For a certain fluid flow field the velocity is given by V = [y² – x(1+ x)]i+ [y(2x + 1)]j+k. Show that fluid flow is irrational.

Q: Consider the function f(x, y) = In(x² + y² + 1). 2 (a) Find Vf(1, 2). (b) Find Daf(1, 2) in the…

A: f(x,y)=ln(x2+y2+1) (a)Gradient vector…

Q: Find a direction in the xy-plane in which f(x, y) = e¹-xy has zero change at the point P(1.0. e).

A: We know directional derivative formula Du→fa, b, c=∇fa, b, c.u^ Lets say u→=cosθ, sinθ be the…

Q: Consider the function f(r, y) = 21 +ry² – 6ry + 5x+2. (a) Show that (1,3) is a stationary point of…

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Q: Let F(t) = (- t + 1, t – 2t", - 2t). Find the line (L) tangent to r (t) at the point (- 15, – 24, –…

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Q: r(t) = 2t²1 - 4tj + 2t³k (a) r(t) (b) r"(t) (c) r(t)r"(t) (d) r(t) x r"(t)

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Q: Find R(t) if R′(t)=⟨sint3,cost2⟩ and R(π)=⟨12,12⟩.

A: According to the given information it is required to find R(t).

Q: B- Let r(t) = (²+2) i + (3t) j .Find (t) such that r(t) parallel to r'(t).

A:

Q: Find a function f such that Vf = 2e²x cos(5y)i - 5e²x sin(5y)j + z³k. f(x, y, z) =

A: Given :-  = 2e2x.cos(5y)i - 5e2xsin(5y) j + z5kNow,

Q: Find the extremum of f(x,y) = 6x²y² +4y² + 4x².

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Q: Evaluate the value of I = f(e6 − 3x)dx + (6xeº + y)dy, r where I is the straight line from (0, 0) to…

A: On the curve Gamma , which is straight line from (0, 0) to (3,0) we have, y=0 and x  ranges from 0…

Q: Given f(x, y) = 4x°y+ 7xy*. Compute:

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Q: Consider f(x, y) = (2x² + xy + 1, xy, x² + y²) and let c = (1,-1). Find Df(c) by definition.

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Q: If f(x, y) = V + y3, find fy(0, 5).

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Q: Find the centroid of the thin plate bounded by the graphs of g(x) = x² and f(x) = -x + 30. Use the…

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Q: Write the Linear Approximation to f(x, y) = x(1 + y)−¹ at (a, b) = (8, 1) in the form f(a+h,b+k)≈…

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Q: 5. Find the linear approximation of f (x, y, z) = x³y² + 22 at the point (2, 3,4). Use this to…

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Q: Suppose that f(x, y) = 2x² + 2y¹ − 2xy Then the minimum is

A: Given function: f(x,y)= 2x4+2y4-2xy partially differentiating f w.r.t. x, ∂f∂x= 8x3-2y partially…

Q: Find f, and f, for f(x,y)=y In (2x+7y).

A: F(x,y) = y ln(2x + 7y) To find : fx and fy

Q: 3. A velocity field v= ui + vj + wk is given as: u = x+2y+3z + 4r°,v= xyz +1 and w = (x+y)z + 21.…

A: u=x+2y+3z+4t2 a1 = ∂u∂t+u∂u∂x     =8t +u    = 16+22    =38 v=xyz+t a2=∂v∂t+v∂v∂y =1+vxz =1+3 =4

Q: Find a function f such that Vf = 2e²x cos(5y)i - 5e²* sin(5y)j + z5k. f(x, y, z) =

A: The given gradient function

Q: Let 7(t) = (- 5t + 2, t² – 4t³, 3tª). Find the line (L) tangent to 7(t) at the point (7, 5, 3) L:(x,…

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Q: Suppose ƒ(x, y, z) = ln(5x² + 6y + 5z²). Find an equation for the level surface of f that passes…

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Q: Estimate zx (0.50, 0) and zx (0, 1.0) and zy (0, 1.0) from the contour diagram for z (x, y) in the…

A: z(0.50,0) = 1z(0.75,0) = 2z(0.50,1) = 1z(0,1) = 1z(0,1.2) = 3z(0,1) = 1

Q: A particle moves in the xy plane so that at any time -2 ≤ t ≤ 2, x = 3 sint and y = t - 4 cost + 1.…

A: A particle moves in the xy plane so that at any time ,  and .What is the vertical component of the…