ycos () - xsin( satisfies the Laplace equuation and boundary conditions u(x, y) = 0 on y = o (x 2 0)(2-Ju(x, y) = On x = 0 (0 < y< 1),4- A- Show that the function u(x, y) = e x/2 %3D %3D %3D ny -xe 2 on y 1(x 2 0),3 u(x, y) ycos u(x, y) - 0 as x → 00.

Q: Let y be the curve with vector form r(t) = (1/(t+ 2), sin(sin(sin t)), e-"). a) Determine whether…

A: The parametric curve is r(t)=⟨1t+2, sin( sin( sin t)),e-t⟩. x(t)=1t+2, y(t)=sin( sin( sin t)),…

Q: (1 point) Evaluate the line integral y dx + x dy where C is the parameterized path x 1 x = t², y =…

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Q: 2. f(x, y) = (x² + y²)² — 2(x² - y²) + 1. Ans.: global minimum 0 at (±1,0), saddle at (0, 0).

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Q: Flux of F(x,y) = 5 x î +3 y ĵ across the circle x? + y? = 1 (anticlockwise) is - 8 T Can not find.

A: Use the Gauss’ Divergence theorem,: Use the Gauss’ Divergence theorem,

Q: 19. Compute the tangent plane at (1.0, 1) for each of the functions 9) f(x, y, z) = x exp(-x² -…

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Q: ds² = dt² – dz² – (1 + 2h(t − z))dx² − (1 – 2h(t – z))dy² - - how that this is a solution to the…

A: Let us see -  the metric given is a solution to the linearized empty space Einstein equations, we…

Q: b) Let X = Cle,e+ 2] and define T:C[e,e+2] → R by re+2 T(z) = | 1(t)dt, r€ Cle,e+2] Show that T is…

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Q: 8. Change to cylindrical coordinates and find the divergence and curl of (a) F xi + yj x² + y² -yi +…

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Q: 5. Find the linear appoximation for the funciton f(x, y) = (ro, Yo) = (3,0). x-y xy+1 at the point

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Q: (x* + y° ) In (x² + y* ). (x,y)= (0,0) f(x, y) = k +1, (x, y) = (0,0)

A: If f is continuous then lim (x, y) tends to (0, 0) =k+1 And using l hospital rule.

Q: Q7. Q7. Consider the vector-valued function 3ry ra(r, y. :) = (V=(* + s°), ) z(2² + y°), r2(r, y, z)…

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Q: (a) r(u) = e" i+e"j. u € [0, 1]. r(u) = (1- u)i, ue [0. 2].

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Q: Find the curl and divergence of F(r, y, z) = (e**, -2e"², 3.xe"). Please show steps/explain

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Q: x(t) = t², y(t) = t³ – 3t, on – ∞ <t < ∞ %3D а. Show that the parametric curve has two tangent lines…

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Q: (a) Find r()of the vector- valued function (b) Evaluate the integral of the following vector- valued…

A: This is a problem of Vector Calculus.

Q: |x'y" -y': y'a - y'= es K(t) = (x2 + у'?

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Q: y be a real constant with y² 1 for the parametric functional S[x, y)] = ['ª dt [√લ + 2y àý + y² –…

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Q: illustrate that the length of a smooth space curve does not depend on the parameterization used to…

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Q: Consider the function f(x,y) over the domain D enclosed by the 1+ x2 + y² polar curve r =2+…

A: (a) Look at the level curves of the function within the domain D.    The boundary of the domain D…

Q: 1 3. (8 points) Evaluate the contour integral dz where C' is the counterclockwise oriented curve | z…

A: To evaluate the contour integral[∮C​z2−2z1​dz,]where ( C ) is the counterclockwise oriented curve (…

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: Given a function f(x, y, z) = ryz° +z°yln (z) – z'y? + 4, find: of (1,2, 1) = (1, 2, 1) = (1,2, 1) =…

A: Given problem based on partial derivative.

Q: Find the linearization L(x,y) of the function f(x,y)=x²-7xy +6 at Po(2,3). Then find an upper bound…

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Q: (b) Let z(r, y) = exp(-|r|) cos r + y², where r + y /9 = 1. Show that z(r, y) has minimum points…

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Q: 56. Let E(x, y, z) = tan-'(xy + 2z) + zex²-y² + z and let r(t) = (h(t), et²-2t, t²-3t+3) be a…

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Q: I grad(f)•dr where C is graph of y = x² + 3x from (0,0) to (2, 14) and f (x,y) = ev/lx*•1}

A: The function is fx, y=eyx+1. Find: ∫Cgradf·dr…

Q: Consider the function f(x, y) = (eª — 4x) cos(y). Suppose S is the surface z = f(x, y). (a) Find a…

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Q: 2. (a) Let z ry, r =r cost and y=r sin t. Compute by using the chain rule and express your answer in…

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