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Trying to work my way through Spivak Calculus on Manifolds.
On page 16, he states
"A function f: R^{}n -> R^{}m is differentiable at a \epsilon R^{}n if there is a lenear transformation \lambda: R^{}n -> R^{}m such that
lim h->0 of |f(a+h) - f(a) - \lambda(h)|/|h| = 0.
"Note that h is a point of R^{}n and f(a+h) - f(a) - \lambda(h) a point of R^{}m, so the norm signs are essential. The linear transformation \lambda is denoted Df(a) and called the derivative of f at a."
on page 20, he states If f: R^{}n -> R^{}m 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^{}n -> R^{}m is differentiable at a \epsilon R^{}n if there is a lenear transformation \lambda: R^{}n -> R^{}m such that
lim h->0 of |f(a+h) - f(a) - \lambda(h)|/|h| = 0.
"Note that h is a point of R^{}n and f(a+h) - f(a) - \lambda(h) a point of R^{}m, so the norm signs are essential. The linear transformation \lambda is denoted Df(a) and called the derivative of f at a."
on page 20, he states If f: R^{}n -> R^{}m 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.