Calculate both sides of Green's Theorem when F = [-x²y, xy²] and y positively oriented circle x2 + y² = R²

Q: By using Green's theorem evaluate £ (Ary – y") dr + (** + 3ry) dy where C is a closed curve starting…

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Q: If (a,ß.r) is a point at which the surface x2+y2-z-2x+42=0 has a horizontal tangent plane, then Iyl=

A: Horizontal tangent plane

Q: ind the equation for the tangent plane to the surface z = In(6x2 +9y2 + 1) at the point (0,0, 0).…

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Q: (b) Verify that Green's Theorem is true for S,3(y² – x) dx + (x² – y)dy given that the curve C is…

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Q: Consider the right triangle withvertices (0, 0), (0, b), and (a, 0), where a > 0 and b > 0.…

A: Given right angle triangle has vertices  (0, 0), (0, b), and (a, 0), where a > 0 and b > 0.…

Q: Let C be a circle lying in the plane x+ z = 2 with center at (1,0, 1). Evaluate the line integr V2…

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Q: Use Green's theorem to evaluate , (5xy + x2 + y?) dx + (x²-y)dy where C is a closed curve that is…

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Q: Find an equation of the tangent plane to the surface given by z² – 2x2 – 2y? = 12 at the point (1,…

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Q: (b) Find the circle of curvature at (0, 1)of the curve y = x3 + 2x² + x +1

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Q: Let YR be the circle of radius R with counter clockwise orientation. Prove that 2 – 1 - lim R00 YR…

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Q: Find the slope of the tangent line to the curve that is the intersection of the surface z = x2 - y2…

A: Calculate point of intersection and differentiate it to get the answer

Q: Let C be the positively oriented circle x2 +y2 =1. Use Green's Theorem to evaluate the line integral…

A: C is the circle  x2 + y2 = 1 The integral to find  ∫c3ydx +9xdy

Q: Calculate the line integral ∫Cy2ds, where C is a part of the circle x=acost, y=asint, 0≤t≤π/2.

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Q: Find the equation for the tangent plane to the surface Z =-8x² - 3y2 at the point (2 1, -35). O 2x +…

A: We can find the answer as below

Q: Use Green's Theorem to evaluate pxcosydx – ysinxdy, where the curve C is the square with 0), (TT,…

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Q: Verify greens theorem form 1 for F=x^3i+yj and c is the circle x^2+y^2=1, oriented anti clockwise

A: Find ∫CFdr, the line integral of F, where C is the circle x2+y2=1, oriented anti clock wise…

Q: Let C denote the circle of radius 1 in R2 centered at the origin, oriented counterclockwise. for…

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Q: use green's theorem to evaluate ∫c 5y^3dx-5x^3dy where C is the circle x^2+y^2=4 counterclockwise

A: Given: ∫c 5y^3dx-5x^3dy where C is the circle x^2+y^2=4

Q: 3) Prove thet the xy-plane with constant curva tue k= urve In is o circle of radious

A: We have to prove that a curve in the xy plane with constant curvature k=1r is a circle of radius r

Q: Let C be the positively oriented circle x2 + y² = 1. Use Green's Theorem to evaluate the line…

A: Given: C is a positively oriented circle x2+y2=1.  Objective of the question is to calculate…

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: Compute F. dr for the oriented curve specified. F(x, y) = (9, y), quarter circle x² + y2 = 1 with x…

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Q: Find an equation of the tangent plane to the surface z = -! -3x? + 3y – 2x + 2y + 2 at the point (2,…

A: We have to find tangent plane

Q: Determine the projection of the following surface on the xz plane. (xz trace) 4x – + = 0

A: Given equation of surface is 4x-y24+z28=0

Q: Suppose that the plane tangent to the surface z = f(x, y) at (-4, 2, 5) has equation z + 5x + 4y =…

A: From the equation of tangent, we can derive: z = -(7 + 5x + 4y) Hence,

Q: Prove that the circle {(x, y), x² + y² = 1} is hot homeomorphic to the interval [-1, 1].

A: We will prove this result by contradiction: Suppose there exists a homeomorphism between -1,1 and S…

Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

A: Greens theorem states that "for closed curve C enclosed a region R then the line integral along the…

Q: Use Green's Theorem to evaluate the integral (x² - y)dax + 3xdy where C is the circle x² + y² = 1…

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Q: Graph the lemniscate curve r2 = sin 2θ in the Cartesian xy-plane

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Q: Verify Green's Theorem in the plane for F = -3y³i+ (3x3 + cos y)ĵ where C is the circle x² + y² =…

A: In this we have to verify green theorem.

Q: Find the equation for the tangent plane to the surface z=-8x2 -4y2  at the point   (2,1,-36)

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Q: OShow that every point (b, c) in the plane lies on a tangent line to the graph of y = x³.

A: Consider the given information. y=x3 Find the derivative of the function. dydx=ddxx3y'=3x2

Q: Suppose a point X lies in the exterior of a disk, and suppose that two lines are drawn from X…

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Q: |Verify that Green's Theorem is true for S,3(y2 – x) dx + (x² – y)dy given that the curve C is…

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Q: (b) Verify that Green's Theorem is true for S,3(y² – x) dx + (x² – y)dy given that the curve C is…

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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: Find the length of the circle of radius r defined parametrically by x=rcost, y=rsint,0≤t≤2π.

A: Here I am attaching image so that you understand each and every step.

Q: By using Cauchy's integral formulas evaluate sin Tz2 + cos TZ² dz (z + 1)(z – 2) C and e2z dz (z –…

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Q: Use Green's Theorem to evaluate (1in z – 2y) dz + (x – ycos y) dy, where C is the circle (x – 4)² +…

A: Given: C is the circle x-42+y-42=9. ∮Cxsinx-2ydx+x-ycosydy. To find: Using Green's theorem, value…

Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

A: Greens theorem

Q: The tangent plane to the elliptic paraboloid z = -2x - 7y? at the point (-3, -4) is

A: Consider the provided question, The equation of the tangent plane to the elliptic paraboloid at the…

Q: Let F = -2yi + 2xj. Use the flux form of Green's Theorem to compute F·n ds where C is the positively…

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Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

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Q: 5. Let C denote the positively oriented circle z- 1= 3. Evaluate the integral: 3 z +2z dz 3. (z- 2)

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Q: 7. Find the area of the portion of the plane z = 2 y that lies directly above the circle x+(y-1) =1…

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Q: apply stok's Theorem to evaluate (Ydz +7dy+ xdz) whereCiB the curve of intersection of x²+ y²+z² =…

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Q: Let C be the positively oriented circle x2 +y2 =1. Use Green's Theorem to evaluate the line integral…

A: The given circle C is  x2 + y2 = 1 Heree the centre of the circle is (0, 0) Radius of the circle r =…

Q: Find the points on the surface xy – z2 = 1 that are closed to the origin.

A: Given surface is xy-z2=1. let xy-z2-1=0⇒z2-xy+1=0 Now distance from origin is…

Q: Use Green's Theorem to find work done by F = 2°yi + (y – 3)3 along the curve c, C is the closed path…

A: We will find out the required solution.

Q: Sketch the reglon enclosed by the glven curves. x = 6y2, x = 4 + 5y2 y y 2- 2- 10 15 20 10 15 20 -1…

A: Here we have,  x=6y2 ............................................. (1) and, x=4+5y2…