Show that T(u, v) = (u² – v², 2uv) maps the triangle = {(u, v) : 0 < v < u < 5} to the domain D bounded by æ = 0, y = 0, and y? = 2500 – 100x. Use T to evaluate Io Vz² + s° dz dy

Q: Let C be the positively oriented square with vertices (0,0), (1,0), (1, 1), (0, 1). Use Green's…

A: Let's solve by using green theorem.

Q: 7. D is bounded by the surface (r2 + y² + z²)² = a²(x² + y² – 22). %3D

A: We have to find volume where D is bounded by the surface (x² + y² + z²)² = a²(x² + y² — z²).

Q: Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition…

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Q: Let S be the solid bounded by the planes z = o and y + z = 3, and the cylinder x2+y2 =2y-

A: answer is in next step

Q: If F= 2y I- 3J +x²k and s the surface parabolic cylinder y= 8x in bounded by the planes OF the 2.…

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Q: Evaluate the line integral xyz ds where C is the line segment joining the points (1, – 1, 1) to (1,…

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Q: 2) Use a change of variables to evaluate ff, zdV; where D is bounded by the paraboloid z = 16 – x² –…

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Q: If E is the solid shape bounded by x 2 0,y 20 ,z 2 0 and the plane 2x + 3y +z = 6 evaluate (x + y)…

A: The given question is solved below

Q: Compute the line integral ∫_C F · dr, where F (x, y) = 〈−x^3, xy〉 and C is part of circle x^2 + y^2…

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Q: Apply Stoke's theorem to calculate S. (x + y – 4)dx + (2x – z)dy + (y + z)dz, where C is the…

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Q: Let D be the half-disc in the xy-plane given by x +y <9. Calculate e-(x2+v)dA

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Q: |Use Green's Theorem to evaluate | -x'ydx+xy°dy where C is the boundary of the C circle x + y =16…

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Q: If F = 2y i - zj + x²k and S is the surface of the parabolic cylinder y²= &x in the first octant…

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Q: Let R be the trapezoid with vertices (1,-1), (3,3), (3,-3) and (9,9). Give a change of variables to…

A: Given that ,  R be the trapezoid with vertices (1,-1) , (3,3) ,(3,-3) and (9,9) . Let , the vertices…

Q: a) Evaluate the line integral TM UTM in terms of 7 where F = cos yi+ rj + ye*k and C is the line F…

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Q: Calculate S," (2x+y) dV where V is a closed space bounded by the cylinder z = 4 – x2 and the planes…

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Q: A solid G is bounded by the paraboloid z = 4 – x2 – y², the cylinder 22 + y? = 4 and the plane z =…

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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: Q:4) Use Green theorem and evaluate the line integral ${Mdx + Ndy} Where M(x,y)=y², N(x,y)=x², C is…

A: See the details solution in below

Q: Use Gauss Divergence theorem to calculate (5zi+ 4yj- 2z k).n dA over the sphere S: 2 + y? + 22 = 9.

A: This is a problem related to vector calculus. Based on the general formula of gauss divergence…

Q: Use the Divergence Theorem to evaluate (5x + 10y + z) ds where S is the sphere x2 + y2 + z2 = 1.

A: In the given problem, S is the sphere x2+y2+z2=1Divergence Theorem:∬SF·n dS=∭Ediv F dVNow, the…

Q: Verify Green's theorem for Q7 [x(x? + y²)dx – y(3x² – y²)]dy Where C is the boundary of the triangle…

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Q: 22. Compute the contour integral (Баллов: 2) Sof (2) dz, z E C, if f (z) = and path is semicircle z…

A: Topic:- application of integration

Q: Evaluate xy dV where E is bounded by the parabolic cylinders y = x², x = y² E and the planes z = 0…

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Q: evaluate $ A. dš over the sphere R = 3 and show whether Divergence theorem holds or not. %3D

A: Calculate surface integral over R=3 or dr=0 ∮A→×ds→=∮R cos2θ,π,0×r2sinθdθdϕ,rdϕdr,rsinθdrdθ…

Q: O By using Green's theorem, evaluate $ (In (2²) + æy² – y²) dæ + (ev + x²y – 3ry) dy, where C is the…

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Q: Use the Green's theorem to calulate So (3x²y – xy? – 3.x²y²) dx + (x³ – 2.x³y+ x²y) dy, where C is…

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Q: Use the Green's Theorem to evaluate the line integral 2xydx + x² – y² dy where C is the curve…

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Q: Evaluate || F.ndS F = xi + yj+(z² –1)k if and o is the curved surface of the cylinder y = \4-x? from…

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Q: 2 Use Green's Theorem to evaluate Jx*ydx-xy dy, where C is the circle 2+ y? – 2601 with…

A: Let be the vector field and C be the boundary of the region D. The Green's theorem states that .

Q: -) Let -y M(x, y) = x2 and N(x, y) = + y? x2 + y? how that ƏN ƏM dx ду ) Show that -y F(x, y) = x² +…

A: Given :-  A vector field for conservative field in the plane ,  ∂N∂x = ∂M∂y To Determine :-  M x,y =…

Q: Calculate F- dS if F(x, y, z) = x²ye²i+ æy?j – 2xye*k, where S is the surface of the box bounded by…

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Q: (b) Let z(r, y) = exp(-|r|) cos r + y², where r + y /9 = 1. Show that z(r, y) has minimum points…

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Q: Let C denote the positively oriented boundary of the square whose sides lie along the lines x =+ 2…

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

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: Use the Divergence Theorem to evaluate (3x + 2y + z2) ds where S is the sphere x2 + y2 + z² = 1.

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Q: Evaluate (x + 5y) dV where E is bounded by the parabolic cylinder y = x² and the planes z = 3x, y =…

A: given is  a partial differential equation in dX , dY, dZ we can partially integrate the given…

Q: Use the Divergence Theorem to evaluate (4x + 7y + z²) dS where S is the sphere x2 + y2 + z² = 1.

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Q: Let W = Span{u} where u -2. Let y = |1, find the distance from y to W L3]

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

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Q: Let S denote the part of the cylinder x2 + y² = 9 that lies between the planes r + y = 1 and x – y =…

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Q: Use the Divergence Theorem to evaluate (4x + 3y + z) dS where S is the sphere x2 + y2 + z2 = 1.

A: The divergence theorem establishes the equality between surface integral and volume integral.

Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

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Q: 7. D is bounded by the surface (a2 + y² + z²)² = a²(x² + y? – z²). %3D

A: Sol

Q: Calculate ⌠⌠F*dS if F(x,y,z)=x^2ye^zi + xy^2j-2xye^zk where S is the surface of the box bounded by…

A: Use projection on xy plane

Q: Evaluate the line integral Levis where C is the half circle x² + y² = 1, y ≥ 0. x² + y² ds

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Q: $ ((x,y) (E,6) )e product Prove 2 xt + 3yt on e?. Gompute inner IL(,2), where the norm i's…

A: We have to prove that f( (x, y) , (t, t) ) = 2xt + 3yt is an inner product on R2 . 1.) &lt; ( 0, 0 )…

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: Evaluate / (7xi – zk) · ndA over the sphere S : x² + y² + z² = | S a) by divergence theorem, (b)…

A: a) By divergence theorem, ∬SA. ndA=∫∫∫V∇.A dV Now,  ∇.A=∂7x∂x+∂0∂y+∂-z∂z=7-1=6 By changing to…

Q: Use Green's Theorem to Calculate the Line Integral ſy²dx + x²dy, where c is the triangle o vertices…

A: Given that, the line integral is ∫y2dx+x2dy and the vertices of the triangle is 0,0,4,0 and 4,4. By…