Math 370 - calculus on manifolds let c be the unit circle

Question

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 =   ₀∫¹ (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=1∑ⁿ f_idx_i be a closed 1-form on Rⁿ. Define a function g by

g(x₁, · · · , x_n) = ₀∫^x_1 f₁(t,x₂, · · · , x_n)d_t + ₀∫^x_2 f₂(0, t, x₂, · · · , x_n)dt + · · ·  + ₀∫^x-n f_n(0, 0, · · · , t)dt.

Show that dg = α.