Use Green's theorem to evaluate F.dr where F(x, y) = xy² i+ (1 – xy³) j and C is the parallelogram with vertices (-1,2), (-1, –1), (1, 1) and (1,4). The orientation of C is counterclockwise.

Q: Calculate F. dĩ, where C is the square of (1,1, 1), (–1, 1, 1), (–1, –1,1) and (1, –1,1), with…

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Q: Find a function r(t) that describes the line through P(3,4,5) that is orthogonal to the plane 2x-z=4

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Q: Let L be a line which passes through the point (-7,0,1) and is perpendicular to the plane x + 2y +…

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Q: Calculate , z² dS if S is the prism whose base is the triangle in the limited xy plane by the x axis…

A: Here given figure is Since prism has 5 surfaces (1)-OAB (2) OADE (3) CDE (4) OBCD (5) ABCE

Q: Find the projection of d on the plane P if a 4 and P has the equation x + y+ z = 1.

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Q: Find the mapping that reflects the plane in the line x+y-1=C O (x,y)>(y+1,x+1) (x,y)→(-x+1,-y+1) O…

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Q: Let r(t) be a positively oriented rectangle with corners (0,0), (0,1), (2,1) and (2,0) and F(r, y)…

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Q: Let L be a line which passes through the point (-4,0, 1) and is perpendicular to the plane x + 2y +…

A: Please refer the attached image for complete solution.

Q: Let T, denote the plane parallel to the xz-plane through the point (1, 2, 3). Let n2 denote the…

A: To solve this problem first of we find the equation of the plane π1 and π2 in standard form and then…

Q: C is the hexagon in the xy-plane with vertices (5, −5), (5, 5), (0, 10), (−5, 5), (−5, −5), and (0,…

A: Given : Vertices : (5,-5) , (5,5), (0,10), (-5,5), (-5,-5) and (0,-10) Integral given :…

Q: find the trace of the plane in the given coordinate plane.  −x + y = 4, xz

A: Find trace of the plane in the coordinate plane -x+y=4 in xz

Q: Let f(x, y, z) = x² + 2y? + 4z2. Find the point on the plane x + y + z = 14 at which f has its…

A: Given: fx,y,z=x2+2y2+4z2and the plane equation x+y+z=14 To find the point x,y,z The Lagrange's…

Q: Let L be a line which passes through the point (-8, 0, 1) and is perpendicular to the plane x + 2y +…

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Q: Let T be the plane x – y – z = 7. Which of the following planes are orthogonal to T and pass through…

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Q: H.W: Find the distance from S (1,1,3) to the plane 3x+ 2y+ 6z = 6

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Q: 2. Use the transformation u = x – 2y, v = 2x + y to find x – 2y 2.x + y Where R is the rectangular…

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Q: If possible, determine the vector, parametric,symmetrical and reduced line r that: a) passes…

A: Given the lines S1:t·1,2,3, t∈ℝ and S2:y=x+2, z=x. To find a line L passing through the point P1,1,1…

Q: Use Green's theorem to find $ x? ydr + xdy over a counterclockwise triangular path passing though…

A: To find the line integral ∮x2ydx+xdy by using Green's theorem, First, we need to draw the given…

Q: Find the linearization L(z, y, z) of the f(x, y, z) = 2,/z° + y° + z° at the point (1, 2, 3). %3D…

A: Consider the provided function, fx,y,z=2x3+y3+z3 At the point 1,2,3 find the linearization. The…

Q: Find the linearization of f(x, y,z) = x + y centered at (8,4, 5).

A: Given:fx,y,z=x+y at 8,4,5Formula:Lx,y,z=fx0,y0,z0+fxx0,y0,z0x-x0+fyx0,y0,z0y-y0+fzx0,y0,z0z-z0

Q: Let R = (c, 1,-1) where c denotes a real number. Let ä denote the plane x + y – z = 2. Find the…

A: Given Information -  The distance from the point R = (c,1,-1) to the plane x+y-z = 2 is 2. We have…

Q: Use the flux form of Green's Theorem to evaluate |(6xy + 8y°) dA, where R is the triangle with…

A: Greens' Theorem : Let D be the region bounded by the curve C. Let F(x,y)=P,Q be the vector field…

Q: a) Evaluate f, (2x – z) dS, where S is part of the plane 3x + 2y + z = 1 that lies above the…

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Q: Find all planes in R3 whose intersection with the xy-plane is the liner( t) = t ( 2, 1, 0) .

A: GIVEN  Intersection with the xy-plane is the line r( t) = t ( 2, 1, 0)

Q: Find the symmetric equations of the L that is parallel to the two planes x+y-z= 2 and z + 3y+ z =…

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Q: Find the linearization L(x,y,z) of the f(x,y,z) = 6sqrtx^3+y^3+z^3 at the point (1,2,3).

A: The linearization of a function f(x,y,z) at a point (a,b,c) is given by the expression;…

Q: Let C denote the triangle with vertices at (0, 0), (0, 1) and (2, 1), oriented positively. Then f(x²…

A: We have to apply the green's theorem here.

Q: Let L be a line which passes through the point (-8, 0, 1) and is perpendicular to the plane x + 2y +…

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Q: Evaluate Sf,(z+ 3y – x²) dS where S is the portion of z = 2 – 3y + x² that lies over the triangle in…

A: Ans :

Q: Find the residues of f(z) = z²-2z (z+1)²(z²+4) at all its poles in the finite plane.

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Q: Find the symmetric equations of the L that is parallel to the two planes x + 2y - z= 2 and x + 6y+ z…

A: Given: Two planes, P1: x+2y-z=2, P2: x+6y+z=-3, Point P1, 2, 3

Q: Let R be the triangle in the xy-plane with vertices (0,0),(0, 1),(1,0). Find C[n(2 – x – y) dA.

A: Given vertices of triangle now we have to find the area as shown below

Q: Use the coordinate transformation u = x + 2y, v = x - 2y to evaluate S (x2 - 4y)dA over the…

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Q: Let L be a line which passes through the point (-6,0, 1) and is perpendicular to the plane x + 2y…

A: Direction ratio of Line is < 1,2,6> Write equation of straight line and solve it with plane to…

Q: Let L be a line which passes through the point (-5,0, 1) and is perpendicular to the plane x + 2y +…

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Q: Let f: R → R be the function which satisfies for all (z. y, z) € R° that z cos(ry) and let C C R' be…

A: Here we have to find out line integral of function f(X,Y,Z).

Q: Find i x w if ||T|| = 3, ||w|| = 4, 0 = and ī, w both lie in the xz-plane (as shown). 3 %3D

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Q: Let L be a line which passes through the point (-7,0, 1) and is perpendicular to the plane x + 2y +…

A: Here we find the equation of line L and also find the intersecting point of line and plane.

Q: Find m, n ∈  R so that the line r X: (n, 2,0) + λ (2, m, m) is contained in a plane π: x -3y + z = 1

A: The direction cosines of the line, r(x)= (n, 2,0) + λ (2, m, m) are <2, m, m> . The direction…

Q: 4. let f(x, y) = x² – xy + y² – y. find the direction u and Duf at Po(1,–1) for which Duf is the…

A: Given:- f(x, y) = x2 -xy+y2 -y To find:- Direction u and Du f at point p0 (1, -1) for which Du f is…

Q: Find the eguation f Plane that Passes through P(12-1,3) and s Parallel to the Plane 3X++6? ans. 3X+y…

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Q: a) Show that the line I = 0, y = t, z =t where te R a) lies in the plane 6z+4y–4x = 0 b) is parallel…

A: In the question it is asked to solve for the planes having equation given below.

Q: Let L be a line which passes through the point (-3, 0, 1) and is perpendicular to the plane x + 2y +…

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Q: Find the linearization L(x,y,z) of the f(x,y,z)= 8 square(x^3+y^3+z^3) at the point (1,2,3)

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Q: c. Let P(x, y) = y² + sin¬1 x² and Q(x, y) = 2xy + 4x. Suppose that C is the parallelogram with…

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Q: b) Use the transformation u = 2x – y, v = x + 3y to find JfR dA, x+3y where R is the rectangular…

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Q: Let L be a line which passes through the point (-4, 0, 1) and is perpendicular to the plane x + 2y +…

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Q: Given Fxy 2) = (z², xy, yz) and C is a triangle in the plane x + y + z = 2 with vertices (1,1,0),…

A: We will use basic knowledge of vector calculus.

Q: Let ƒ (x, y) = x² - y² . Find all all points (x, y) such that Vf(x,y) =±(1, 1).

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