Sketch the triangle defined by A and compute LA fdA for A = {(x, y) E R²;0< x

Q: C: r(1) = (1 – cos t)i + sin tj Evaluate F. dr, where F(x, y) = (y + 1)i + (3xy² + 1)j 1- and C, is…

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Q: Scalar line integrals Evaluate the following line integral along the curve C.

A: Given integral is Parametric equation of semicircle is

Q: | f(x, y, z) ds where f(x, y, z) = xy and C is the elliptical helix given %3D by x = 2 sin t, y = 3…

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Q: Let F (r, y, z) = sin y i+r cos yj- sin z k.

A: Given,        F(x, y, z)=sinyi+xcosyj-sinzk.

Q: Let /:R'R. S(x. y.2) = (e' tan y,x' In y). Compute D(f).

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Q: Use Green's Theorem to evaluate the line integral |(v - x)dx + (2x – v)dy for the given path. C: x =…

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Q: Prove that fe, r*dx + rydy = 0 where Co is the line segments from (0, 1) to (0,0) by using…

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Q: Evaluate the line integral f e-*(cos y dx - sin y dy), where c is the rectangle with vertices (0,0),…

A: The line integral function is ∫Ce-xcosydx-sinydy. The vertices of the rectangle is 0, 0, π, 0, π,…

Q: The integral ∫C[(3x2y+y2)dx+(x3+2xy)dy] is independent of the path. Evaluate the integral where C is…

A: Given: R=ti+(t2+t-2)j   for 0&lt;=t&lt;=2

Q: Evaluate | f(x,y) ds where Cis the portion of the curve y = In x from the point (1,0) to the point…

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Q: compute the arc length function s(t) = | ||r'(u)|| du for the given value of a. %3D r(t) = (12, 212,…

A: Given that, rt=t2,2t2,t3a=0 From the given equation we get, ru=u2,2u2,u3 a=0

Q: Q.1B) Prove that: I = f, Cos () dxdy sin(1) %3D 2 where R the region bounded by x+y 1,x = 0,y = 0

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Q: Compute the line integral Sc 2.xy dx + x² dy along the following curves. (a) C1 along the circle x2…

A: In given question that is a conservative vector field mean exact integration in which intrgration…

Q: 3. For all a? + y + 2? > 0, verify that the velocity potential 1 $(r, y, z) = Vr? + y² + z2 is a…

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Q: Evaluate the line integral , F dr where F (3 sin x, -5 cos y, xz) and C is the path given by r(t) =…

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Q: Evaluate the integral x² dx + y? dy where C consists of the arc of the circle x2 + y² = 4 from (2,0)…

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Q: Compute the line integral f 2xy dx + x2 dy along the following curves. (a) C1 along the circle x2 +…

A: A detailed solution is given in steps 2 and 3.

Q: 1. Let C be a path along the upper half of a circle from the point P(2,0) to the point Q(-2,0).…

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Q: Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A: It is an simple and easy question on vector calculus  Can be easily solve by use of green's theorem

Q: Verify Green's Theorem by evaluating both integrals ƏM y2 dx + x2 dy = dA ду for the given path. C:…

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Q: Evaluate the line integral F. dr where F = (3 sin x, 3 cos y, 10xz) and C is the path given by r(t)…

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Q: At what points (x, y) in the plane are the functions continuous? a. g(x, y) = sin (1/xy ) b. g(x, y)…

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Q: Q.1B) Prove that: Iff, Cos) dxdy - = sin (1) where R the region bounded by x + y = 1, x = 0, y = 0

A: We use the change of variable to prive the given equality

Q: Evaluate the line integral f. F. dr where F = (2 sin a,–5 cos y, 10xz) and C is the path given by…

A: We will evaluate the given line integral.

Q: Find the sum of the residues of f (z) = sin z at its poles inside the circle |z| = 2. z cos z

A: Residue theorem∮f(z)dz=2πi(R1+R2+.........Rn)             =2πi(sum of the residues at the poles…

Q: Let f(r, y, 2) = 2ry + 2. Evaluate f(1, y, 2) dz where C is the portion of the helix r(t) = cos ti+…

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Q: Y – 5x |dA x + y 100 cos

A: We have to evaluate the double integral ∬ 100cosy-5xx+ydxdy over the region D where D is the…

Q: Evaluate the line integral ,F. dr where F = (sin x, –2 cos y, 5xz) and C is the path given %3D by…

A: Integral F.dr = integral F(r(t)).r'(t) dt

Q: The value of the line integral | F(r).dr when C F(F) = [-y, -xy]and C is the upper half circle arc…

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Q: integrate ƒ(x, y, z) = sqrt(x2 + z2) over the circle r(t) = (a cos t)j + (a sin t)k, 0&lt;= t…

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Q: Find the equation for the Tangent Plane on the given surface coS ax – x²y + e*z+ yz = 4 Po(0, 1, 2)

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Q: 39. Show that the graph of r =t sin ti + t cos tj + t'k lies on the paraboloid z = x² + y².

A: Here we compare r as its basic equation to prove this.

Q: Find Laplace treins form for Ý +4Y+3Y = e y(o) =y co) = 1 %3D %3D

A: Formula Used:L{y''}=s2Y(s)-sy(0)-y'(0)L{y'}=sY(s)-y(0)L{eat}=1s-aL{tneat}=n!(s-a)n+1

Q: Evaluate x*dx C + xy dy, where C is the triangular curve consisting of the line segments from (0, 0)…

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Q: Evaluate the integral c (xy+z^2)ds where c is the arc of the helix x=cos t ,y =sin t, z=t which…

A: To evaluate: The integral ∫c (xy+z2) ds where C is the arc of the helix x=cos t, y = sin t, z = t…

Q: Sl, Cos () dxdy sin (1) Prove that: I = JJ. x+y where R the region bounded by x+ y = 1,x = 0,y = 0

A: Here we have to prove that       ∫∫R Cos{(x-y)/(x+y)}dxdy = sin(1)/2  Where R is the region bounded…

Q: 18. Integrate f(x, y, z) = – Vx² + z² over the circle r(t) = (a cos t)j + (a sin t)k, 0 <t< 2m.

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Q: Evaluate |(2y+cos x)dx + (5x+ sin y)dy along the positively oriented curve C. C is the boundary of…

A: We have to evaluate the integral ∫2y+cosxdx+5x+sinydy along the positively oriented curve C. Where C…

Q: Use Green's Theorem to evaluate the integral Sc(5 sin y – 3y, 2 + 5x cos y) · dr where C is the line…

A: Use Green's Theorem to evaluate the integral ∫C5sin(y)-3y,2+5xcos(y)·dr→, where C is the line…

Q: Find the diih of y integral tanar 230 e dn dy a) | shso b) c) O sysl-z 1-2 d) Nove of theoe

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Q: (7) Let w(x, y, z) = xy cos z, where x = t,y = t², and arcsin(t). %3D dw Find dt

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Q: Find the value of the closed loop integral z3 + sin z -dz (z – i)3 taken around the boundary of the…

A: GIven  ∮z3+sinz(z-i)3dz   To compute this integral we have to use the following theorem. Theorem:…

Q: Use Green's Theorem to evaluate the line integral (y - x)dx + (2x - y)dy for the given path. C: x =…

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Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: Consider the given function, xy+y2-3x-3=0 now differentiate with respect to x,…

Q: Consider the following functions: e e f1(x) = sin(x1) cos(x2), æ € R? f2(x, y) = x' Y , æ, y E R"…

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Q: Find..aW and. au at. the. paint.(u.,.v.).(-2D). i.W.In (x+yz)X=u.e.sinu. y=uě.cosu.. and..z=u.e.

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Q: Given the force field F = (2xze* - 4, cos°z, x?e**: - 3ycos?z sin z) Solve for the work done by F in…

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Q: Evaluate line integral F. dr, where F(x, y) =(2xy)i + (x²+3)j, and C is the path given by r(t) = t =…

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Q: Consider F and C below. F(x, y) = 6xy? i + 6x²y j (re), e + cos{ nt), Gnt)), osts1 C: r(t) = (t +…

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