By A. Blechman
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Additional info for Mathematics Primer for Physics Grad. Students [Vers. 2.0]
It is also true that this space also forms a vector space itself. It is so important in linear algebra, it is given a name. We call it the dual space of V, and denote it as V ∗ . The vectors (functions) in the space are called covectors, or 1-forms (hense the 1 superscript). Armed with this knowledge we can go back and redefine what we mean by “tensor”: consider the set of all multilinear (that is, linear in each of its arguments) transformations that send k vectors and l covectors to a number: T : V × · · · × V × V ∗ × · · · × V)∗ → R k times l times The set of all such transformations is called Tkl and the elements of this set are called tensors l of rank .
To see this, notice that at first, you might assume that ∇2 A = (∇2 Ai ) for each component Ai . In Cartesian coordinates, this is fine. But in curvilinear coordinates, you must be extra careful. The true definition of the ith vector component is Ai = A · ei , and in curvilinear coordinates, ei has spacial dependence, so you have to take its derivative as well. We will take the derivative of a unit vector and discuss its meaning in the next chapter. 8). The usual Laplacians are on the cover of Jackson.
1 Form Operations Wedge Product In order to define the standard basis for Λk (Rn ), we needed to introduce a new operation, called the wedge product (∧). Let’s look more carefully at this new form of multiplication. 9) So the wedge product defines a noncommutative product of forms. 10) Looking at this as an antisymmetric product on forms, you might be reminded of the cross product. This is exactly right: the cross-product of two vectors is the same thing as a wedge product of two 1-forms. We will prove this explicitly soon.