Show that if |sin z| is bounded along the straight line {(x, y) : ax - By = 0}, then a = 0, i.e., ZER.

Q: Show that the line integral is independent of path by finding a function f such that Vf = F. 2xe-Ydx…

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Q: Find the line integral of f (x, y,z) = -x + y + z over the straight-line segment from (1, 2, 3) to…

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Q: Let Li be the straight line segment from (0.0,1) to (1,a, 8) and let F: (z+ax, ay, x> Calculate the…

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Q: Show that the function u e(xy) + Vry V(y/x) provides the quation = 0. dy?

A: Again differentiate,

Q: Show that the line integral is independent of path by finding a function f such that Vf = F. 2xe-Ydx…

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Q: 2)Evaluate ex dA , Ris bounded by y = x2,y = 1

A: By using Fubini's theorem we solve the given problem.

Q: Show that F(x, y) = x² + 3y is not uniformly continuous on the whole plane.

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Q: Evaluate the line integral x'z ds, where C is the line segment from (0, 5, 2) to (4, 6, 5).

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Q: Evaluate the line integral [x(1 с where C is given by the line r(t) = (1 + 3t, 2t), 0≤ t ≤ 1. x(1 +…

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Q: Show that the integral is independent of the path, and use the Fundamental Theorem of Line Integrals…

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Q: Evaluate the line integral f. (2x – 4xy)dx + 3xy – 2x³y)dy over the straight path L joining point…

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Q: use the Fundamental Theorem of line Integrals to evaluate F. dR, where F (*.y) = < y. - X and C is…

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Q: a) Evaluate the line integralZ F dr C in terms of t where F = cosyi+xj+ye²>k and C is the line…

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Q: Evaluate the line integral / x°z ds, where C is the line segment from (0, 2,4) to

A: Line integral: - The function to be integrated is determined along a curve in the coordinate system…

Q: Prove that the function f : R? → R, f(x, y) = sin(/x² +|sin y|) is uniformly continuous on R?.

A: The time interval between two waves is known as a Period whereas a function that repeats its values…

Q: Evaluate the line integral ∫c x3 z ds, where C is the line segment from (0,8,8) to (5,7,6).

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Q: b. x sin y dA, R is bounded by y = x², x = 0 and y = T.

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Q: Evaluate the line integral xyld x + (x²+y)dy] +. where c is the saruare farmed by the lines

A: Let us consider a line integral ∫CP dx+Qdy then it can be written as  ∫D∫∂Q∂x-∂P∂y dA  Where, P and…

Q: The tangent function f (x) = tan x is integrable from 0 to T. O True O False

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Q: Evaluate the line integral [F-ds where F = 2xyi +(x-y³)j and C is (a) y =x from (0,0) to [10] (1,1)…

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Q: Show that the line integral is independent of path by finding a function f such that Vf = F. 2xe-Ydx…

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Q: Evaluate the line integral ∫Cydx+xdy along the curve y=x2 from the point O(0,0) to the point A(1,1)…

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Q: Show that the line integral is independent of path by finding a function f such that Vf = F. 2xe-Ydx…

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Q: Let F(x, y, z) = z* +x+y sin x +x In z 0, where z =f(x, y), then at (0,0,1) and %3D are equal to:

A: Given,          Fx,y,z=z4+x+ysinx+xlnz=0 , where z=fx,y

Q: Find the line integral of f(x,y,z) = -x+y+z over the straight-line segment from (1, 2, 3) to (0, −1,…

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Q: Evaluate the triple integral. 10x dV, where E is bounded by the paraboloid x = 5y? + 5z2 and the…

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Q: c) Let f(r, y) = Vlry| Prove that f(r, y) is not differentiable at point (0, 0).

A: Given a function : f(x,y)=xy We need to show that f(x ,y) is not differentiable at (0 ,0). Let's…

Q: Let 0 be the angle between the line y = mx +band the x-axis. Prove thatm = tan 0.

A: The question should be :                 Let θ be the angle between the line y=mx+ b and the x-axis…

Q: || x dA, R is bounded by y = x, 2x + y = 6 and the y-axis. R

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Q: Find the line integral of f(x,y,z) = -x + y+z over the straight-line segment from (1, 2, 3) to (0,…

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Q: a) Evaluate the line integral S. (2x – y) dx – (x + 3y) dy, where C is a straight line from (1,1) to…

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Q: Show that the line integral is independent of path tan y dr + r sec y dy where C is any path from…

A: Given that ∫C tany dx +x sec2y dyShow that the line integral is independent of path and evaluate the…

Q: (b) Let F = (-yx² , xy²). Let D be the annulus 1< x? + y² < 4, and let C be its boundary. Evaluate…

A: Greens theorem

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

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Q: Show that UV=12 given, UX = 8, and XY = 10 using the intersecting %3D Secant-Tangent Theorem. * U

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Q: Use the local linear approximation of (1 – x)2º at xo O to estimate (0.99)2º. %3D

A: The linear approximation of f(x) at x0 is given by L(x) = f(x0) + f'(x0) (x-x0)

Q: . Show that the function u = p(ry) + Vry v(y/x) provides the quation = 0. ду? %3D dx²

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Q: (b) Let F = (-yx², xy²). Let D be the annulus 1 < a2 + y² < 4, and let C be its boundary. Evaluate

A: Value of the given integral is

Q: Show that the function u = p(xy) + Vry v(y/x) provides the quation = 0. dy?

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Q: Evaluate the line integral ∮C x dy−y dx where C is a circular, closed path defined by x^2+y^2=16

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Q: Solve the problem with Neumann oundary conditions, - Uxx = 0, x, t > 0 u(x, 0) = f(x) ux (0, t) = 0…

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Q: The path integral fv· dr, where v = (sin y, x – cos y) and C is the perimeter of the triangle formed…

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Q: If f(z) = sin(z 2) = U(x, y) + iV (x, y), determine U and V . Verify that the C-R conditions are met…

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Q: 2. Use the Fundamental Theorem of Line Integrals to evaluate fe F.dR, where F(x, y) = (x²y? –…

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Q: a) Evaluate the line integral S. (2x - y) dx – (x + 3y) dy, where C is a straight line from (1,1) to…

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Q: Evaluate the line integral y dx + x dy where C is the parameterized path x = = ² , y = t° , 3 -ydx +…

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Q: Evaluate the line integral | F•ds where F = 2xyi +(x- y²)j and C is (a) y= x from (0,0) to (1,1) (b)…

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Q: Study the continuity of f(x,y) = xe-Vx at p(1,0) ey.

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