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I learned that if i have a function $f:R^p\to R^q$ i can write differential of function like $df(a)(u)=\frac{\partial f}{\partial x_1}(a)u_1+\frac{\partial f}{\partial x_2}(a)u_2+...+\frac{\partial f}{\partial x_p}(a)u_p$, where $u=(u_1,u_2,...,u_p)$ and $pr_i:R\to R, pr_i(u_1,...u_p)=u_i, \forall u\in R^p$. And i saw notations like this: $df(a)(u)=\frac{\partial f}{\partial x_1}(a)dx_1+\frac{\partial f}{\partial x_2}(a)dx_2+...+\frac{\partial f}{\partial x_p}(a)dx_p$.
Why we use this notation and how we get from $u_p \to dx_p$. $dx_p$ mean differential of $x_p$ or is just a notation?

MathLearner
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    See my answer to Help understanding expression for the derivative of a function. – peek-a-boo May 13 '22 at 21:11
  • @peek-a-boo .Thanks. I understand until this "This is an equality of linear transformations $R^n$→R. Or, if we go one step further, we can suppress the point of derivative evaluation to write simply". Why we can supress the point of derivative evaluation? I know that df(a) is a linear transformation, but if we remove point a, what is df? Is all linear transformations? Sorry for my english.. – MathLearner May 13 '22 at 21:36
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    If $U\subset\Bbb{R}^p$ is open and $f:U\to\Bbb{R}^q$ is differentiable then we have the mapping $df:U\to\text{Hom}(\Bbb{R}^p,\Bbb{R}^q)$, i.e $df$ is a function with domain $U$ and is linear-transformation-valued; it is the mapping $a\mapsto df_a$. So, $df_a=\sum_{i=1}^p\frac{\partial f}{\partial x^i}(a),dx^i_a$ is an equality of elements of $\text{Hom}(\Bbb{R}^p,\Bbb{R}^q)$. If you omit the $a$, to get $df=\sum_{i=1}^p\frac{\partial f}{\partial x^i},dx^i$, this is an equality of linear-transformation-valued functions on $U$, i.e equality of functions $U\to\text{Hom}(\Bbb{R}^p,\Bbb{R}^q)$. – peek-a-boo May 13 '22 at 22:15
  • @peek-a-boo oh. Ok. I think i got it. So if i have equality of f(a)=g(a) for any a then functions are equal so i can remove the parameter? Right? – MathLearner May 13 '22 at 22:27
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    Yes exactly, that's the definition of equality of functions. – peek-a-boo May 13 '22 at 22:32

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