Sometimes a function of several real variables is written with a vector $\mathbf{x}=(x_1,x_2, \dots, x_n)$ such that $$ f(\mathbf{x})=f(x_1,x_2, \dots, x_n) $$
In linear algebra I learned that a vector is a vector in the form (with the standard basis) $$\mathbf{x}=(x_1,x_2, \dots, x_n)=x_1\mathbf{\hat{e}}_1+x_2\mathbf{\hat{e}}_2+\cdots + x_n\mathbf{\hat{e}}_n$$ Is this the same thing in multivariable calculus?
I.e. I mean is \begin{align} f(\mathbf{x})&=f(x_1\mathbf{\hat{e}}_1+x_2\mathbf{\hat{e}}_2+\cdots + x_n\mathbf{\hat{e}}_n)\\ &=f(x_1,x_2, \dots, x_n) \quad \text{?} \end{align}