Prove Green's theorem for the rectangle defined by xo < x < x1 Yo < y < Y1 (a) directly, and (b) using the result for triangles. and

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

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

Q: A set W is defined as consisting of all the planes in R^3 that are parallel to the plane x+2y-3z=1.…

A: A set W is defined as consisting of all the planes in R3 that are parallel to the plane x+2y-3z=1.…

Q: Use Green's Theorem to calculate ( - sen(a))da + (2a + 3)dy, where C is the triangle with vertices…

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Q: Sketch the Surface z./4-x* and verify that fay -lyx.

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Q: Prove that the volume of the sphere x² + y² + z² = a² is V =÷na³

A: In this question, we find the volume of sphere by using triple integral

Q: Find c > 0 c > 0 such that the area of the region enclosed by the parabolas y = x 2 − c 2 y = x 2 -…

A: To find the value of c given area between the curves.

Q: 3. Prove that a flat surface of revolution is part of a plane, cone, or cylinder.

A: prove that a flat surface of revolution is part of plane or cone or cylinder. solution a flat…

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: E. Determine the set of points satisfying the condition described by |z – 2i| + |z + 4i| = 10. Then…

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Q: The half-space consisting of the points on and below the xy-plane

A: Concept:

Q: 6. Given a rule x2 + y2 = 9, its image rule x2 + y2 – 2x + 6y + 1 = 0 under a translation, state a…

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Q: The intersection between the surface x + y + z = - 1 and the circle x2 + y2 = 1,z = 0 is

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Q: 3. Verify Euler's formula for the plane graphs shown below. B (a) (b)

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Q: Verify Green's Theorem where F1=y^2-7y , F2=2xy+2x and the curve C is a circle x^2+y^2=1

A: Given: F1=y2-7y , F2=2xy+2x and C is a circle i.e., x2+y2=1 As, we have:…

Q: 2.2 Find upper bound of z + 4 -dz (z – 3i)3 + 1 where C is the circle |z – 3i| = 4.

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Q: · Let A the area of the triangle ABC is given by ("1, ул), В %3D (x2, Y2), and C = (x3, y3) be…

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Q: #5

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Q: Let a point a = (2, 4, 5). %3D a) Find the distance from this point to the xy-plane.

A: Consider the point

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: Formulate an verify Green theorem when [P, Q] = [2yx, y² + 2x] and the curve y is the positively…

A: According to the given information it is required to formulate Green theorem. When

Q: Prove that the circles x² + y2 + z2 - 2x + 3y + 4z - 5 = 0, 5y + 6z +1 = 0 x² + y2 + z2 - 3x - 4y +…

A: We will use equation of sphere passing through sphere S  and plane P  S+λP =0

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: A surface is called bounded if there exists M &gt; 0 such that everypoint on the surface lies at a…

A: Explanation: The possible Quadric surfaces are Spheres, Ellipsoids, Hyperboloid of one sheet,…

Q: Sketch the surface 2./4- and verify that fry - lyx

A: Given surface is  z=4-x2-y2 we have to sketch this and verify that fxy=fyx

Q: Consider a sphere E with center at the origin and radius equal to 1, following the Beltrami-Klein…

A: Substitute the parametric coordinates of the line in the equation of sphere and solve to get the…

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

A: According to the given information it is required to calculate both sides of the Green’s theorem.…

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: prove Theorem 4P.1 Theorem 4P.1 There are exactly six lines in the four-point geometry.

A: The known fact is that in four point geometry, any two distinct points have exactly one line on both…

Q: Consider the IVP{ ty = V1-Y", . Test Existence and Unique- y (1) = 0.5. ness Theorem by sketching a…

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Q: 2.2 Find upper bound of z +4 -dz - 3i)3 +1 where C is the circle |z – 3i| = 4.

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Q: AB and AC are two chords of a circle of radius r such that AB = 2AC. If p and q are the distances of…

A: Given in a circle of radius r, there are two chords AB and AC such that AB=2AC. Also, the distance…

Q: 11) Example: of R' (t) to find the curvature Use the Theorem in 10 to

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Q: 1. Find the distance between the two planes - 2y + z=4 and z-2y+ :-1 HI

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Q: Try to visualize the set of all points (x, y, z)in a coordinate space that are twice as far from the…

A: a) Suppose the distance between two points  A(x, y, z)  and the origin   Plug into the distance…

Q: Verify Green's theorem Where C is the boundary of the triangle formed by x=3 , y=0 , x2= 3y

A: Verify Green's theorem Where C is the boundary of the triangle formed by x=3 , y=0 , x2= 3y

Q: XydA, D is the triangular region with Vertices (0,0), Cu,0], and Cor) SketchD onthe XY planel

A: 8/3

Q: Show that for every real number k the plane x - 2y + z + 3 + k(2x - y - z + 1) = 0 contains the line…

A: Show that for every real number k the plane x - 2y + z + 3 + k (2x - y - z + 1) = 0 contains the…

Q: Find the area of on of the triangle that can drawn between the planes 3x – 6y – 2z = 15 - and 2r + y…

A: For any triangle that will lie between two planes, one point has to lie on a plane and the other two…

Q: Let y = Vxy, y(xo) = yo- %3D %3D Find all points (x0, yo) in the xy-plane where an existence and…

A: Given that  Y'= √xy Y(Xo) =Yo

Q: Find the point on the sphere X+y+=4 and the point on The sphere x+y ansi- 2 Selution:-

A: To find the point on the sphere which is farthest from the given  point (1,-1,1).

Q: paraboloid z= 1 - (2x^2) -y^2 and plane z + x + y =4 find the shortest distance between them

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Q: Find the area of on of the triangle that can drawn between the planes 3x – 6y – 2z = 15 and 2r + y -…

A: The equations of the two planes are: 3x-6y-2z=152x+y-2z=5 As, 5,0,0 satisfies the equation of plane…

Q: (i) Sael-²dz, where C is the anticlockwise unit circle |2| = 2. (ii) ſadz, , where C is the…

A: As per our guidelines we supposed to answer only three subpart in case of a multi part question.

Q: Calculate both sides of Green's Theorem when F = [-x²y, xy²] and y positively oriented circle x? +…

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Q: Sketch the surfaces HYPERBOLOIDS z2 - x2 - y2 = 1

A: Sketch the surfaces HYPERBOLOIDS z2 - x2 - y2 = 1

Q: 2 Let w = z². Find the image of the first quadrant in z-plane

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Q: 7. Sketch the surface 1≤z≤2-x² - y²

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