By L. V. Tarasov

An interactive textbook of Calculus offered as a discussion among the writer and the reader.

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**Extra resources for Calculus: Basic Concepts for High Schools**

**Sample text**

Explain in the above light which of the examples of immersed submanifolds given in [Ch. 17] are integral curves and which not, for a suitable line subbundle of the tangent bundle. We will now show that immersed integral manifolds do not admit pathologies of the kind we pointed out in [Ch. 17]. 6. Proposition. If go : N -> M is any integral manifold and L is any other manifold with a map f : L --3 N, then f is differentiable if and only if cp o f is. Proof. If f is differentiable, the composite coo f is of course differentiable.

Indeed we have f (x) - f (a) = f f i t (tx + (1 - t)a)}dt 7-{f axi f (tx + (1 - t)a). dt (txi + (1 - t)ai)dt J(xi - ai) f 1a axZ f (tx + (1 - t)a)dt. 2. Locally Free Sheaves and Vector Bundles 29 The above considerations motivate the following definition. 4. Definition. A homogeneous first order differential operator on a differential manifold M is a linear operator on A(M) satisfying the Leibniz rule. Equivalently, it is a sheaf homomorphism A -* A which satisfies the Leibniz rule. A linear differential operator of order at most one on functions is of the form f H D f + gyp.

Lemma. If w is any nonzero element of Ar(V), then the linear map V -* Ar+1(V) given by v - vnw has kernel of dimension < r. Moreover, the kernel is of dimension r if and only if w E Ar(W) for some r-dimensional subspace W of V. Proof. In fact, it is obvious that if w belongs to Ar(W), then every element of W is in the kernel of the above map. Let {ei},1 < i < n, be a basis of V such that the first r of these generate W. In other words, el A ... A er can be taken to be w. Let v = E aiei; then vnw = 0 if and only if ai = 0 for all i > r.