By Klaus Jänich

Translated by way of L. Kay

Classical vector research bargains with vector fields; the gradient, divergence, and curl operators; line, floor, and quantity integrals; and the fundamental theorems of Gauss, Stokes, and eco-friendly. smooth vector research distills those into the Cartan calculus and a common type of Stokes' theorem. This basically smooth textual content conscientiously develops vector research on manifolds and reinterprets it from the classical perspective (and with the classical notation) for three-d Euclidean house, then is going directly to introduce de Rham cohomology and Hodge thought. the fabric is obtainable to an undergraduate scholar with calculus, linear algebra, and a few topology as necessities. the various figures, routines with certain tricks, and assessments with solutions make this booklet rather appropriate for a person learning the topic independently.

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What can be done if it is? 2. This is an exercise in the definitions. No ideas are needed here. You "only" have to prove both directions ==* and {::= directly from the definitions. 3. Maybe you have no idea where to start with "state more precisely" and are muttering that I should have formulated the problem more precisely. The convenient phrase "in a canonical way" says something meaningful only if it's clear what way we're really talking about. An n-dimensional real vector space is, in any case, not an n-dimensional manifold according to the strict wording of the definition.

15. The coordinates xil- in the Ricci calculus 44 ________________~c=ha~p~te=r~2~~Th~e~1a==n~ge=n=t~S~pa==c~e__________________________ a -;- E aXIL and abbreviated aIL E TpM TpM. 16. Coordinate basis (aI, ... Lth coordinate curve); and finally, as a derivation, aJ1. 17. The vector field aIL on U t:p(M) ---+ cP J---+ JR, a( h- l ) CPa:J1. Lth partial derivative of the downstairs function. And this is just what the Ricci notation aJlcp, despite its superb terseness, suggests unambiguously: What possible meaning could applying a/axJl to a function cP defined on a manifold have, other than first expressing the function in the coordinates xl, ...

4. 5. For every tangent vector v E r;ub M to a submanifold M C ]RN we can find a curve a in M with (X(O) = P and &(0) = v. which I call (a) the geometric, (b) the algebraic, and (c) the "physical" definitions. We use all three. The order is irrelevant, and we begin with (a). We start with the intuitive idea of a tangent vector v to a submanifold of]RN and ask ourselves how we can characterize it without using the ambient space, so as to get a generalizable version of the definition. Well, isn't any such v the velocity vector of a curve a that lies entirely in M, and doesn't such an a contain enough information about v?