Differentiable Geometry, diff structure for manifold.

Differentiable Geometry. Diff structure for manifold. -------------------------------------------------------------------------------- Let S^1 a subset of R^2 be the unit-circle; that is, S^1 = { (x,y) element of R^2 : x^2 + y^2 = 1}. (a) Explicitly find a differentiable structure for S^1 ( and indicate what the dimension ...continues

Proofs : Collinear and Distinct; Boomerang Quadrilateral

1- Prove that if AF/FB = AF'/F'B where A, B, F, F' are collinear and distinct then F does not have to equal F' 2- Suppose that the sides AB, BC, CD and DA of a quadrilateral ABCD are cut by a line at the points A' B' C' D' respectively, show that AA'/A'B * BB'/B'C * CC'/C'D * DD'/D'A = +1

Proof : Incircle and Triangle - Menelaus Theorem

The incircle of triangle ABC touches the sides BC, CA, AB at the points x,y,z respectively. yz is produced to meet BC at k. Show that Bx/Cx = Bk/Ck.

Angles/Pounds

A right triangle has one angle that measures 90 degree. If one of the acute angles is 2 times the size of the other, what is the measure of the smallest angle? Explain your answer. The cake recipe calls for 14 ounces of cream cheese. How much of the cream cheese will be left over? Show your answer in pounds and ounces.

Differentiable Manifolds : Let M be a differentiable manifold such that M is orientable and connected. Prove that there exist exactly two distinct orientations on M.

Let M be a differentiable manifold such that M is orientable and connected. Prove that there exist exactly two distinct orientations on M.

Differentiable Manifolds and Diffeomorphisms : Let M and N be differentiable manifolds. Let alpha: M -> N be a local diffeomorphism. Prove that if N is orientable, then M is orientable.

Let M and N be differentiable manifolds. Let alpha: M -> N be a local diffeomorphism. Prove that if N is orientable, then M is orientable.

Proofs : Quadrilaterals

1- Prove that in a cyclic quadrilateral we have opposite angles supplementary without introducing the center of the circle. 2- Prove that a quadrilateral is paralleogram if it is simple and opposite angles are congurent.

Finding Diameter of a Circular Table Given its Circumference

A circular table measures about 0.9 m around its edge. What is the approximate length of its diameter.

Prove that a non-simple quadrilateral can be inscribed in a circle <=> opposite angles are equal (both directions).

Prove that a non-simple quadrilateral can be inscribed in a circle <=> opposite angles are equal (both directions).

Show that the number of diagonals in a polygon is never the same as the sum of the measures of the exterior angles, one per vertex, of the polygon.

Show that the number of diagonals in a polygon is never the same as the sum of the measures of the exterior angles, one per vertex, of the polygon.