Evaluate f(x, y) dS. f(x, y) = xy S: r(u, v) = 4 cos u i + 4 sin u j + vk 0

Q: Use an appropriate form of the chain rule to find dz/dt 1. z = 3x?y; x = t and y = t2 2. z = 3 cos x…

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Q: Let f(x, y, z) = ln () Vxy – sin (cos (zy)) , evaluate ху %3D дх ду

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Q: Let I = f x*cos(x) dx. The approximation of I using the two-point Gaussian quadrature formula is:

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Q: d Calculate f (r(t))| dt , where l1=x/2 f(x, y, z) = xy + z? and r(t) = (cos t, sin t, t)

A: Given, fx,y,z=xy+z2  and  r(t)=cost,sint,t To find ddtf(r(t))t=π2

Q: Let /:R'R. S(x. y.2) = (e' tan y,x' In y). Compute D(f).

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Q: Find Wr 0 if W = f(x,y) x=r cos 0 and y = r sin 0, then prove that r Wr 0 =yfx + xfy + xy(fyy-fxx) +…

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Q: Calculate dw/du and dwldv using the following functions: = f(x, y, z) = 3x² – 2ry + 4z2 = x(u, v) =…

A: Apply chain rule

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: Given that F(x, y, z)=sin xi+ cos yj + xzk. Evaluate ſ. F.dr where C is given by r(t)=( r', –f,t) ;…

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Q: Q2) A) Verify that wxy wyx for w x sin y+y sin x +xy. %3D R) Show that f(x v) = e-2y cos 2r…

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Q: (r, y, z) = sin 2x i + xz cos y j+ e*y iv F,

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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: dw If w = f(x,y,z) = In(x² +y² z?),x = cost y= sin t ,z = 4VE , then is : dt + ww.

A: Given  w = f(x,y,z) = ln(x2 + y2 + z2) x = cos(t) , y = sin(t) and z = 4t Substituting the values of…

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: Find u+ Hy:given that u(x,y) s tan(x+ux+ 11y") 11- x2-y?

A: The partial derivative of a multivariable function with respect to a variable is the derivative of a…

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: af du (a) f(x, y) = sin-¹ (b) f(x, y): = x+y (c) f(x, y) = In ( t = 1 and u = 2 af and where Ət 2.…

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Q: Find the work done by force F over the given paths in the direction of increasing t. F = zi + xj +…

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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: Find the holomorphic function f (2) = u(x, y) + iv(x, y) (z = x + iy) with f(0) = 0, given that v(x,…

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Q: Integrate f(r, y, z) = for 0 <t< V3. + y² over the curve r(t) = [cos(t) +t sin(t), sin(t) – t…

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Q: Find f, fy, and f. f (r, y, z) = z In (a*y® cos(2z)) f%3= fy = %3D fz =

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Q: The function f(z) =e² is: O contiuous in C discontiuous in C Ou(x,y) =e*cos(y) Ou(x,y)=e*sin(y)

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

A: Follow the steps.

Q: 22. u = Vr² + s², r = y + x cos t, s=x + y sin t; ди ди ди əx’dy when x = 1, y = 2, t = 0 dt

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Q: /1+x² + y² dS where S is the helicoid: r(u, v) = u cos(v)i+ u sin(v)j + vk, with 0 < u < 5,0 < v< 4n…

A: Let I = ∫∫S1+x2+y2dS and r(u,v)=u cos(v)i +u sin(v)j+vk We know that dS=ru×rv du dv Consider ru =…

Q: Find fæ, fy, and fz. f(z, y, z) = z In(a"y cos(5z) 4.3 = z In (x*y° cos(52)) fr = fy = f: =

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Q: Give the function z f(x, y)=4x²+ 3y, x(t)=sin tand y (t)= cos t. zp =2sin t.cos t 2.1 Show that dt…

A: The derivative of the function can be calculated by differentiating the function with respect to the…

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

A: Our aim is to find the derivative of the above expressions.

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: Given the function f(x, y) = y + e**y, where x = In(s) and y = sin(s2t), find the partial %3D af af…

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Q: Given f(z, y) = 7x° cos (y*), find fzy(z, y) = | fyy(T, y) = [

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Q: Integrate f(x, y)= y cos xy over the rectangular 0 sxsm, 0 s ys 1*

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Q: Let f (x,y) = sin(xy) + x² ln(y). fyx at (0,) is equal to: %3D А. 33 В. О С. 3 D. 1

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

A: We know, ∂2f∂y∂x=∂∂y∂f∂x

Q: %3D a) Find fyxyx for f(x,y) = In(xy) – cos (xy) at (1,t).

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Q: at at 4) Let t = ,u = x² – y² ,v = 4xy³. Find· ay əx

A: To find: The value of ∂t∂x, ∂t∂y. Given information: t=uv, u=x2-y2 and v=4xy3 Concept used: Quotient…

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…

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

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Q: does not exist. (b) If u = x cos y+ye", where x=s+2r and y = st , prove that ди ди dt = (s – 21) (x²…

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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: 3xy dV, where E is bounded by the parabolic cylinders y = x² and x = y and the planes z = 0 and z =…

A: It is given that, y=x2 and x=y2 and the planes z=0 and z=7x+y. Define the region E as follows.…

Q: Calculate Sf(F(t)) | (t)) dt f √ √ √x + 1 (x+y+z) ds where C is the helix r(t) = (cos(t), sin(t),…

A: What is Line Integral: A line integral in mathematics is an integral in which the function to be…

Q: Find of /dy if f(x, y) = y sin(xy) 1 + xycos(xy)csc(xy) csc(xy) B) xy cos(xy) + sin(x) A) O

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Q: Let f(r, y) = x sin ry. Fi (a) fz(x,y) (b) fy(r, y) Sy(x, y) (c) fyy(", y) YY (d) fyr(r, y)

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Q: The function f(z)=e² is: „(x.y) =e*sin(y) O contiuous in CO u(x.y) =e*cos(y) O discontiuous in CO

A: Use the property of complex function to break in real and imaginary part. Here also use the concept…