(a) Prove that d θν θ E) - e, de ди дө dw a0 - dt ax ax dt ax ax Әх ду Әх др given that + u- + v -+w at' "ax ay dt de

Q: Let r be a differentiable vector function of t. Show that if r . (dr/dt) = 0 for all t, then | r |…

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

A: Given, the slope field:

Q: Use Part 1 of the Fundamental Theorem of Calculus to find g(x) = 1 (a) dt t3 + 5 g'(x) = (b) h(x) =…

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Q: Let P(z) = | tin(t) dt. Use part 1 of the fundamental theorem of calculus to determine if P(z) is…

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Q: Suppose that r(t) = cos³ t i+ sin³ tj+et k, show that d dr (r.- dt dt d?r X at2 dr d'r. r.at at3…

A: Given: rt=cos3t i+sin3t j+e-t k we have to show that ddtr·drdt×d2rdt2=r·drdt×d3rdt3

Q: Suppose that z = xy² and x and y are differentiable functions of t with x = 1, y = −1, dx/dt = −2,…

A: Given  z=xy2 Also at t=-1 x=1y=-1dxdt=-2dydt=3

Q: 2. What is the relation between dV /dt and dr/dt if V = ()7r? %3D

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Q: In Exercises 39-42, with f (x) as in Figure 11, let A (z) = f(t) dt and B (z) = f (t) dt y f(x) 0.…

A: From the given graph and the integral , we know that  A'(x)=f(x) The minimum value of A(x) occurs…

Q: d Calculate ri(t) · r2(t)] and d ri(t) x r2(t)] first by differentiating dt' dt the product directly…

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Q: Find the local linearization of f(z) =z² at 9. lo(z) = %3D

A: Given : f(x) = x2   x0 =  9

Q: Find the value of dr/dt at t = 0 if r = (u2 + 7)^(1/3) and u^2t + u = 1.

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Q: Show that if f(2) is analytic for |z| <1+e, then for any z = rei® with r < 1, 5. eit z dt = 0 1–…

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Q: d d Calculate ri(t) · r2(t)] and ri(t) × r2(t)] first by differentiating dt dt the product directly…

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Q: Evaluate the integrals

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Q: - (a x r) = a x r' for any constant vector a. dt Show that

A: To prove: ddt(a×r)=a×r'. Consider L.H.S. Differentiate with respect to t using product rule.…

Q: Use Green's theorem to evaluate (2y dx-x dy) around the closed curve C. 27 (2,8) 8. 4. C. -2 (1,-1)

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Q: Compute N(0) if r(t)= and prove that T(0) and N(0) are orthogonal

A: Given rt=&lt;t sin t+cos t,2t2,sin t -t cos y&gt; Then vector is given by r→t=t sin t+cos…

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: Show that the isoclines of dy/dt= t are vertical lines. Sketch the slope field for −2 ≤ t ≤ 2, −2 ≤…

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Q: f(x, y) =(e-2y)Cos2x

A: The Laplace equation is: ∂2f∂x2+∂2f∂y2=0

Q: Compute d/dt f (r(t)) at the given value of t for f (x, y, z) = xz − y^2, r(t) =〈t, t^3, 1 − t〉at t…

A: Given: The function fx,y,z=xz-y2and rt=t,t2,1-t, To find: ddtfrt at t=-2,

Q: Verify by differentiation that a) / t dt : V1+t2 /1+t² + C; _ b) tan(x)dr = – In(cos(x)) +C, –1/2 <…

A: Given to find ∫t1+t2dtPut 1+t2=udifferentiate 2tdt=dutdt=du2therefore…

Q: Assume that z = ƒ(x, y), x = g(t), y = h(t), ƒx(2, -1) = 3, and ƒy(2, -1) = -2. If g(0) = 2, h(0) =…

A: We use the partial derivatives to find dz/dt  Then plug x=g(t) and y=h(t)

Q: The slope field oY = f(t,y), with initial condition f(0) = 1, is shown. Find limf(t.y). dt - 10 「 ンン…

A: On the slope field draw the graphical solution with initial condition f(0)=1:

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: 10. Para resolver la siguiente ecuación diferencial: y" + 2y' + y = 0 con y(0) = 1; y'(0) = 1 Se ha…

A: The solution of differential equation is given by using Laplace transform and partial fraction…

Q: a) If u=(x,7), where x = 0(1) and y=p(1), then that %3D prove du du dr ди dy dt Əx dt ây dt If…

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Q: -dt. dF = (dr · V)F+ at

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Q: Show that if the vector field F = Pi + Qj + Rk is conservative and P, Q, R have continuous…

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Q: Exercise 11.3 Prove the equality [* | f(x) dx = fm({x : [ f(x)| ≥ t}) dt, 0 where m is Lebesgue…

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Q: 1. Evaluate the integral ab- a + b à) + a(b a)-2(à ab - bla dt in which a, b are the derivatives of…

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Q: dri(t) · r2(t)] and ri(t) x r2(t)] first by differentiatin e dt luct directly and then by applying…

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Q: I. Consider the vector-valued function In(2t + 1) Ŕ(t) = ²₁+ î+ t 1. Determine the domain of R. 2.…

A: This is a problem of Vector Calculus.

Q: psin z F(#) = [" t cos(t) dt (1) (a) . Compute F(T). Let

A: The solution are next step

Q: EXERCISE 6. dy +y dr where A is an arbitrary constant () - (+*) + A, Given that prove that tan

A: Solve By substitution

Q: 1. Detanmne the rusutt OF a. Gradi ent f(x.y,z) ex sin cy) In (7) b. Divergenie =7? C Curl F=y2 t…

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Q: Suppose that z = f(x, y) is differentiable at the point (1024, 256) with f-(1024, 256) = 4 an dz…

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Q: (b) Show that the function f(2) =u +iv, where S(2) = **(l+1) – y'(1-1) x² + y² = 0, z = 0 Satisfies…

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Q: If w = x²y - y²z³,x = rCost,y = rSint, dw z = re' , (r = 2)and(t = 0)then dt

A: Given equations - w = x2y-y2z3       ...............(1)x = rcost        ............(2)y = rsint…

Q: Let f(x,y,z) = x² sin y + y e, where x = t +3,y = net-1 and z In 2t. %3D %3D at t 1 is: dt -2 O None…

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Q: Use Green's theorem to evaluate (2y dx - x dy) around the closed curve C. y (2,5) 4 3 2 C 1. -2 -1…

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Q: d d Calculate #ri(t) - r2(t)] and [ri(t) × r2(t)] first by differentiating dt dt the product…

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Q: Let g(x) 8 In(t) dt. Use the Fundamental Theorem of Calculus tc g'(x) =

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Q: -if f₁ (t) properties of Fourier transfor eat alt) JF.T = $ fit) ējut dt = - feat. ejust dt е ماز-…

A: Topic:- calculus

Q: Let r be a di¡erentiable vector function of t. Show that if r # (dr/dt) = 0 for all t, then  r  is…

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Q: Let F = (t+ 4, sin(4t) , t°) Find the indefinite vector integral F(t) dt = ( +C1, +C2, +C3)

A: Given, F = (t+4, sin (4t), t³) ∫F(t) dt = ∫(t+4, sin (4t), t³) dt ∫F(t) dt =( ∫(t+4)dt , ∫sin(4t)dt…

Q: if f(x,y) =x2y÷x+y and y(t)=  (x(t),y(t)) is parametrized curve in domain of f with y(0)=(2,-1) and…

A: The first step in solving the problem is to obtain the parametric equation. The initial values can…

Q: Assume x and y are functions of t. Evaluate dy,dt for each of the following .xey = 2 - ln 2 + ln x;…

A: It is given that, therefore the time derivative of the given function is, But it is also given…

Q: 2. Find the parametrization of the tangent line to the space curve r(t) = (-t3, In(t), e² ) at t =…

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Q: Let g(u, v) = u² + v³, 2(t) = cos(1), v(t) = In(t). Compute dg (please use only the dt your…

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