- (19,8) (10,0) (19,8) ) Show that So fdx + gdy = Sao.0 24x?y dx + 24y2x*dy is independent of path: af dy dg %3D Sao.o 24x?y*dx + 24y²æ*dy = r(19,8) (10,0)

Q: 1. Use an antiderivative to show that for every contour C extending from a point z₁ to a point 22, 1…

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Q: Evaluate the line integral ſc 2y dx + 6x dy where C is the straight line path from (4, 1) to (6, 5).…

A: Given query is to find the integral.

Q: Let P denote the path that travels along the graph of y = x2 from (0,0) to (1,1). Let f(x,y) = √1 +…

A: Let P denote the path that travels along with the graph of y=x2 from (0, 0) to (1, 1). Let f(x,…

Q: Calculate the line integral f -y²dx + xdy where ŕ is the path y² = 2x − x² such that x>1 and y > 0.

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Q: Evaluate the integra ∮C -y3 dy + x3 dx for any closed path C.

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Q: Determine the Linearity of this DE. * ay dy ニ X Linear O Non-Linear

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Q: Evaluate eVf •dr where f(r,y, 2) = 4rye² and C is the path parameterized by r(t) = cos² t i +…

A: Given: fx,y,z=4xyezrt=cos2ti+sin2tj+tantk0≤t≤π4 Formulas used: ddxxn=nxn-1ddxex=exddxfgx=f'gx·g'x…

Q: Show that the path given by r(t) = (cos t,cos(2t), sint) intersects the xy-plane infinitely many…

A: Consider the given path rt = cost , cos2t , sint  Then, xt=cost , yt=cos2t , zt=sint

Q: Consider thế iñifial a. 2y"- 30y" + 150y' – 234y = 0, b. y"+ 6ay' + (9a² + 1)y = 0, y(0) = -3, y(0)…

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Q: Use chain rule to find az at (u, v) = (1, 1) if

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Q: Find the Jacobian of following functions f(x,y,z) = 3xy – yz + 2y³z² – 5xyz³ g(x,y, z) = xyz – x³yz…

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Q: 15. What is the correct piecewise parametrization of a path C created by starting from C₁ of…

A: Introduction: The parametrization is a very significant part that is related to vector calculus.…

Q: Compute x 2z ds where C is the line segment C from (0,6,–1) to (4,1,5).

A: The given line segment from   to   can be represented parametrically as: , , and , for    Using the…

Q: Prove thut:f focK, 2) Y is a contimans findion then, g=XxX-> YxY 3 3 n K2) = (F(), PU=) Einction is…

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Q: Suppose that a duck is swimr - perature is given by the formı v²sin x. Find dT/dt in two chain rule…

A: Given query is to find the derivative of function T.

Q: 14 Verify the given linear approximation at a = 0. Then deter-

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Q: b) Show that the implicit Trapezoidal method h Wi+1 = Wi +5 f (ti+1, Wi+1) + f(ti, w;), is…

A: The solution is given below in the next step:

Q: 4Ux+Uy=U² , U(x,0)=1/1+x²

A: In the given partial differential equation with the given initial condition. 4ux+uy=u2u(x,0)=11+x2

Q: = In z are two linearly indepen- of ry" + y' = 0, then the partic- - ry" + y' = f(x) can be written…

A: We have to choose the correct answer from the given options according to the given condition

Q: 3.2 Evaluate where z = x + iy with y the path on line from z = 2+i to z = 1 + 2i.

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Q: Compute Jx²z ds where C is the line segment from (0,6,–1) to (4,1,5).

A: The objective is to compute ∫Cx2zds where C is line segment from  0,6,-1 to 4,1,5

Q: 17. Integrate f(x, y, z) = (x + y + z)/(x² + y² + z²) over the path r(t) = ti + tj + tk, 0 < a < t<…

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Q: Use Green Theorem to evaluate P xydx+edy where Cis C the path from (0,0) to (1,1) along the graph of…

A: The correct option is (d). Partially differentiate M and N w.r.t y and x respectively.

Q: let z- 57 dy .. dy By chain Ruie

A: The objective is to find the second derivative using chain rule.

Q: Find the general equation f the plane passing through P(-!,3,-2) and having n(-2,1,-1) hormal. O so

A: The objective is to find the general equation of plane.

Q: Compute (ax + by² + cz) dS over the sphere x +y +z² = 1.

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Q: tnd the kes reetor projection of anto r if

A: Given : u→=5i→+2j→ and v→=6i→-3j→

Q: The gradient vector of the function f defined by f(x,y)= In(1+7x² +5y²) is equal to Select one: O…

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Q: Evaluate I = Jc(sin ¤ + 2y) dx + (2x + y) dy for the nonclosed path ABCD in the figure. D А 3 (0,0),…

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Q: 2. Compute Ja F · ds for F c(t) = (t , In(1+ t²), e*) for 0 <t < 1 VV, where V(r, y, z) = 4x² In(1+…

A: Find your answer below

Q: Show that Sa fdx + gdy = Sa - (16x*y²dx + 8yxʻdy) is independent of path: (10,5) –2) (10,5) ,-2) 4…

A: We are given a line integral as follows : ∫1, -210, 5 fdx+gdy=∫1, -210, 5 -(16x3y2dx+8yx4dy)   First…

Q: Minimize c = s + t + u subject to 3s + 2t + u ≥ 48 2s + t + 3u ≥ 48 s + 3t + 2u ≥ 48 s ≥ 0, t ≥ 0, u…

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Q: Show that there is no hamiltonian path in Cay ({(1,0),(0,1)}: Z3 +Z2) from (0,0) to (2,0

A: Given problem is : Show that there is no hamiltonian path in Cay ({(1,0),(0,1)}: Z3 +Z2) from (0,0)…

Q: Suppose that u € C² (R²) and u(x, y) = 0 everywhere on the boundary of the square ((x,y): 0≤x≤ R,0 ≤…

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Q: 1 Check xyz? dx +a²z² dy + x²yz dz for path independence. 1 Does xyz? dx +x²z² dy + x²yz dz Yesv…

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Q: of the b) nd y- Fons c) with point.

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Q: 2. Calculate (y+ z)dx+(x+z)dy + (x+ y)dz , where C consists of line segments from (0,0,0) to (3,2,4)…

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Q: bot A de groutne conitant. est Cheafuntis uh that f cark3 and f'4)=-s. g(xl=In (f(x)) + A conx)

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Q: If C is the line segment from (1, 4, 2) to (3, 2, 1), evaluate the line integral y+ 2) ds.

A: Express the line segment in its parametric form and then use the formula below to evaluate the line…

Q: Integrate h (x. y) = yi+xj over the indicated path: r(u) = ui+ u²j. ue [0, 1).

A: Given hx,y=yi+xj and ru=ui+u2j, u∈0,1 We have to integrate hx,y=yi+xj along the path ru=ui+u2j…

Q: Use Green's theorem to evaluate s cy? oy)dx + (y²rlo4") dy where Cis s hoon belbw, N y = 2-x² G,1)

A: The given path contains 4 curves: C1:  from (1, 1)  to  (-1, 1)C2:  from (-1, 1)  to  (-1, -2)C3:…

Q: Evaluate I = Se(sin æ + 7y) dæ + (3x + y) dy for the nonclosed path ABCD in the figure. А 3 (0,0),…

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Q: Determine whether the work done along the path C is positive, negative, or zero. /c/ zero negative…

A: For calculating the work done from the field we will have to see the direction of both vector field…

Q: Construct a linear map L(z) = az + 6 that will send the imaginary y-axis to the line v = - u

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Q: Compute I ds over the path C : F(t) = (t, 2t,ť²) from (0,0,0) to (4, 4, 4).

A: The answer is given below:

Q: Determine whether the work done along the path C is positive, negative, or zero. negative positive…

A: Consider the provided graph, Determine whether the work is done along the path c is positive…

Q: Verify the chain rule for jacobian, if y2 u = x + , v = * y2 %3D

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Q: Show that Si2 f dæ + g dy + h dz = S - (4xy³z*dx + 6y?x?z4dy + 8z°y³x²dz) is independent of path:…

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