Calculate the integral

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Reference no: EM131456462 , Length: word count:1250

CALCULUS ON MANIFOLDS

(1) Let C be a circle of radius r centered at 0 oriented counter clockwise. Check that

Cxdy,

gives the area of circle.

(2) Let C be the unit circle centered at 0 oriented counter clockwise. Calculate the integral

C -(y/x2+y2)dx + (x/x2+y2)dy

and check that it is nonzero. Conclude that this 1-form is not exact.

(3) Given w = fdx + gdy + hdz such that w Λ dz = 0. What can we conclude about f, g, and h?

(4) (a) Let w = Fx + Gdy + Hdz and C be the straight line connecting (0, 0, 0) to x0, y0, z0.  Then show that

C w =   01 (F(tx0, y0, z0) + G(x0, ty0, z0) + H(x0, y0, tz0))dt.

(b) Show that, if F, G and H are C1 and w is closed on R3 then w is exact.

(5) Find a solution of dξ = w, when w = zdx Λ dz + dy Λ dz.

(6) Let α = i=1n fidxi be a closed 1-form on Rn. Define a function g by

g(x1, · · · , xn) = 0x_1 f1(t,x2, · · · , xn)dt + 0x_2 f2(0, t, x2, · · · , xn)dt + · · ·  + 0x-n fn(0, 0, · · · , t)dt.

Show that dg = α.

Reference no: EM131456462

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len1456462

4/10/2017 1:38:53 AM

Subject: Fwd: Calculus on Manifolds [Don't delay and provide accurate solution]. Word count/calculation: 1250. References: 2. You are encouraged (and strongly recommended) to work together on the homework problems.

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