By Andrew McInerney

ISBN-10: 1461477328

ISBN-13: 9781461477327

Differential geometry arguably bargains the smoothest transition from the traditional college arithmetic series of the 1st 4 semesters in calculus, linear algebra, and differential equations to the better degrees of abstraction and facts encountered on the higher department by means of arithmetic majors. this present day it really is attainable to explain differential geometry as "the research of constructions at the tangent space," and this article develops this aspect of view.

This ebook, not like different introductory texts in differential geometry, develops the structure essential to introduce symplectic and speak to geometry along its Riemannian cousin. the most target of this booklet is to convey the undergraduate pupil who already has a high-quality starting place within the common arithmetic curriculum into touch with the great thing about better arithmetic. specifically, the presentation right here emphasizes the implications of a definition and the cautious use of examples and structures that allows you to discover these results.

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**Additional info for First Steps in Differential Geometry: Riemannian, Contact, Symplectic (Undergraduate Texts in Mathematics)**

**Example text**

Hence S is linearly dependent. 2 illustrates the fact that deciding whether a set is linearly dependent amounts to deciding whether a corresponding homogeneous system of linear equations has nontrivial solutions. 1. The reader is invited to supply proofs. 3. Let S be a finite set of vectors in a vector space V . Then: 1. If 0 ∈ S, then S is linearly dependent. 2. If S = {v} and v ̸= 0, then S is linearly independent. 3. Suppose S has at least two vectors. Then S is a linearly dependent set of nonzero vectors if and only if there exists a vector in S that can be written as a linear combination of the others.

Vn ), then εi (v) = vi . 5, εi is the projection onto the ith component. 1 gives a standard method of writing a linear transformation T : Rn → Rm as an m × n matrix. Linear one-forms T ∈ (Rn )∗ , T : Rn → R, are no exception. , as row vectors. For example, the standard basis B0∗ in this notation would appear as [ε1 ] = 1 0 · · · 0 , .. [εn ] = 0 0 · · · 1 . We now apply the “dual” construction to linear transformations between vector spaces V and W . For a linear transformation Ψ : V → W , we will construct a new linear transformation Ψ ∗ : W ∗ → V ∗.

17. 8. In that example we showed that S did not span R3 , and so S cannot be a basis for R3 . 2, we showed that S is linearly dependent. A closer look at that example shows that the rank of the matrix A = v1 v2 v3 v4 is two. 5 Linear Transformations 21 for W = Span(S) can be obtained by choosing vectors in S whose corresponding column in the row-echelon form has a leading one. In this case, S ′ = {v1 , v2 } is a basis for W , and so dim(W ) = 2. ” In the case of linear algebra, such functions are known as linear transformations.

### First Steps in Differential Geometry: Riemannian, Contact, Symplectic (Undergraduate Texts in Mathematics) by Andrew McInerney

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