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A short draft passage, apparently from Phil's curvilinear-systems tensor document (a May 2015 update file), numbered section 7.5. It defines an equation as covariant if it keeps the same form in x-space and x'-space. Three examples illustrate this: Newton's law F = ma under rotations, the dot product equation A·B = π as a scalar, and the outer product T^ab = U^a V^b. The text ends at a 'STOP' note, so the section on contraction itself is not included.

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7.5 Covariance and Contraction of a Pair of Indices As we shall discuss in more detail below in Section **, an equation is said to be covariant under the transformation x' = F(x) if it has "the same form" in both x-space and x'-space. The "same form" means that the equation looks the same but everything is primed in x'-space. Example 1: Newton's Law F = ma is covariant under rotations (x' = F(x) = Rx), and in x'-space this law takes the form F' = m'a' which has the same form as the equation in x-space F = ma. Once we know that F and a are contravariant vectors and m is a scalar, this conclusion is automatic from (2.3.2), F = ma F' = m'a' proof: F' = RF = R(ma) = m Ra = m a' = m' a' where the last m = m' follows since mass is a scalar under rotation. Thus, Newton's Law has the same form when it is examined in two frames of reference related by a rotation. Example 2: Consider the equation A B = π where A and B are contravariant vectors and is the covariant dot product defined in (5.10.1), A B ≡ abAaBb . It was shown in (5.10.2) that the quantity A B transforms as a scalar under general transformation x' = F(x) so that A' B' = A B. Since the number π is also a scalar under any transformation (it is a constant), one could say that π' = π. Then A B = π A' B' = π' , equation is covariant. What we see here is that an equation is covariant IFF both sides of the equation transform as the same tensorial tensor under the transformation of interest. In Example 1, both sides of F = ma transform as contravariant vectors under rotations, and in Example 2 both sides of A B = π transform as scalars under a general transformation. Example 3: Consider the outer product (7.1.1) Tab = UaVb where U and V are contravariant vectors. We show in (7.1.2) that Tab transforms as a contravariant rank-2 tensor. Both sides of this equation transform in this way, so in x'-space the equation becomes T'ab = U'aV'b. The equation is covariant. STOP and hold on this idea. __________________________________________________________________________