Proof $ df(x,\,y) = {\partial f \over \partial x} dx + { \partial f \over \partial y } dy $

Question

Proof differential of multivariable function $$ df(x,\,y) = {\partial f \over \partial x} dx + { \partial f \over \partial y } dy $$

Answer 1

You are in a 2 dimensional manifold. By definition, $\mathrm d f$ is a $1-$form, and thus written (uniquely) as $$\mathrm d f=a\,\mathrm d x+b\,\mathrm d y.$$

Now, by definition $$\mathrm df(\partial _x):=\partial _xf,$$ and $$\mathrm d f(\partial _y):=\partial _yf.$$ This yield $a=\partial _xf$ and $b=\partial _yf$.