Verify that the conclusion of Clairaut's theorem is valid, that is, that uxy = uyx u = x4y3 - y3 this is? True or False

Q: Use Green's Theorem to find f. [(2x – y+ 4) dx + (5y+ 3x – 6) dy] around a triangle in the xy plane…

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Q: 1. Use an antiderivative to show that for every contour C extending from a point z₁ to a point 22, 1…

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Q: a2 If a;j = aji (ajX;X;) əxyəx; are constant, calculate %3D

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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: Use Green's theorem to evaluate Sc(y² + x³) dx +xª dy where C is the perimeter of the square [0, 1]…

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Q: then show that u and v satisfy the Cauchy-Riemann equation everywhere for ( i) and for all z # 0 in…

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Q: Find ôz|ðu and ôzlôv for z=4e* In(y), x=ln(ucos(v)), y=usin(v) at the point (u, v) = (2, 7 | 4).

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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: Prove that flx)= (i+x)? + X and diverge Litx)? both

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Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann equations if ux = v, and u,…

A: Given u = x2-y2v = 2xy Then ux = 2xuy = -2yvx = 2yvy = 2x

Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy'dy, where C is the triangle with…

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Q: Prove that A <m AUB where AUB = {0x | r E A}U{ly|y € B}.

A: Solution: Let us consider two sets A and B. It may be possible that A∩B≠∅. But we want to construct…

Q: Find the work done by the force F = x²yi– xy²j along the paths shown from (1, 1) to (4, 2). (4, 2) a…

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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: = a + zy", z - w* + w, y I = uu" + w", y u+ ve fi az Find when u = 1, v 2. w=0 = %3D H.

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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: du If u = x²y + y²z + z?x then at (1,1,0) is ax %3D -

A: Given that u=x2y+y2z+z2xTo find :∂u∂x  at 1,1,0Solution :u=x2y+y2z+z2x∂u∂x=2xy+0+z2∂u∂x=2xy+z2

Q: Determine if u and v u = (cos 0, sin 0,–1); v= (sin 0.-cos 0,0) are orthogonal

A: We have to determine if u and v are orthogonal   The given vectors are u=cosθ, sinθ, -1v=sinθ,…

Q: 2. Let W be the union of the first and third quadrants in the xy- plane. That is, let W = xy 0 a. If…

A: According to our company gardening can answer only one question. If a specific questions are asked…

Q: Show that u(x, y) is harmonic in some domain. u(x, y) = 2x – x³ + 3xy?

A: since Uxx+Uyy=0, then U is harmonic

Q: 6. () Let r(ø, 0) = a sin ø cos đi+a sin ø sin 0j+a cos øk be the vector equa- tion of a sphere of…

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Q: 7 Prove : id Is perpendicular line to ra duic at its outer end point ,then t ang ent to the cirde is

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Q: f(r, y, z) = byz In(4ry) %3D fz = fy = た=

A: Given:

Q: 3. Find the circulation of F 2xi + 2zj +2yk around closed contour %3D

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Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy*dy, where C is the triangle with…

A: The given triangle C with the vertices (0,0),(1,0) and (1,2) is as as shown below.

Q: U tt = quna %3D u(x, o) = Cos(x) Ut (X, o) = los (2x) %3D

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Q: Find the work done by the force F= xyi + (y - x)j over the straight line from (1,2) to (3,6).

A: we have to find the work done by the given force

Q: If F(x,y) is a differentiable function and the equation F(x, y) = 0 defines (y) as a function of…

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Q: biveniJN/AN, Dis the midpoint of JA Prove: AJON E A ADM A.

A: Solved below

Q: 4-j2 z = tanh¯(4e-j3)*

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Q: The work done by the force F = in the displacement along the path C: r(t)=from the point (1,0,0) to…

A: To find the work done by a force

Q: Find up +Uy given that u (x,Y)s. (x,x)s

A: Given that ux, y=tanx3+mx2y2+my4m-x2-y2 To find ux+uy we follow the following steps calculate…

Q: Find aw in terms u and v if du w= In(x2+y2 +z²), x = ue" sin u, y = ue' cos u, z = ue"; (u, v) =…

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

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Q: Uzz + Uyy = u %3D tt u(г, у, 0) — 0 и (х, у, t) — 2? -у?

A: In this question, we have find the velocity of the membrane at given time and point.

Q: 2л where a, b > 0. dx = - ile* + b Prove that %3D

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Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann equations if ux = v, and u,…

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Q: The spherical equivalent of the vector B = yi + (x + z)j located at (-2,6,3) is given by

A: The variables of the spherical coordinate system r,θ,ϕ are given as.…

Q: Evaluate the line integralf, (xy + z)ds along the line segment C from (1,2,3) to (-1,1,5).

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Q: 5. Show that the equation of the tangent plane to f(r, y, z) = c at (ro, Yo: 20) is given by the…

A: This is the question from vectors.

Q: Find the work done by the force F (x; y) = xyi-(y-x)j over the straight line from (1,1) to (2,3)

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Q: Find the work done by the force F= xyi + (y - x)j over the straight line from (-1,- 2) to (2,1).

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Q: Evaluate the line integral ∫Cx^3z ds, where C is the line segment from (0,3,8) to (8,4,6).

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Q: 8. Use Green's Theorem to evaluate px*dx+xydy where C is the triangle with vertices (0,0), (1,0),…

A: As we know from the Green's theorem: ∫CMdx+Ndy=∫∫R∂N∂x-∂M∂ydA And The expression for the equation of…

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

A: We know, sin z = sin x cosh y + i cos x sinh y, Where, cosh and sinh are hyperbolic functions.

Q: Qz: IF U=h cx,g )= Lnce+ ě).show that %3D Sy

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Q: Show that f (z) = e? is not analytic using Cauchy-Riemann equations.

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