Q4. Let C be a portion of a curve from (2,0) to (2, 2). Let F(r, y) = (tan(2²) + e²", 2.re²y + cos y). (a) Is F conservative? Show all the steps of your work. (b) Compute fc F · dr.

Q: Integrate f(x,y) = x + y over the curve C: x2 + y? = 9 in the first quadrant from (3,0) to (0,3).

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Q: (3) Compute f ds over the curve specified. (a) f(x, y, z) = 2²,r(t) = for 0 for 0 < t < 1

A: Consider the equation

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A: Differentiate both sides with respect to t.

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A: Given that, Fx,y=8xy3+3i+12x2y2+2e2yj⋯⋯(1) C: rt=2sin5ti+2tπsin85tj 0≤t≤π2S Let, F=Mi+Nj Where,…

Q: 2. Find the divections..in.w.hic . fCx,y) = X- a.. Increases.ms..rapidly.at. paint.C1). b. Decrenses…

A: Solution: The given function is fx,y=x22+y22    ......(1)

Q: b) Evaluate S(2y + 4z)dx + (2æ + 3z)dy + (4x + 3y)dz, where C is the curve given by r(t) = for 0 <t…

A: given ∫C2y+4zdx+2x+3zdy+4x+3ydz where C is the curve given by rt=t3,2sinπt2,3cosπt2 for 0≤t≤1…

Q: et C be a portion of a curve from (2,0) to (2,2). Let F(r, y) = (tan(2?) +2, ) Is F conservative?…

A: In the given question, the concept of integration is applied. Integration Integration is a method of…

Q: F(x, y) = (8xy + 3)i + (12x²y² + 2e²")j is conservative by finding a potential function f for F, and…

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Q: Use linear approximation to find y = L (x) of the function f (x) = = COS X about the point xXo = JT…

A: Use formula of  linear approximation

Q: Q9. Find the equation of the tangent plane to f(x, y) = x ln(y) +x²y+3 at the point (x, y) = (2, 1).…

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Q: Find the length of the curve y = Vcos 2t dt from x = 0 to x = T/4.

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Q: F(x,y) = (4xy^3 + 9)i + (6x^2*y^2+2*e^(2y))j  conservative by finding a potential function f for F,…

A: let f be the potential function

Q: Use Green's Theorem to find the work done by F(x,y)=x(3x+y)i+xy^2j in moving a particle from the…

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Q: Differentiate: y(x) along the curve xy =ex Show all working out.

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Q: dy (5) Obtain the integral curve for +y cotx dx = e*cscx passing through the point (0, 0).

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Q: Evaluate the line integral / (2x + 4y)dx + (2x – 9y)dy along the curve C : x = 4 cos t, y = 8 sin t…

A: Solution is given below:

Q: (a) A beam is fixed at its left end x = 0 but is simply sup- ported at the other end x = L. Show…

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A: Given parametric functions are: dxdt=3t+2dydt=t2-4

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Q: (4) Evaluate sin(yz)dr+xz cos(yz)dy+ry cos(yz)dz, where C is any curve joining the point (0, 0,0) to…

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Q: Find the path of a heat-seeking particle placed at point P on a metal plate whose temperature at (x,…

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Q: 5. Consider the saddle surface r = y- 2, find the point where the tangent plane is parallel to x+y+z…

A: If two vectors a1, b1, c1 and a2, b2, c2 are parallel, then it can be written that a1, b1, c1=ka2,…

Q: 1. Evaluate S. (2x+ y² – z) ds where C is the piecewise smooth curve below. (0, 1, 1) (0, 0, 0). (0,…

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Q: -2y The line integral (e sin x) dx+(x' +3y)dy with path F (t)= (8cost, 8 sint) such that 0sts2n. Set…

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Q: Evaluate the line integral (5x + 8y)dx + (3x – 5y)dy along the - curve C : x = 10 cos t, y = 20 sin…

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Q: Use a local linear approximation of the function f(z) = VI cos[(V2r)] at some approximate v3.98…

A: To use local linear approximation of the below function to approximate 3.98cos7.963π. fx=xcos2x3π

Q: Find the symmetric equations for the normal line LN to the surface S:f (x,y) = 5VT +y? at the point…

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Q: Consider the curve C in RS parameterized by r(t) = sin(2t) î + cos(t) ĵ+t k, con 0 <t < 2n An…

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Q: 3. Find the work done by F over the curve in the direction of increasing t. = (2,x, y) = (sin t, cos…

A: Write the vector field F=zi+xj+yk The equation of the curve. rt=sinti+costj+tk Differentiate…

Q: Integrate f(x,y) =x -y over the curve C: x² + y = 16 in the first quadrant from (0,4) to (V8,18 - y)…

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Q: Q1: Use Green's theorem to evalute f. (xy + y2)dx + x²dy by the curve y = x and y = x2.

A: Given problem:- Use Green's theorem to evaluate ∫c (xy + y2 )dx + x²dy  by the curve y = x and y =…

Q: Evaluate S. (2xy – x²)dx + (x + y²) dy such that c is a curve bounded by y? – x , y – x² ?

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Q: Evaluate the line integral. { 4) ( y2 dx + x dy; C is the curve x =t2 - 1, y = 4t, 0 st s 1

A: The given line integral is ∫Cy2 dx + x dy; C is the curve x=t2-1, y=4t, 0≤t≤1

Q: Compute / f ds over the curve specified. (1) f(x, y, z) = z², r(t) = for 0 < t < 2

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Q: 1. If F(x, y) = (3+2xy)i + (x² – 3y²)j. Evaluate F. dr, where C is the curve given by r(t) = e'…

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Q: computes curve length for 0

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Q: (a) Calculate the line integral (4x+2y²)dy where C' is the curve y = 2³, starting from z = 0, y = 0,…

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Q: B) Evaluate the line integral F-dr, where F (x, y) = x²y³i+ x°y°j and C is the curve r (t) = (t³ –…

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Q: 4. Integrate f(x, y, z) = Vx² + y² over the involute curve (cos t + t sin t)i + (sin t – t cos t)j,…

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Q: Integrate f(x,y) =x+y over the curve C: x² + y = 49 in the first quadrant from (7,0) to (0,7).

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Q: 11 Calculate the line integral f Gy)ds for the given junc rion and path xiyl=xy KHt)=<3+,2t=-37 for

A: r(t)=&lt;3t,2t-3&gt;=&lt;0,-3&gt;+t&lt;3,2&gt; x(t)=3t  and y(t)=2t-3 ds=32+22dt=13dt 0≤t≤1

Q: Using an appropriate linearization, the local linear approximation of is f(x) = T+ cos r at 2.

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Q: Using an appropriate linearization, the local linear approximation of f(z)= T+ sin z at 27 is

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Q: Ext find the anen of the ragion bounded on the leftby the Curve yx, on the right by the line y and…

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Q: Q2) A) Verify that wxy = Wyx for w = x sin y + y sin x+ xy. B) Show that f(x, y) = e-y cos 2x…

A: Here we have to find the partial derivative of the functions with respect to x and y seperately…

Q: Show F(x, y) = (2æy³ + 8)i + (3x²y² + 2e2")j is conservative by finding a potential function f for…

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Q: 3. Find S. F · dr where F(x,y, z) = (3y²e*, 6ye* F(t) = (sin t, ,2-) for 0 <tsn. 1+3) and C is the…

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Q: Evaluate [xy dx+(x+y)dy along the curve y=x² from (−1,1) to (2,4). [Verify using Mathematica]

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