By Nomizu K., Sasaki T.

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**Example text**

Let F be a function from an open set U ⊂ Rn to Rm and let a ∈ U . We call F diﬀerentiable at a if there exist a linear transformation L : Rn → Rm and a function R deﬁned in a neighborhood of a such that F (a + v) = F (a) + L(v) + R(v), with R(v) = 0. v v→0 lim If F is diﬀerentiable at a, the linear transformation L is denoted by dFa and is called the diﬀerential of F at a. Notations for the diﬀerential vary widely. Though we will consistently use dFa for the diﬀerential of F at a, some authors write dF (a) instead.

2)), and suppose that the vector variable z = (z1 , . . 3. Differentiation Rules: Functions of Class Cr 25 as a function of the variable y (this function corresponds to f ). With Leibniz’s notation, one writes the chain rule as ∂zi = ∂xj n k=1 ∂zi ∂yk ∂yk ∂xi for all i = 1, . . , m and j = 1, . . , p. Implicit in this notation is that when evaluating ∂zi /∂xj at a point a ∈ Rp , one must calculate ∂zi ∂xj n = a k=1 ∂zi ∂yk ∂yk y(a) ∂xi a . Suppose a function f is diﬀerentiable over an open set U ⊂ Rn to R .

Changing types of frames sometimes makes diﬃcult integrals tractable or makes certain diﬃcult diﬀerential equations manageable. In Chapter 1 in [5], we used the {T , U} frame to study the local properties of a plane curve x(t). The vector T (t) is the unit tangent vector T (t) = x (t)/ x (t) , and the unit normal vector U (t), is the result of rotating T (t) by π/2 in a counterclockwise direction. This is a moving frame that is deﬁned in terms of a given regular curve x(t) and, at t = t0 , is viewed as based at the point x(t0 ).