brin, matthew g - introduction to differential topology

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Ngày tải lên: 27/03/2014, 11:54

... relations ∂ j g q =Γ i jq g i ,wenotethat 0=∂ j (g p · g q )=−Γ p ji g i · g q + g p · ∂ j g q = −Γ p qj + g p · ∂ j g q . Hence ∂ j g q =(∂ j g q · g p )g p =Γ p qj g p .  Remark. The Christoffel symbols ... that g i (x) · g j (x)=  g ik (x )g k (x)  ·  g j (x )g  (x)  = g ik (x )g j (x )g k (x) =g ik (x)δ j k = g ij (x), and thus the ve...

Ngày tải lên: 17/03/2014, 14:24

... use the chart map to lift the result back up to M. In other words, define (ff ff +gg gg )(t) = x -1 (x(ff ff (t)) + x(gg gg (t)) and (¬ff ff )(t) = x -1 (¬x(ff ff (t))). Taking their derivatives ...         r 2 -2 (x 1 ) 2 -2 x 1 x 2 -2 x 1 x 3 … -2 x 1 x n -2 x 2 x 1 r 2 -2 (x 2 ) 2 -2 x 2 x 3 … -2 x 2 x n …………… -2 x n x 1 -2 x n x 2 -2 x n...

Ngày tải lên: 27/03/2014, 11:52

... y both in G 1 or G 2 f 1 (g) · f 2 (g) −1 for g ∈ G. Then one defines the amalgamated product G 1 ∗ G G 2 as F/R. There are natural maps g 1 : G 1 → G 1 ∗ G G 2 , g 2 : G 2 → G 1 ∗ G G 2 obtained ... commutator subgroup. Let G b e any group, and let C (G) be the subgroup generated by elements of the form ghg −1 h −1 , g, h ∈ G. The subgroup C (G) is obviously...

Ngày tải lên: 31/03/2014, 16:26

...         r 2 -2 (x 1 ) 2 -2 x 1 x 2 -2 x 1 x 3 … !-2 x 1 x n -2 x 2 x 1 r 2 -2 (x 2 ) 2 -2 x 2 x 3 … !-2 x 2 x n … … … … … -2 x n x 1 !-2 x n x 2 !-2 x n x 3 … ! r 2 -2 (x n ) 2         -x 2 x 1 0 0 … 0 ... 1 r 4         r 2 -2 (x 1 ) 2 -2 x 1 x 2 -2 x 1 x 3 … !-2 x 1 x n -2 x 2 x 1 r 2 -2 (x 2 ) 2 -2 x 2 x 3...

Ngày tải lên: 24/04/2014, 17:08

... space consisting of its eigenvectors is m-dimensional. An eigenvalue with multiplicity 1 is called simple. Using the same terminology, we see that the eigenvalues λ n for the eigenvalue problem ... D. Because the general term u n decays exponentially in D ε , it is possible to differ- entiate (5.26) term-by-term to any order with respect to x and t. The corresponding series converges un...

Ngày tải lên: 16/02/2014, 15:20

... applied to more challenging problems. In Fig. 1.4 we have plotted the exact and numerical solutions of the problem (1.10) on page 6 using u 0 =2.1. Even though this problem is much harder to solve ... doing this we are able to derive very sim- ple solutions without having to deal with complications related to boundary values. We also restrict ourselves to one spatial dimension in...

Ngày tải lên: 31/03/2014, 15:56

... |B ∈B} is a σ-algebra, called the σ-algebra generated by X. This is the smallest sub-σ-algebra of U with respect to which X is measurable. Proof. Check that {X −1 (B) |B ∈B}is a σ-algebra; clearly ... differential equations as well. This is all too much to expect of undergrads. But white noise, Brownian motion and the random calculus are wonderful topics, too good for undergraduates to m...

Ngày tải lên: 31/03/2014, 15:56

... |B ∈B} is a σ-algebra, called the σ-algebra generated by X. This is the smallest sub-σ-algebra of U with respect to which X is measurable. Proof. Check that {X −1 (B) |B ∈B}is a σ-algebra; clearly ... differential equations as well. This is all too much to expect of undergrads. But white noise, Brownian motion and the random calculus are wonderful topics, too good for undergraduates to m...

Ngày tải lên: 08/04/2014, 12:24

... engineer. The engineer has to program in high-level languages to solve problems. He or she also should be able to read assembly-language programs to understand what a high-level ... concepts of programming of a microcontroller in high-level language and assembly language. It also covers the principles of good programming practice through top-down design...

Ngày tải lên: 18/04/2014, 11:06