Download e-book for iPad: A brief on tensor analysis by James G. Simmonds

By James G. Simmonds

ISBN-10: 0387906398

ISBN-13: 9780387906393

ISBN-10: 3540906398

ISBN-13: 9783540906391

During this textual content which steadily develops the instruments for formulating and manipulating the sector equations of Continuum Mechanics, the maths of tensor research is brought in 4, well-separated phases, and the actual interpretation and alertness of vectors and tensors are under pressure all through. This re-creation includes extra routines. moreover, the writer has appended a piece on Differential Geometry

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Example text

An important example is the analytisches Gebilde of an algebraic function. In Sect. 3, we shall give another construction of this. Hence we recommend that readers who are interested in this basic example should skip the analytische Gebilde and proceed directly to Sect. 3. One of the simplest examples of a “multivalued √ function” is the square root. By choosing, for example, the principal branch z, we can obtain uniqueness, but we obtain √ only an analytic function on the slit plane C − . Other branches such as − z on C − have equal validity.

For this, we can assume (possibly after ˜ in X are equal. We can also shrinking V and V˜ ) that the images U and U assume that V and V˜ are connected. Then the chart transformation γ is a topological map ∼ γ : V −→ V˜ . For every a ∈ V , there has to exist an ω(a) ∈ L such that γ(a) = a + ω(a). Since ϕ is continuous and L is discrete, the function ω(a) = γ(a) − a has to be locally constant. Since V is connected, ω must be constant. Hence the chart transformation is a translation; in particular, it is analytic.

The Riemann Surface of an Algebraic Function 37 of two complex variables is called irreducible if it cannot be written as a product of two nonconstant polynomials. Every polynomial can be written as a product of finitely many irreducible ones: P = P1 · · · · · Pm , Pj irreducible. If f : D −→ C (D ⊂ C a domain) is an analytic function with the property P (z, f (z)) ≡ 0, then Pj (z, f (z)) ≡ 0 for some j (1 ≤ j ≤ m). For this reason, we may assume in what follows that P itself is irreducible. We associate with P a plane affine algebraic curve N = N (P ) = {(z, w) ∈ C × C; P (z, w) = 0}.

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