Let X=(x1,x2) and Y==(y1,y2) belongs to R^2. Verify that = 5(5x2y2+ x1y1-2x1y2-2x2y1) is an inner product on R^2

Q: calculate th integral SSo(x+y) dydx where D is th reigon that deter mined in thees Curves : Y = x, Y…

A: Solution is given below:

Q: Show that every positve colufion at Ay + B, k=osl,2 bounded, %3D JK is i-e, osysB-if A<1. 1-A

A: Question: Objective is to find the upper bound of     yk+1=Ayk+B, k=0,1,2,... where yk is positive.

Q: Compute the scalar line integral x – yz ds where C is the line seg- ment from (0, 1, 1) to (1,0, 1).

A: Given  ∫C x-yz   ds    where C is line segment from (0,1,1) to ( 1,0,1)

Q: 2. Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on…

A: An equilateral triangle can be inscribed in the epitrochoid and that its centroid is onthe circle of…

Q: Find the centroid (¯x,¯y) of the region bounded by: y=8x2+9x,   y=0,   x=0,   and    y=0,   x=0,…

A: Given data: The first function given is f(x)=8x2+9x. The second function given is g(x)=0. The given…

Q: Show that F = (2xy + z³)ï + (x²)j+3xz²k is a conservative field. Find the scalar potential and the…

A: Given F = 2xy+z3 i ; x2j + 3xz2kThe objective is to show that given vector fieldis coonservative…

Q: Evaluate the triple integral ∭ExydV where E is the solid tetrahedon with vertices…

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Q: Calculate line integral F-dr where F(x, y) = (7x + y, 5x + 2y) and C is a rectangle with vertices…

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Q: Show that the vector field F = 2xyi + (x - 2 y)j is conservative and evaluate the line integral %3D…

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Q: Find the divergence of the vector field F→(x,y,z)=xyzi→−yzj→+xzk→at the point (8,3,2) .   Enter…

A: The given vector field is : Fx,y,z=xyzi-yzj+xzk To find the divergence at 8,3,2

Q: . The solid bounded by z = 1 – x², z = 5 – 2x², y = 0, and y – z = 6. |

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Q: Compute the flux of the vector field (x, - xy*), out of the rectangle with vertices (0,0), (5,0),…

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Q: The directional derivative of the field u(x, y, z) = x² – 3yz in the direction of the - vector (i +…

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Q: Find the centroid of the region that is bounded by the line Y = 3X, the line X 2 and the X axis.

A: To find the centroid

Q: A region in the xy-plane is bounded by y = 2x + 2, X = y + 2, y = -2, 2 y = 2. and

A: Given curves are y=2x+2 and x=y22+2 The limits are y=-2 and y=2 We isolate x from the first curve,…

Q: Use Green's Theorem to evaluatc the integral x²ydx + xydy where C is the rectangle with vertices (0,…

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Q: Find the centroid (x¯,y¯)(x¯,y¯) of the region bounded by: y=7x^2+2x,   y=0,   x=0,   and   x=9?

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Q: Find the centroid of the region bounded by the parabola y= 8x and the line y=2x. O (0.8,2) O (1,4) O…

A: We need to find centroid of the region.

Q: Find the centroid of the area bounded by x^2 - 4y + 8 = 0, the lines x=0 and x=4, and the x-axis.

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Q: 4) Find the centroid of the region that is bounded by the X axis, the Y axis and the line that…

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Q: Find the curl of the vector field F(x,y,z) = x²zi – 2xzj + yzk and the point (2, -1, 3).

A: Let the given vector field is :

Q: Is = xjx + yjy+zźz3 an inner product on R'?

A: No it is not an inner product.  Inner product looks like a dot product.

Q: Find the area of the region cut from the plane x+4y+8z=9 by the cylinder whose walls are x=y2 and…

A: Given, x+4y+8z=9 x=y² x=18−y² Therefore, The surface is  f=0 with f= x+4y+8z-9

Q: Use Ring Method Rotate the region bounded by x = (y – 3)2 and x = 16 about the x = -5.

A: Let's find.

Q: Calculate the volume under the elliptic paraboloid z = 4x² + 6y2 and over the rectangle R = [-2, 2]…

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Q: Determine the coordinates of the centroid of the shaded area. 3″ 6" 2"

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Q: 68. Solid bounded above by 3x² + 3y² + z² = 45 and below by the xy-plane

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Q: Express (5x + 7y, 8x + 5y, 0) as the sum of a curl free vector field and a divergence free vector…

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Q: Show that the vector field =(-2 x²y xy²¹ is conservative by finding all its potential functions.…

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Q: Calculate the volume under the elliptic paraboloid z = 4x2 + 6y2 and over the rectangle R = [-2, 2]…

A: The solution for the above question is shown below.

Q: + 1 -оу-х -oy=x2

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Q: Show that 1 y - X 1. Z-y X-Z = 1. %3D alx-y) alz - x)

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Q: Compute the flux of the vector field (x3, – xy³), - out of the rectangle with vertices (0,0), (3,0),…

A: Given: F=x3 ,-xy5with rectangle with vertices (0,0) , (3,0) , (3 ,2) , (0,2) To find flux of vector…

Q: Find the centroid (¯x,¯y)of the region bounded by: y=4x2+7x, y=0,   x=0,   and   x=3 ¯x=    ¯y=

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Q: Find the centroid of the region that is bounded by the line Y = 3X, the line X = 2 and the X axis.

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Q: Show that the vector field F = 2xyi +(x² - 2y)j is conservative and evaluate the line integral (F•dš…

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Q: Express (5x + 7y, 8x + 5y, 0) as the sum of a curl free vector field and a divergence free vector…

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Q: Calculate line integral F-dr where F(x, y) = (7x + y, 5x + 2y) and C is a rectangle with vertices 0,…

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Q: Calculate the volume under the elliptic paraboloid z=3x2+7y2 and over the rectangle R=[−2,2]×[−4,4

A: Explanation: We know that,  Volume =∬zdydx By using the double integral concept, the volume of the…

Q: Let i = (x²yz, ry z, zyz?). Compute curl i

A: The given vector is: u→=x2yz, xy2z, xyz2

Q: Find a vector potential field for Ġ = [6xz+x³, –3r²y – y², 4.x + 2zy – 32²], i.e. find F such that V…

A: The solution is below.

Q: Solve by Cramer’s rule         3x  + y   + 4z = 11         4x  – 4y + 6z = 11…

A: Topic = Determinants As per guidelines , only one question can be answered out of different multiple…

Q: #1. Evaluate (2x – 5y) dA, R where R is the parallelogram with vertices (0,0), (2,3), (–1,4) and…

A: The given region R is a parallelogram with vertices as shown in below graph. We have to use change…

Q: 2. Solve for the centroid of the shaded region. -zolx-1)

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Q: Consider the region in the first quadrant bounded by y=x^2, the y-axis, and y=4. -the y-axis using…

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Q: 6. Show that the system with vector field -x + sin(y) F(x, y) : -y+ cos(x), admits no periodic…

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Q: Compute the divergence and curl of F= -(z³-3x)j+4y²k+x²yi

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Q: Find the Jacobian for the transformation x = 3u + v and yY= u+3v. NEXT, use this transformation to…

A: We have given transformation x=3u+v and y=u+3v Jacobian J=dxdudxdvdydudydv

Q: Calculate of A= Xzi- 2x*yzj+2yz'K at ( 1,-1,2). the Curl and Divergence

A: Given A = xz3i-2x2yzj+2yz4k Compare it with A = Axi+Ayj+Azk Hence Ax = xz3Ay = -2x2yzAz = 2yz4 Then…

Q: Use Ring Method Rotate the region bounded by y = x2 – 2x + 4 and y = x + 14 about the x – axis.

A: The given curves are y=x2-2x+4 , y=x+14 and x -axis To find: the volume the region bounded by the…