(a) Let C be the curve of intersection of the surfaces x^2 = 2y and 3z = xy. Find the length of C from the origin to the point (6, 18, 36).

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A: Slope field of y' =1/y

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A: The total work done by a vector field, F→ moving a particle along the curve r→(t) is calculated…

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A: Consider the provided information, The work done by force F in moving particle along the path C is…

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Q: F(x, y. =) = (y+: sin x, x+: cos y, cos x-sin y) and G(x, y, 2)={y-:sinx, x+zcos y, cos x+sin y} 1.…

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

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Q: Show that F = (2xy + z³)ï + (x²)j+3xz²k is a conservative field. Find the scalar potential and the…

A: Given F = 2xy+z3 i ; x2j + 3xz2kThe objective is to show that given vector fieldis coonservative…

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Q: Given the vector field F = (In(x + 2z), xy, xyz). Calculate curl F(-1,1,1).

A: Solution:-

Q: The continuity equation for incompressible fluids states that the divergence of the velocity vector…

A: The given statement is The continuity equation for incompressible fluids states that the divergence…

Q: Find the gradient vector field of f бу f(x, у, 2) = X COS Z (бху) (6y бу -sin бх -sin бу i cos k…

A: Given,

Q: 4. Find the flow of the velocity field F = V(xy²z³) along the line segment from (1,1,1) to (2,1,…

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Q: Show that the line integral is independent of path. | 2xe Ydx + (2y - x²e)dy, Cis any path from (1,…

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Q: (15) Find the divergence and curl of i = (ryz)î + (3x²y)ĵ + (æz² – y²z)& at point (2, –1,1).

A: Given v→=xyzi^+3x2yj^+xz2−y2k^ We have to find the divergence and curl of v→ at point 2,−1,1. If…

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Q: Explain why a potential function for a conservative vector field is determined up to an additive…

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Q: The motion of a liquid in a cylindrical container of radius 2 is described by the velocity field…

A: Given,F(x,y,z) = -19y3i + 19x3j + 7k⇒curl F = ijk∂∂x∂∂y∂∂z-19y319x37          = 0i + 0j +…

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Q: Find the gradient field associated with the function φ(x, y, z) = xyz.

A: To find the gradient field associated with the function  φ(x, y, z) = xyz.

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A: Given that The surface S is simple and F is twice differentiable To show ∫∫scurlF.ds=0

Q: .) Find the work done by field F = (-y,-x) along curve C: counter clockwise along semicircle y = v4…

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Q: Describe the different methods available for evaluating the integral of a conservative vector field…

A: To Describe: Different methods available for evaluating the integral of a conservative vecctor field…

Q: Let F(x, y, z) = (y² + 2z, 2xy + 4, 2x), and let the curve C be described by x = for 1<t < 2. t, y =…

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Q: Find the line integral F.dr of the vector field F=ax] over a curve C defined by a C rectangle in the…

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Q: 4. Verify that the two integrals in the circulation form of Green's Theorem are equal along a circle…

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Q: Consider the force field F = (3x²yz - y + z, x³z - x, x³y + x). (a) Verify that it is a conservative…

A: A vector field will be conservative if its curl is zero . If f is potential function of F then ∫abF…

Q: Flow curves in the plane Let F(x, y) = ⟨ƒ(x, y), g(x, y)⟩ be defined on ℝ2.

A: The vector field Fx,y=fx,y,gx,y.

Q: Verify Stoke's theorem for the vector field F = (2x - y) I- yz*J-yzK over the upper half surface of…

A: Stokes theorem is used

Q: Therefore, div(curl G) = sin(y) – sin(y) + 8 = Since div(curl G) is different from 0, we conclude…

A: Given that:div(curl G)=siny-siny+8=

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Q: . Verify Stoke's theorem for the vector field F = (2x - y) I– yz²J -y z K over the upper half…

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Q: a. Show that the line integral Sc (yz - 2z) dx + (xz+3) dy + (xy - 2x) dz is independent of the…

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Q: The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In…

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Q: How much work is required to move an object in the vector field given as F = yi+3x j Along the lower…

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Q: If the vector field A = âx3x² + âyy – â̟5z³ is given, express à in cylindrical and spherical…

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Q: 4. Verify that the two integrals in the circulation form of Green's Theorem are equal along a circle…

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Q: Compute the outward flux of the vector field F = (5xy, 5y²) for the region enclosed by the curves y…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: i = e-* sin x

A: Given dxdt=e-xsinx. We equate the RHS of the differential equation above, e-xsinx=0 As e-x≠0, we…