What does multivariable differentiability imply?

Question

I have a multivariable scalar function $\psi$ and it is differentiable everywhere. What does it imply?

Does it only imply that the value of $\nabla\psi$ does not change if we rotate the coordinate system?

Does it imply anything else?

Answer 1

Intuitively, differentiable in any circumstance (where it makes sense) means "locally approximately linear".

In the case of a function $\psi:\Bbb R^n\to\Bbb R$, it means that at any point $p\in\Bbb R^n$, there is a linear transformation $\nabla\psi(p):\Bbb R^n\to\Bbb R$ such that $\psi(p+v)\approx \psi(p)+\nabla\psi(p)v$ for any small $v\in\Bbb R^n$ (where $\approx$ has a rigourously defined meaning). This $\nabla\psi(p)$ is called the gradient of $\psi$ at $p$.

Think about the special case $n=1$, where the gradient goes by the name "derivative". Given a function $f:\Bbb R\to\Bbb R$, and a real number $x$, the derivative $f'(x)$ is a real number such that $f(x+h)\approx f(x)+f'(x)h$. The paragraph above is merely a generalisation.

And, of course, we can generalize further, to functions $\Bbb R^n\to\Bbb R^n$. This time the linear transformation is represented by the Jacobian matrix, where all entries are the partial derivatives of all the component functions (its rows are the gradients of the component functions).