f z = f (x, y), where x = r cos 0 and y = r sin 0, show that a2 z 1 a²z 1 dz ar2 ar2 p2 aA2 r ar

Q: Continuity for Three VariablesAt what points (x, y, z) in space are the functions in Exercises 35–40…

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Q: Find Fdr F = cos(x+ z)i + sin(y + z) j + sin(z + x) k r = ti – 3tj + tk

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Q: Obtain the general solution using separation of variables. dz = a(cos 0 dz + z sin 0 d0) %3D

A: Separation of variables.

Q: Determine the solution of the differential de +y cot x = cosx O a. ycos I : sin'z +e o b. ycos x =…

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Q: If u(x, y) = e-2* sin(2y) and f(x) = u(x, y) + iv(x, y) is analytic, then v(x, y) = %3D %3D

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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: Let v(x, y) = cos(4x)sinh(4y). Find f(z)with Im(f(z)) = v(x, y). I

A: Given function is v(x,y)

Q: For F = (x sin y, tan(xz), ye-²), find rot F. A. (-x sec?(xz) – e²zy¯-1,0, z sec²(xz) – x cos y) В.…

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Q: If f(z, y) = 22e cos z+9 cot y, then the total differential of f at (0, ) (0. is: O df = 22 dz + 18…

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Q: af Find of f (x, y) = sin %3D dx 1+ Y COS 1+y () Option 1 sin 1+ y 1+ y Ontion 4

A: Partial derivative of a function in several variables with respect to one variable is calculated by…

Q: (0. ) : If f(x, y) = 12e cos x + 3 cot y, then the total differential of f at is : J, df = -12 dx 6…

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Q: = f (x, y) is differentiable and x=u cos v and y =u sin v. Show that z = Oz Oz 1 + ди + и

A: According to question given that z=f(x,y) is differentiable x=u cos(v), y=u sin(v)

Q: Calculate ôz/ðu and dz/ðv given the following functions: 2x – Y z = f(x, y) = x(u, v) = e²u cos 3v,…

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Q: F. dr, where F(x, y, z) = y sin zi + z sin æj + x sin yk and r(t) = cos ti + sin tj + sin 5tk, 0 <t…

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Q: dz Find when u= 1, v=0 if z= cos(xy) – xsin(y), x= u² + v2, y= uv. dv A) 2 B -1 D) 1 E -2

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Q: sin x S ft) dt = Choose... 4. If f(t) dt = c, then / f(t) dt = Choose...

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Q: f(x, y, z, w) = (1+x+ sin(z − 2y), ez-w, 2z+tan(w+x²)).

A: we have to find the approximation of ƒ at the point P = (0, 0, 0, 0).  For this we use the proper…

Q: Q1:- dz at t=al4 at Calculate z = f(x, y) : 4x² + 3y?, x = x(t) = sin t, y = y(t) = cos t %3D

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Q: f(z) = Cosz %3D ,0< Izl f(2) = sinz 3D 0< zl %3D f(z) = e-V, 0< Izl %3D

A: Note: As per our company guidelines we are supposed to answer the first question only. Kindly ask…

Q: aw Find for w e" sin p where ax p = sin x q = z2 In y x- at (x,y,z)3글) %3D z2 e sin-1 x = Hint: dx…

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Q: find ow/ar if w = (x + y + z)², ar www w x = r – s, y = cos(r + s), z = sin(r + s) at the point (r,…

A: See the attachment

Q: (w = xy + z, x = cos(t), y = sin(t), z = t)

A: Follow the steps.

Q: Find the potential P(x,y) attached to the exact ODE 2x sin(y) +x² cos(y) y' = 0

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Q: b) Find the potential P(x,y) attached to the exact ODE 2x sin(y) + x² cos(y) y' = 0 %3D

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Q: dy z = 3e*S, y = 3z2 + sin(z) S i5I dx %3D

A: Given,           z=3ex and y=3z2+sinz

Q: If f(x, y) = 19e CoS I +7 cot y, then the total differential of fat (0.) is: 4 O df= 19 dr + 14 dy…

A: Let the function is f(x, y)=19e-xcos x+7cot y Total differential is defined as df=fx(x, y) dx+fy(x,…

Q: fze [sin(3x)]-m(x, y, z), where m(x, y, z) = Part C. yz-1 %3D

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Q: Determine if F=(e* cos y + yz)i + (xz-e* siny) j+(xy+z)k is conservative. If it is conservative,…

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Q: (e) z = x? + 2y2 where x = sin(t), y = 3sin(t) and t = dz dz answer: dt 19sin(2t) , = 19 dtlt=! (f)…

A: (e) Given z=x2+2y2 where x=sin(t),y=3sin(t) and t=π4. (f) z=cos(x)sin(y) where x=πt,y=2πt+π2 and t=3…

Q: (2y sin a cos x – y + 2y²e#v° )dx – (x – sin² æ – 4xye*v* )dy = 0 Let M be the coefficient of dr and…

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Q: Utt || for 0 < x < T u(x, 0) = x? u:(x, 0) = }cos(8x) Uz(0, t) = u„(r, t) = 0

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Q: f(x, y) =- sin xcos y, then

A: The given function is: f(x,y)=-sin xcos y To find ∂2f∂x2. Here, note that y is constant w.r.t. x.

Q: If f(r, y) = 22e cos z +2 cot y, then the total differential of f at (0, ) %3D is: O df = 22 da +4…

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Q: A)lf y = sin¯¹ (3) prove Zyy + z = 0 B)Z = 1+ Ini, find [21.

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Q: Find a potential function for F(x, y, z) = (y² cos z, 2xy cos z, –ry² sin z). %3D COS

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

A: Given  w = xsin(y)+ ysin(x)+ xy We need to verify that, wxy= wyx

Q: II. Let f(x, y, z) = z° sin a + Determine 2 dydzərəz'

A: We know, ∂2f∂y∂x=∂∂y∂f∂x

Q: Given f(x,y,z) = 7xyz where x = sin(2t), y = cos(2t) and z = -1t. Find df/dt at t = -1. %3D Give…

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Q: Exercise 5 Normal. , e-x² dx = \TT -00 Use 1? = (Se-x* dx)( S,e-y² dy) = L. Le-(x²+y²) dx dy = Sp2…

A: Find the valve of the integral

Q: S(t) dt = In(x) cos(r) + sin(3) e-² sin(3) "S(t) dt, find f(r). ven

A: The expression is given by ∫sin(3)xftdt = ln(x)cos(x)+∫sin(3)xe-t2ftdt                            To…

Q: Calculate dz/du and dz/dv given the following functions: 2x-y x + 3y' = f(x, y) = = Z = x(u, v) =…

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Q: dv dv If V = ÷f(0) and x = r · cos 0 , y = r · sin 0 . Prove that x -+y÷= -V dr dy

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

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Q: Given that F(x, y, z)=sin x i+cos yj+ xzk. Evaluate , F.dr where C is given by r(t)={r',-f,t) :…

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Q: aw (a) Calculate ди aw and dv using the following functions: x + 2y – 4z = f(x,y,z) = 2х — у + 32 x…

A: Partial derivative of multivariable function with respect to any of the variable is obtained by…

Q: If f(x, y) the total differential of f at (0, ) i 16e Cos x + 7 cot y, then the total differential…

A: Total differential