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krcmd1
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Trying to work my way through Spivak Calculus on Manifolds.
On page 16, he states
"A function f: R[tex]^{}n[/tex] -> R[tex]^{}m[/tex] is differentiable at a [tex]\epsilon[/tex] R[tex]^{}n[/tex] if there is a lenear transformation [tex]\lambda[/tex]: R[tex]^{}n[/tex] -> R[tex]^{}m[/tex] such that
lim h->0 of |f(a+h) - f(a) - [tex]\lambda[/tex](h)|/|h| = 0.
"Note that h is a point of R[tex]^{}n[/tex] and f(a+h) - f(a) - [tex]\lambda[/tex](h) a point of R[tex]^{}m[/tex], so the norm signs are essential. The linear transformation [tex]\lambda[/tex] is denoted Df(a) and called the derivative of f at a."
on page 20, he states If f: R[tex]^{}n[/tex] -> R[tex]^{}m[/tex] is a linear transformation, then Df(a) = f.
His proof states, on p 21:
lim as |h| -> 0 of |f(a + h) - f(a) - f(h)| / |h| =
lim as |h| -> 0 of |f(a) + f(h) - f(a) - f(h)| / |h| = 0
Here's what confuses me is that it seems to me that in the second limit, the second f(h) represents f(a)(h), while the first f(h) is f of h.
by the way, how do you use tex to write |h| -> 0 below lim?
Thank you, in advance.
On page 16, he states
"A function f: R[tex]^{}n[/tex] -> R[tex]^{}m[/tex] is differentiable at a [tex]\epsilon[/tex] R[tex]^{}n[/tex] if there is a lenear transformation [tex]\lambda[/tex]: R[tex]^{}n[/tex] -> R[tex]^{}m[/tex] such that
lim h->0 of |f(a+h) - f(a) - [tex]\lambda[/tex](h)|/|h| = 0.
"Note that h is a point of R[tex]^{}n[/tex] and f(a+h) - f(a) - [tex]\lambda[/tex](h) a point of R[tex]^{}m[/tex], so the norm signs are essential. The linear transformation [tex]\lambda[/tex] is denoted Df(a) and called the derivative of f at a."
on page 20, he states If f: R[tex]^{}n[/tex] -> R[tex]^{}m[/tex] is a linear transformation, then Df(a) = f.
His proof states, on p 21:
lim as |h| -> 0 of |f(a + h) - f(a) - f(h)| / |h| =
lim as |h| -> 0 of |f(a) + f(h) - f(a) - f(h)| / |h| = 0
Here's what confuses me is that it seems to me that in the second limit, the second f(h) represents f(a)(h), while the first f(h) is f of h.
by the way, how do you use tex to write |h| -> 0 below lim?
Thank you, in advance.