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

Q: Q2/ find dz ond dZ as function ne U and v if of Z=tan (중), X=u Cos V , 9-u sinV

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Q: Iffis of class c? and z= f(y.x), con u= x°y°,v = 5x+1. Verify that| :4xy kxy+ 4x³y° + 10xy- af дхду…

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Q: Q:Ifx = sin 6, y = 0,y= cos p6, prove that (1-x )y, xy,+ p°y= 0, dy and y1 d dy where y, %3! %3! de

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Q: dw dw and ds 3 Assume that w =f(s° +t) and f'(x) = ex. Find dw at dw do II

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Q: dz Find when u= 0, v= 1 if z= sin(xy) + ysin(x), x = uv, y3Du² +v?. du

A: Given  z = sin(xy) +ysin(x)              x = uv and y = u2 + v2  Substituting x and y in z we have…

Q: Find af af and f(x, y) = etry In(6y) af 4yln(6y)e^(4xy) af 1/ye^(4xy)+4xIn(6y)e^4xy dy f(x, y) = cos…

A: In the first part the answer is correct but you have mistakenly not taken bracket for 1/y . You can…

Q: in term of t, if w -+, x= cos t,y sint and z in term of t. if w = xy, x = cost and y = sint, whent=.…

A: 4. Given data is : w=xz+yz , x=cos2t , y=sin2t  and z=1t To express dwdt in terms of t

Q: If z = f(x, y), where x = r cos 0 and y = r sin 0, show that a²z 1 a°z 1 əz r дr дх? dy? дr?

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Q: Find ▼ · (V × F) if F(x, y, z) = 8 sin xi + 8 cos(8x − 9y)j + 10zk. V · (V × F) = [

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Q: f(r,t) = cos(3r) exp(4t) %3D fi(x,t)+ frz(x,t) = f(r,t) %3D

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Q: If z = cos (r + cos y), show that az Fz ar ərðy dy dr

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Q: Prove that A <m AUB where AUB = {0x | r E A}U{ly|y € B}.

A: Solution: Let us consider two sets A and B. It may be possible that A∩B≠∅. But we want to construct…

Q: dw when r= 3 and s = - ar 3 if w = (x+ y + z)“, x=r-s, y= cos (r + s), z= sin (r + s). Find

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Q: Find V · (V × F) if F(x, y, z) = 7 sin xi + 10 cos(4x – 5y)j+ 5zk.

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Q: If z= t(x,y), where X = rCOse and y= rsine, %3D Show that 12 dz 12 dz dz dr dz del + dx

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Q: where yco,t) = y (5,t) = = 2Jx2 and y cx, o) y (X,o) = 10 sin 4T X

A: We solve the differential equation by the method of saperation of variable. In this method we…

Q: Find ▼ · (V × F) if F(x, y, z) = sin xi + 7 cos(9x - 10y)j + 5zk.

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Q: 3. Given: y = f(u) and u = g(x), find dy/dx: а) у %3D cos u, и%3D—х/3 b) у %3D tan u, и%3D 10х — 5…

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Q: π/3 So, f²ef/³ t In u tans ds du dt

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Q: QS A) Find IF y = - Ve IF y =-Vt and t sinx

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Q: 6. If w=xy + yz + zx. x =r cos0, y = r sin0, z= r0, Find dw/dr, dw/dO, when r = 2,0 =x / 2.

A: Answer

Q: du Q. Show that du and dy for u = In Vx2 + y? %3D dy and v = tan

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Q: Be w = (u2 + v2)3/2, where u = e-t sin(θ), v = e-t cos(θ), calculate: ∂w/∂t and ∂w/∂θ

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Q: Q2/ Find ow/or if w = (x+y+z)³, x=r-s, y = cos(rs), z = sin(r + s) at the point (r, s) = (0,1)

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Q: Given f(x, y) = 2e" sin(4y) v (0, then V %3D 2

A: THE EXPLANATION HAS BEEN GIVEN IN THE ATTACHED FILE.

Q: a²f If f(x,y) = e" sin (r + y), then prove that дхду дудх %3D

A: Hi! You have posted two non-related subparts. As nothing is specified, we are answering only the…

Q: az Find when u = 0 and v dv 117 if z(x, y) = sin x + cos y, x = u •v, and y =u + v.

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Q: If y= Vx (x-1), find dy (Use rules, not a XP de finition

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Q: 01. If w f(u, v), u = cosx and v = sinx, show that: d'w dx2 = v? fuu + u? frv - 2uvfur - ufu - vf,

A: Given w=f(u,v), u=cos x and v=sin xso we see that dudx=-sin x =-v   and dvdx=cos x=u.By chain rule,…

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

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Q: If u- f(x) where y=J x+y%*34 show thal do

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Q: Q1. If f(x,y, z) = cosh?(Vx² In y) + tan-(e2y²vz) + sec³(sin vVE In x²) Find fx, fy, fz» fyx

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Q: Let f(x, y) = 2e sin(4) of д х of dy Check Answer

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Q: Q:Ifx= sin 6, y = 0,y = cos pe, prove that (1-x )y,-xy,+p°y = 0, %3D dy where y, dy and y1 d

A: Here given that x=sinθ, y=cospθ Taking derivative of both x &amp; y with respect to θ, we get…

Q: Suppos analyts s? show work. you

A: Given that f is analytic. And z≤1 And f(0)=0 And f(5)≤5 The formula for derivative is,…

Q: 2- FIND f (t) or fis) AS APPRO PRIATE A) L}Je costdt COS ) y+2メ=8(t-)0)00)=イ

A: Here we use basic definition of Laplace transform

Q: dw If w = x2y sinz - 3zey, find dz 6. O a. dw =x?y cosz - 3exy dz O b. dw =2x2y cosz- exy dz O c. dw…

A: Find the partial derivative of w  with respect to z

Q: So Evaluate F. dr where C F = (– 2z, – y, – 4x), and C is given by 7(t) = (sin(t), cos(t), t), 0 < t…

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Q: Given f(x, y) = 7æ° cos (y*), find .9 COS fzy(x, y) = fyy (x, y) = %3D

A: Given  f(x, y) = 7x9 cos(y8) To find fxy, fyy

Q: For u = ex cos y,(a) verify that ∂2u/∂x∂y = ∂2u/∂y∂x;(b) verify that ∂2u/∂x2 + ∂2u/∂y2 = 0.

A: The given  function is, u=excosy (a) To verify ∂2u∂x∂y=∂2u∂y∂x Solution: We have u=excosy…

Q: Find V · (V × F) if F(x, y, z) = 10 sin ci + 4 cos(10x – 11y)j + 6zk.

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Q: Qz: IF U=h cx,g )= Lnce+ ě).show that %3D Sy

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Q: Q5: Find W,e if W = f(x,y), xr cos6 and y =r sino, then prove that r Wre =-yf + xfy+ xy(fyy-fex) +…

A: The given problem is to find the double partial derivative of the given multi variable function and…

Q: for w = x sin y +y sin x + xy. W yx Q2) A) Verify that wry

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Q: 1. Differentiate F(t)= e4t° sin sin 2t [Hint: y = e", u = 4t³ sin 2t.]

A: In this question we have to solve the differentiation of the function.

Q: Find LT for f(t) = sin²(wt), f(t) = cos²(wt) %3D %3D

A: We need to find the Laplace transform of ft=sin2wt and ft=cos2wt Now, before finding the Laplace…