OF

A: