The velocity field for a fluid is given by V=(X2t3)i+(2XYT)j. The convective acceleration in the y-direction at point (1,1) and t=1 is equal to

Q: When body is an equilibrium ,the net force is vanish

A: We have to check whether the given statement, "When body is an equilibrium, the net force is vanish"…

Q: Let x and y be differentiable functions of t, and lets = √(x2 + y2) be the distance between the…

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Q: The force exerted by our home planet is GMx if x R, measured on a unit mass at a distance x from the…

A: We are given the function h(x) as the following. Notice that GMxR3 is continuous everywhere. And…

Q: Let R(t) = et+e be a particle position moving in the plane Then the acceleration at time zero is

A: We have the position vector given in terms of time t.                  Rt=et i+e-t j To find the…

Q: X p(x, y, z)=xz²+sin (y) ex

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Q: vesify thie equation yzdx -z?dy tx ydz = 0 ir integsable PSove Hhat J a) - (-1) J, (2)

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

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: Let R(t) = e'i+ ej be a particle position moving in the plane. Then the acceleration at time zero is

A: Given, The position of moving particle is R(t)=et i+e-t j.

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: When body is an equilibrium ,the net force is not vanish

A: False. The given statement is false. We know that: There are two conditions that must be met for an…

Q: If F= represents the flow of a liquid through a parabolic cylinder S: y=x, 0sxs2, 0szsl. That is…

A: If F→=y2,-x2,z2 represents the flow of a liquid through a parabolic cylinder S:y=x2,0≤x≤2,0≤z≤1.…

Q: A body of mass m falls from the rest in a medium offering resistance proportional to the square of…

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: Find the tangential and normal components of the acceleration vector. r(t) = cos(t) i + sin(t)j + tk…

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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 tangential and normal components of acceleration at the given time t = 1 for the space…

A: Given rt→=ti^+2tj^-3tk^t=1

Q: A slender rod of constant density lies along the line segment r(t) = tj + (2 - 2t)k, 0≤ t ≤ 1, in…

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Q: Compute the flux of water through parabolic cylinder S: y = x', from 0 < x < 2, 0 < z< 4 if the…

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Q: 2. Show that f(t) =1 – sint defines a smooth flow on the manifold S = ( (cost,sint) ), but %3D g(t)…

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Q: Find the total differential of W = x^4 y^3 z^2 + x^3 Y^2 + x^2 z +y + 3 at (2,1,1)

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Q: Let X, Y be jointly Gaussian, standard, and Cov (X,Y) = y. Find pdf of Z = X +

A: Answer:-  Given that, X and Y have jointly Gaussian distribution.

Q: Consider the slope field shown. (a) For the solution that satisfies y(0) = 0, sketch the solution…

A: Given,  the slope field :

Q: Find the tangential and normal components of accelerationfor a particle moving along the circular…

A: We know that, tangential and normal components of acceleration for a particle moving along the…

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

A: Given: ψx,t=Asinwt+kx

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

A: The given plane curve is rt=ti+5tj, t=2.

Q: An object rotates with angular acceleration, a, defined as dw a = dt where w is the angular…

A: Solving two ordinary differential equations, we get the result.

Q: . Calculate the flux integral fF.n ds, where C is the positively oriented circle of radius 1…

A: Given function is, Fx,y=x,2y The formula for calculating the flux of the integral ∮F·nds is,…

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

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Q: Suppose F = ∇ * A, where A = (y + sqrt(z))i + e^(xyz) j + cos(xz) k. Determine the flux of F outward…

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

A: The acceleration vector is,                              a =ax + ay +az  where…

Q: 8) Find the moment of inertia about the x-axis of a thin plate bounded by the parabola x = y - y?…

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Q: It takes 1700 J of work to stretch a spring from its natural length of 1 m to a length of 6 m. Find…

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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: Evaluate /|/ (x² + y² + z²)² dV where E is the solid ball centered at the origin with radius 2.

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Q: Consider the unsteady flow u= u0, v = kt, w = 0 where uo and k are positive constants. Show that the…

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Q: For the velocity field that is linear in both spatial directions (x and y) is           V-›= (u, ? )…

A: Calculate the vorticity vector. In which direction does the vorticity vector point?

Q: . Compute the tangential and normal components and express the acceleration a in the form a =…

A: The given displacement vector is r(t)=(et cos t)i + (et sin t)j +(2t)k at t=0. The components of the…

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 flux of 3 – e")i+ (y³ + sin z)j+ (z³ – unded above by z = V4 – x² outward orientation.

A: We have to find the flux using the divergence therom according to the given information

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: Given velocity potential, determine the velocity components u and v. P = a cos xy

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Q: Figure Q4 shows a block with mass of 765 N resting on a horizontal plane. Calculate the force F…

A: Given: W=765N a=4.2m/s2μ=0.18θ=25° We know that ∑Fx=ma…

Q: A particle is accelerated along a curved path of lengthl = 4.0 m under the action of an applied…

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Q: Using an appropriate linearization, the local linear approximation of f(z)= T+ sin z at 27 is

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Q: he position x(t) of an object moving in 1-D on the a - aris in an inverse-square force held (such as…

A: As given by the question There are given that the equation of inverse-square force field is:…

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: A spring is stretched 1.5 inches by a 2-pound weight. If the weight is pulled 3 inches below the…

A: This is a problem of differential equation.

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: