47-48. Line integrals from graphs Determine whether fF. dr along the paths C, and C, shown in the following vector fields is positive or negative. Explain your reasoning. F- dr b. F. dr а. C, 47.

Q: A particle moves in a velocity field V(x, y)= (x-y,x+y). If it is at position (-4, 2) at time = 2.…

A: Given: Velocity field V(x, y) = x2-y2, x2+y2, Partical is at (-4, 2) when t = 2. To find: Position…

Q: d) A vector field is given by F(x,y,2) = 2xj+ (x² - y(1-y))k . 2(1-e"?) Find the work that the…

A: To solve d and e: A vector field is given by, The parameterized curve C is given by, The interval…

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: 3. (a) Determine the value of coefficient a for which the vector field F = (ayz + 22, 2.02?, 6ryz² +…

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Q: 9) Check whether the vector field F = (-cos x cos y, sin x sin y, - sec² z) is a gradient vector…

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Q: A hot-air balloon has the truncated spherical shape. The hot gases escape through the porous…

A: Image is attached with detailed solution.

Q: 2.10 (a) Express the vector field H = xy'za, + x'yza, + xyz'a. in cylindrical and spherical…

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Q: The flow lines (or streamlines) of a vector field are the paths followed by a particle whose…

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Q: 41.) Use given grid to sketch the vector field F = (y, -x). Include vectors at all integers points…

A: Given vector field is F=12y,-x

Q: 3. а. Given a potential function/- Ax'y*z' - Bx"y'z -Cx'y'z, where the coefficients 4.B.c, are the…

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Q: 3. The figure shows a vector field F and two curves C1 and C2. Are the line integrals of F over C1…

A: The curve C1 starts from a point in clockwise direction as in the same direction of the vector…

Q: (b) Given a potential functionf = Ax-2y³z² + Bx³z² – Cxy, where the coefficients A, B,C, are the…

A: Here A = 0, B = 2 and C = 8

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…

Q: 4. Suppose that an insect population, z(t), is controlled by a natural predator population, y(t). a)…

A: Since the sub parts are not specified so i am solving first three parts. (a) It is given that an…

Q: 1. Find the line integral in a vector field a curve consisting of 2 parts. First, the straight line…

A: Given F→=y,x+2y We have to find the ∫CF→⋅dr along the given path.

Q: 10.) Use the Fundamental Theorem for Line Integrals to find the work done by the conservative vector…

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Q: Given the vector field v=⟨0,2xz+3y^2,4yz^2 ⟩. Find the line integral of the path from (0,0) to…

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Q: Convert the second-order differential equation d²y Ź-y = 0 into a first-order system in terms of y…

A: Let, v=dydtThen, d2ydt2-y=0or, ddtdydt=yor, dvdt=yThus, the 1st order system is dydt=v,  dvdt=yAnd…

Q: 8. Show that the vector field, -y a² + y?' x² + yj²' ) is not conservative in R³, despite the fact…

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Q: 6. Show that a constant force field does zero work on a particle that moves once uniformly around…

A: We have to prove that constant force field does zero work on on a particle that moves once uniformly…

Q: VII. - ) Let C be the curve consisting of the line segment from (0,0) to (2,0), the the portion of…

A: Given that a curve consisting a line segment from (0,0) to (2,0) and a portion of circle from (2,0)…

Q: F - () (3x²y²z and F = 2r³yz Q.)11.1 Be (a) Explain why vector fields are conservative. (b)…

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Q: A particle moves in a velocity field V(x, y) = (x², x + y2). If it is at position (x, y) = (1, 8) at…

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Q: A particle moves along line segments from the point (2,0,0) to the points (2,2,2) (0,2,2), and back…

A: Let C covers all the line segments. Consider C1 be the line joining the points (2,0,0) and (2,2,2).…

Q: Apply fundamental theorem for conservative vector field to find the line integral fF dr where F =…

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Q: 3. (a) For the vector field defined by F(x, y, 2) = yz²i – zĵ+ xyk, Evaluate the line integral ſ. F…

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Q: 52. Which of (A) or (B) in Figure 12 is the contour plot of a potential function for the vector…

A: By the equality ∇ϕ=F and since the gradient vectors are perpendicular to the level curves.

Q: 6. Use the fundamental theorem of line integrals to evaluate 2x dr + 2y dy where C is the line…

A: Let's understand the Fundamental Theorem of Line Integrals:

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: Can you help with 16 please?

A: Hey, since there are multiple sub-parts posted, we will answer first three sub-parts . If you want…

Q: 2. Letr =√x² + y² + z². (a) For n ≥ 1 and r> 0, express V(r-n) in terms of r and the radial vector…

A: Given  solution(a): Radial vector field  solution(b):

Q: F ds for the given vector field F and the oriented surface S. In other words, find the flux of F…

A: From the given information,

Q: Find a potential function for the vector field F(x, y) = (6x sin y, 3 cos x ), if it exists. A. No…

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Q: Find potential 9, for the conservati ve field the vector F (x, y,a)- (2xe? - y*sinx) î * sinx) î +…

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Q: Find a potential function for the vector field F(r, y) = (22ry + 11y- 3, 11z? + 11r). f( x, y) =…

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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: Q1. What is vector field and potential function? Sketch the gradient field of p(x, y) = x² + y² -k².

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

Q: tch the vector field with its graph. F(x, y) = yi y 事 5 -5- -6+ y 1信 I 5 5 -5+

A: Given  F(x,y)=yi Option (1) is correct by given graph

Q: 9. (~Ch16 Review #13) Find the work done by the conservative vector field F(x, y) = along the curve…

A: given a conservative vector field  Fx,y=4x3y2-2xy3,2x4y-3x2y2+4y3 and a curve  rt=t+sinπt,2t+cosπt…

Q: 9. Find the flux of the vector field B(x, y,=) shown in the left figure through the square S of - yỉ…

A: The general form of a vector valued function in three dimensions is given by…

Q: (4) Two scalar differentiable electrical potentials F(x, y, z) and G(x, y, z) superpose in space.…

A: Prove the following property for the potential function F and G

Q: 1) F=Cy+2xz)i+ (x+ z²+1)j +(2yz+x)K Give the Vector field. Protected, ☆Fis the vector fied…

A: According to the given information first it is required to check whether the given vector is…

Q: (b) A potential function for the vector field F (x, y) = (2x + 1)(x² + x + 1)² 7+ (2y + 1)(y² + y +…

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Q: EXAMPLE 4 Line integral of a conservative vector field Evaluate |(2ry – z) i + (x² + 2z) j + (2y –…

A: The line integral of a conservative vector field is  C is a simple curve from A(-3,-2,-1) to…

Q: Calculate the line integral of the vector field F -3 i+ 2j along the line from the point (4, 0) to…

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Q: This is a calculus 3 problem. Please explain each step clearly, no cursive writing.

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Q: 5. Find a potential function corresponding to the vector field F(x, y, z) = (2x, 3Y, 4z).

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Q: Time left 1. Given that the vector field F(x, y)= (2x-3y, 2y-3x> is conservative in the xy-plane. If…

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