1) Prove Green's Theorem for F = 3xy'i - 4xj around the curve C bounded %3D 160 by y = 2x and y = 4x. Ans = 3 2)Verify Green's Theorem for F = (y',6xy?) around the curve C bounded 64 by x = 4 and Vx = y. Ans 5 %3D 3)Evaluate F.dr where C is the circle x' + y? = 4 transversed counterclockw ise, F = (-y, xr).Ans 87 %3D %3D 4)Evaluate F.dr where C is around the curve C bounded by y = and y = x transversed counterclockwise, F = (y²,x² ).Ans = 15

Q: Evaluate Se (5xy + x² + y²) dx + (x² - y)dy by using Green's theorem, where C is a closed curve that…

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Q: etermine Z J f(x,y) ds with curve C is the curve of the circle x+y² =4 from (0, 0) to (- nd fly d

A: ∫c f(x,y) dsC ;  x2+y2=4   from   (0,0) to  (-3   , 1)and  f(x,y) =x(y+1)+y

Q: A circle is the set of points equidistant to a given center. Suppose distance is defined as d((x,y),…

A: Solution is written in below steps.

Q: Evaluate [(y² - 4xy, 2xy - 2x²). C where C is the line from (0, 0) to (1,5) followed by the line…

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Q: 8. For the curve given by r(t)=√√² +8 i + In(t² + 8)j + tk 2 find: a. r'(t)

A: Introduction: For a vector curve rt=ati+btj+ctk, the tangent line at the point of contact t=t0 is…

Q: c) A curve has a parametric equation x = 2t - 1 and y = t* -t'. dy Evaluate dx? 1 at t = 2

A: Given, The parametric equations x=2t3-1, y =t4-t3.

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A: Given, F(x,y)=x2ex,60x2y2 The curve C1 consists of the line segment Ca from 0,0 to 1,1, followed by…

Q: Let F = -yi+xj and let C be a curve consists of the line segment C₁ from (0,0) to (4,0) followed by…

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Q: 5. nts) Let F =, use Green's Theorem to evaluate F. dr where the curve C is C the curve shown in the…

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Q: Use Green's theorem to evaluate O -x²ydx + x°dy, where C is the circle x? + y? = 4 with a…

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Q: Find T, N, and x for the space curve r(t) = (6e'cos t) i + (6e'sin t)j+ 4k. T(1) = Di+Oi+k (Type…

A: Given r→t=6etcosti^+6etsintj^+4k^ We have to find the T, N and κ for the given curve. Differentiate…

Q: 5) Let F(x, y, z) = (z − x², xz, y²) and C be the curve of intersection between x² +3y² 1 and z = x²…

A: Let us first parameterize curve C.

Q: I, Use Stoke's theorem to evalueate S.Fidr, where F Cau2)= - it +x3 + 2î aind C is the Ceurve of…

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Q: 6y dA, R is the quarter-disk in the fourth quadrant enclosed by the curves r+ y = 16, r = =0, and y…

A: Explanation of the answer is as follows

Q: Let C be the curve consisting of line segments from (0, 0) to (1, 1) to (0, 1) and back to (0, 0).…

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Q: Let F(x, y) == and let C be the circle r (t) = (cost)i + (sint)j, 0 ≤ t≤ 2π. A. Compute მი Note:…

A: Step 1: Step 2: Step 3: Step 4:

Q: Which of the following is/are points on the curve y-x Br+ 22x - 24x +9 whore the tangent line is…

A: We need to find the points on the curve where the tangent line is parallel to the x-axis.

Q: Let C be the curve connecting (0,0,

A: Given that C be the curve connecting  (0, 0, 0) to (1, 4, 1) to (3, 6, 2) to (2, 2, 1) to (0, 0, 0).…

Q: 4. Consider the parametric curve x = cost, y = sin cost, te [-T, π]. (a) Show that (0,0) is a point…

A: Consider the parametric curve given by  and , where  belongs to .a) To show that  is a point of…

Q: Let F=-yi+xj and let C be a curve consists of the line segment C₁ from (0,0) to (4,0) followed by…

A: