The motion of a fluid particle is defined parametrically as follows, r(t) = sinh(t), and y(t) = cosh(t). Show that d²y dr? 1

Q: Find the work done by the force F = zi + xj+ yk in moving a particle along the curve x = t y = t²…

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Q: Evaluate the unoriented line integral (22 + y° + z²) ds (2x² along the parametric curve a : [0, 1]…

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Q: Suppose (X(t))t>o is a standard Brownian motion. Let Y(t) = eX(t). Find (a) E(Y(t)). (b) Cov(Y (s),…

A: A standard Brownian motion Xtt≥0 is a Gaussian process N0,t that satisfies the following properties:…

Q: Evaluate the line integral rye dy where C has parametric equations y = t 2 = x =t 0st<1.

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Q: ди ди ду ду Find the cauchy-riemann equations of, f (z) = | z | sin z. əx' dy'dx' ðy,

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1. In…

Q: For which regions of flow is the Laplace equation ∇-›2? = 0 applicable? (a) Irrotational (b)…

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Q: 3. For all a? + y + 2? > 0, verify that the velocity potential 1 $(r, y, z) = Vr? + y² + z2 is a…

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Q: Determine the acceleration (vector and magnitude) of the fluid particle occupying the point (-2, 1,…

A: Accleration is given by  a→ =DV→Dt= ∂V→∂t+(V.→∇)V→Here, V→ =2xy i^+xz j^+yz k^Hence, ∂V→∂t= ∂2xy…

Q: Find the parametrization of the tangent line to the space curve r(t) = (e" , sin(t), (t + 1)-³) at t…

A: Given that  r(t)=(e^(4t),sin(t),(t+1)^(-3)) at t=0 We have to find out the parameterization of the…

Q: Find the coordinates of the centroid of the curve x = e^t cos t, y = e^t sin t, 0&lt;=t&lt;= pai.

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Q: find a parametrization c(t) of the curve satisfying the given condition.  y = x 2, c(0) = (3, 9)

A: The given function is y=x2 and c(0) =(3,9) Let x(0) =3 and y(0)= 9 Take x =t +x(0) Thus, x = t+3

Q: Find the flow of the velocity field F = x /(y + 1) i + y/( x + 1) j, where velocity is measured in…

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Q: Find the unit tangent vector T(t) and a set of parametric equations for the tangent line  r(t) =…

A: The given space curve rt=e2ti+cost j - sin3t k. Its derivative is r't=2e2ti-sint j - 3cos3t k. At…

Q: Evaluate e dz where C is the line segment from 0 to 2+2i, parametrized by z(t) = t + it?.

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Q: Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0&lt;=t &lt;= pai.

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Q: consider a particle moving on a circular path of radius b described by r(t) = b cos(ωt)i + b…

A: Consider a particle moving on a circular path of radius b given as, rt=bcosωti+bsinωtj Where ω=dudt…

Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: ω=xydx+12ydy To find: I=∫αω  where α:[0,ln2]→ℝ2 with α(t)=(e2t,e-2t) ∴x=e2t,y=e-2t Now,…

Q: Show that ¥(x,t)= Asin(wt+kx) describes a wave motion

A: Given: ψx,t=Asinwt+kx

Q: Let r(t) = (4sin 21)i+ (4cos 2t)j–(e“)k, be the position of particle in space at time . Find the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: valuate the line integral Scy da +a dy where C is the parameterized path a = t2, y = t°, 1<t<4. Scy…

A: We have to find the given integral.

Q: Show that the function z = e* sin y satisfies Laplace's equation a²z a?z = 0 дх? ду?

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Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. (a) Find a vector parametric equation for the parabola y =…

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Q: y = sec t, x = cosec t %3D %3D

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Q: Given the force field F = (2xze*z – 4, cos³z, x²e**z – 3ycos²z sin z) Solve for the work done by F…

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Q: 6. Given the space curve r(t) = (t, 2et, e2t), find the tangential and normal components of…

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Q: Find the work done by force F(x, y) = 4yi + 7xj + (x + y)k in moving an object along curve r(t) =…

A: To find the workdone along the given path

Q: Show that the function f(x, y) = e-2y cos 2.r satisfies the Laplace equation.

A: use partial derivative to check whether it  satisfy the equation or not

Q: Show that the function f(x, y, z) = x² + y? – 2z² satisfies the Laplace equation.

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Q: Find the flow of the velocity field F = (y + z)i + xj - yk, where velocity is measured in meters per…

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Q: Find the circulation of the Fİeld F(x; y) = 3xi+yj around the circle r(t) = (cos6t)i + ( sin6t)j;…

A: We need to find the circulation of field F→=3xi^+yj^  around the circle r→t=cos6ti^+sin6tj^ where…

Q: Consider a particle moving on a circular path of radius b described by r(t) = b cos(wt)i + b…

A: For the given particle moving on a circular path of radius b, we have to answer the following...

Q: Suppose that the polar coordinates of a moving particle at time t are (r(t), 0(t)). Prove that the…

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Q: find a parametrization c(t) of the curve satisfying the given condition. y = x2, c(0) = (3, 9)

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Q: Let W(t) be a standard Brownian motion. Find P(0 0).

A: Let, X1 = W(1) + W(2) X2 = 3W(1) - 2W(2) E(X1) = E[W(1) + W(2)] = 0. Var(X1) = Var[W(1) + W(2)] = 1…

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

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Q: Find the mass and center of mass of a wire of constant density d that lies along the helix r(t) = (2…

A: Given information:

Q: Uo (r² + a?) cos 0. u(r, 0) = ) Hence show that the velocity components of the flow are ди Ur = ax…

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Q: 13. Given the space curve r(t) = (e' cos t , e' sin t , e“), find the tangential and normal…

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Q: dw Let w =xycosz, where x = t°, y =ť, and z = arccost. Find dt

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Q: Calculate the work done by F(x,y) = ⟨2x + ln( 2y2 −1), arctan(y4 +2y +1) + 4xy / (2y2 − 1⟩ along the…

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Q: Find the tangential and normal components of acceleration at the given time t = 0 for the space…

A: It is given that space curve is  rt=et sin t i+et cos t j+etk Find the tangential component and…

Q: Consider the 2nd order equation d²y + Sin dy + y? +y = 0. dt, Choose the right assertion A) the…

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Q: Suppose F(2, y) = -y sin(æ)i + cos(2)j. (a) Find a vector parametric equation for the parabola y =…

A: Given F→(x,y) = -ysin(x)i→+cos(x)j→ a) Let us assume x=t then y=x2=t2 by differentiating r→(t) with…

Q: The surface differential, dS, corresponds to: A) u/2u² + IdA в) и? (4u? + 1) dA C) uv4u² +1dA D)…

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Q: For the velocity field V =2xyi + 4tz?j – yzk, The acceleration at the point ](2, 21, 1) att = 2] =…

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Q: Which of the following represents the arc length of the vector function r(t) = (sin 2t, e, ) on the…

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Q: Given the force field F = (2xze* - 4, cos°z, x?e**: - 3ycos?z sin z) Solve for the work done by F in…

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Q: Find the tangential and normal components of acceleration at the given time t for the space curve…

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Q: Given: vector-valued function R(t) whose derivative R' (t) = (4t, 6t, -12t) (b) Reparametrize R…

A: Let's reparameterize vector value function R(t).

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.