Let X=ℝ2 and define d,:ℝ2×ℝ2→ℝ d2((x1 ,y1),(x2,y)2)) = max{|x1 - x2|,|y1 - y2|}. Draw the neighborhood N(0;1) for d2 where 0 is the origin in ℝ2.

Q: Let f (x, y) = 3xy − 6x − 3y + 7. Find the absolute maximum and absolute minimum values of f over…

A: Given: f(x,y)=3xy-6x-3y+7   Find: The absolute maximum and minimum values.

Q: Calculate the line integral f -y²dx + xdy where ŕ is the path y² = 2x − x² such that x>1 and y > 0.

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Q: 3. Show that the line integral 4. x + y)dx + dy C is independent of path and evaluate this integral…

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Q: 3. Compute the inner product g (v,w) of v = + x. 2 əx and w = x.-+ 2y. Əx ay' Where the line element…

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Q: 2) Use Green's Theorem to compute f F df where F= (3x +4y)i + 2xyj and C is the triangular path from…

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Q: Let f(x, y) = xcosy- y'e*y. fry at (1,0)is equal to: %3D | -2/3 -2 1

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Q: 28. Evaluate by Green's theorem (x - cosh y) dx + (y + sinx) dy, where C is the rectangle with…

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Q: Transform the BVP 0<x<1 u (0, t) =t u(1, t) = 1 into new one with homogenous boundary conditions, we…

A: To find -

Q: Show that the line integral | 2.xe- dx + (2y – a²e¬") dy is independent of path and evaluate the…

A: Here we have to show that the given line integral is path independent and evaluate the line integral…

Q: 3. Calculate the line integral S F - d7, where F = (x²y³, x³y?) and C is the path given by C 7(t) =…

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Q: Suppose that a point (X,Y) is chosen at random from the region S on the xy-plane containing all…

A: Consider  two random variables x and y, suppose x is a sex of a person as well as  whether…

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

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Q: If z = f(u) + g(u), where u = - and v = x y. Prove that y %3D x: Zx - y· Zy = y2 · Zyy – x2 . Zxx…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

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: Evaluate the integral ∫c(2x-y)dx+(x+7y)dy along the path C, defined as y=9-x2 from (0,9) to (3,0)

A: Given ∫c2x-ydx+x+7ydy  where y=9-x2  from 0,9 to 3,0 Parametrize C by xt,yt=t,9-t2 and 0≤t≤3

Q: Show that the integral S. (3x²y + y)dx + (x3 + x)dy is independent of path, if it is, then evaluate…

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Q: If z = f (u) + g(v), where u = = and v = x ·y. Prove that Q1: x· Zx – y· Zy = y² · Zyy – x² · Zxx…

A: We have z=fu+gv, and u=xy  and v=x.y We need to use the chain rule and prove x.zx-y.zy=y2.zyy-x2.zxx…

Q: 3) Let F = (y+ev=,8x+ cos(y*)) Use a theorem to compute ſ F •dT where c is the boundary of the…

A: Given that  F→=y+ex,8x+cosy3 To compute ∫cF→⋅dr→ where c is the boundary of the planar region…

Q: Find the absolute maximum and minimum values of f(x, y) = 3xy−6x−3y + 7 on the closed triangular…

A: Find out the absolute maximum and minimum values of  f(x,y)=3xy-6x-3y+7 for the maximum and minimum…

Q: Find the bounded solution of âyở y x> 0, t>0 where y (0, t) = 1, y (x, 0) = 0.

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Q: Let z = f(x, y) = x² – y² + xy + 2. Consider the triangular region in the xy-plane with corners at…

A: The given function is z=fx,y=x2-y2+xy+2

Q: 2) Use Green's Theorem to compute f F. df where F=(3x +4y) i + 2xyj and C is the triangular path…

A: Use green's theorem

Q: Find the bounded solution of au Əx = 2 อน at + U_if_U(x, 0) = 6e-³x. 1

A: Given: The equation is ∂u∂x=2∂u∂t+u and the given initial condition is ux,0=6e-3x. To find: The…

Q: 43. Find f, 1dS, where s is the portion of the surface deter- mined by z = x2 – V3y that lies above…

A: Find ∫s1 dS where S is the portion of the surface determined by z=x2-3y that lies above the region…

Q: Use the transformation u = - -6x +y, to evaluate the integral [ ( (5x • y) dx dy, where R is the…

A: use the transform u=-6x+y, v=5x+ywe evaluate ∫∫R 5x+y dx dywhere R is the parallelogram bounded by…

Q: Consider: J²u J²u + = 0, əx² Əy² ди ər u(x,0) = 0, (PDE) (BC1) (BC2) (BC3) u(x, y) remains bounded…

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Q: By using Green's Theorem, evaluate the integral (1+xy') dx +x²y dy, where R is the closed R path…

A: From the given problem:   I=∫R1+xy2dx+x2ydy   From Green's theorem:   ∫F1dx+F2dy=∬∂F2∂x-∂F1∂ydxdy…

Q: 2. A solid G is bounded by the paraboloid z = 4-²-y², the cylinder x² + y² = 4 and the plane z = 5.…

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Q: Use Greens Theorem to evaluate: §. (2x + ev´)dy - (4y2 + e*")dx where C is the boundary of the…

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Q: Suppose that u € C² (R²) and u(x, y) = 0 everywhere on the boundary of the square ((x,y): 0≤x≤ R,0 ≤…

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Q: If z = f(u) + g), where u = = and v = x· y. Prove that y x: Zx - y Zy = y? Zyy – x2 - Zxx using the…

A: We have to prove this

Q: Find the surfaces on which the function u = 2 (x2 + y²)/z is constar

A: Since u is a constant so, x2+y2z=4a , where a is an arbitrary constant.

Q: (3) +. R Ris bounded by the semicircle y = V2x – x² and the line y = x

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Q: Let, T: R3 → R³;T(x, y, z) = (2x + y, y – z,2y + 4z) Find the rank and nullity of T.

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Q: 4. Let f(x, y, z) = e² + zy² and let C be a straight line segment connecting the points (4,2,1) and…

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Q: 2. Let i = y?x + xŷ + z²ż, and let S be the paraboloid z = x2 + y? with 0 < z < 1. Show that fci. de…

A: Given  v→=y2x^+xy^+z2z^ and S be the paraboloid defined by  z=x2+y2      , 0≤z≤1 The boundary C of…

Q: Find the Wronskian of a given set {yı, y2} of solutions of 6. y"3(x21)y- 2y 0, given that W(T) = 0.

A: Given the differential equation and the value of the wronskian at pi as:

Q: Evaluate the line integral §. xy²dx + 2x²y dy over triangular path with vertices (0,0), (2,2) and…

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Q: Evaluate the line integral y² dx + x²y dy where C' is the rectangle with vertices (0,0), (5,0), (5,…

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Q: 2. Let z = sin(xy) + x siny,x=u² + w² and y = uw. 2a. Compute Vz= when u = 0 and w = 1.

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Q: Verify Green's Theorem by evaluating both integrals ам √₁ y ² dx + x² dy = √₂√ √ (³x - dA ax ду for…

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Q: Show that the line integral / 2re" dx + (2y – x²e¯") dy is independent of path where C is any path -…

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Q: .The primal form of SVR with soft margin is M 1 min |w+CE(6i + &;) i=1 s.t. y) – w"x1) – b < e+ §i,…

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Q: Solve ut = upx in the interval [0, 5] with u(0, t) = 0, and u(5, t) = 1 for t > 0 and u(x, 0) = 0…

A: Hi! As per norms, we will be answering only the first question. If you need an answer to other then…

Q: Consider a potential problem in the half-space defined by z 0, with Dirichlet boundary conditions on…

A: The green function solution to the potential problem with this boundary for Dirichlet boundary…

Q: Use Green's Theorem to evaluate S. axy dy + cxy dy, where C is a counterclockwise oriented…

A: Given : a=2, c=8 We have to use Green's Theorem to evaluate            ∫C axy dy+cxy dy where C is a…

Q: 4) Let f(r, y, z) = In(r + 2y + 3z). Find Vf = . %3D

A: Given: f(x, y, z)=ln (x+2y+3z)

Q: 3 Find the bounded Solution of : = xt メ>。,t>。 >y(メノ) where y cx, o) = %3D and y co,t) = 0 . Ansi] y=…

A: