For the trivial line bundle E over the manifold M = R onsider the connection 1-form w =-xzdt + (x² – yr)dx + (x +zt?)dy+(z²t – y)dz. Compute the curvature 2-form F = -dø and show dF =0

Q: 5. Give an example of a curve C C xy plane so that its curvature as a space curve is different from…

A: The given: A curve  in the -plane defined by parametric equations and .The objective: To…

Q: a) Start by finding a single vector function that represents the intersection of the surfaces z = x?…

A: Given surfaces are z=x2x2+y2=4 Now, the equation x2+y2=4 represents a circle centered at origin with…

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: 2. Find the equation of the tangent plane to the surface z = 3++ at point (-4, 3).

A: Topic- application of derivatives  Tangent to surface

Q: Prove that the circle {(x, y), x² + y? = 1} is not homeomorphic to the interval [-1, 1).

A:

Q: Let H be the segment of the hyperbola x-y°= | Connecting (1,0) to (2, J3). Evaluate the line,…

A: Given  H be the segment of hyperbola x2-y2=1 connecting the points (1,0) to (2,3)  we have to…

Q: 22 Consider the linear differential equation Select one alternative: ○ (x³y)' = x² ○ (x³y)' = x³ ○…

A: The given linear differential equation is ,               dydx+3xy = x (.)  Solution of a first…

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Q: a) Use Green's theorem to compute f F dr where c is a circle of radius R oriented CCW and F(x, y) =…

A: Green's theorem:∫C[Pdx+Qdy]=∫∫T∂Q∂x-∂P∂ydxdy where T is the region bounded by a simple closed curve…

Q: i) Calculate length of the curve r(t) = acost î+ asint ĵ+ 4t k from (a,0,0) to (a,0,87).

A: As per our company guideline we are supposed to answer only one question kindly post remaining…

Q: 1. Use Green's Theorem to evaluate |F. dr, where F(r, y) = (3y- esin, 7x + Vy + 1) and C is the…

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Q: Let P be the tangent plane to the surface z-2x2+y2 at the point Q=(1,1,3). Which of the following…

A: Tangent plane to given surface

Q: b) Calculate dz z² - 1 S ₂ Along the positive direction of the curve |z| = 2 Hint: The curve |z| = 2…

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Q: 8. let C be the curue space ret) = (e*,2et ,2et ),0s ts lnca) the rength of C is

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Q: 2. Consider the line r(t) = (4t+1,3, 2 - t) and the ellipsoid + + z? = 1. What is the distance from…

A: Consider the line  and the ellipsoid To find  distance from the point (-3,3,3) on the line to the…

Q: Find the curvature k of the circle C:(x – 2)2 + (y +3) = 25 - at the point (5, 1).

A: circle equation is x-22+y+32=25 and point 5,1find curvature k

Q: Q. find the curvature K of each plane curve traced out by the vector function r=r(t) = (3t – t3)i +…

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Q: Find the curvature of the space curve. r(t) = - 3i + (3+2t)j + (t +8)k %3D

A: The given curve    r(t)=-3i+3+2tj+t2+8k

Q: Evaluate la z*–3z++6 dz where C is any simple closed curve containing -i Sc (z+i)³ Select one: O a.…

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Q: Q5. Prove that the circle {(x, y), z² + y² = 1} is not homeomorphic to the interval (–1, 1). 1

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