What is a vector valued in a function?

[Jhenrique]

What is a vector valued in a function? For example: $f(\vec{r})$; or a vector valued in another vector, as: $\vec{f}(\vec{r})$. What this means? How this kind of calculus is done?

 

[ShayanJ]

The vector $\vec{r}$ is called a displacement vector.A vector from origin to some point.In such uses,they are interpreted as showing that point.So you should interpret it as a function which associates to point $\vec{r}$ a scalar $f(\vec{r})$ or a vector $\vec{f}(\vec{r})$.

 

[D H]

What is a vector valued in a function?

A function is something that maps one space into another. If that other space is a vector space, the function is a "vector-valued function". For example, position as a function of time. The input to the function is time, a scalar. The output is a vector. Thus $\vec x(t)$ is a vector valued function. The input can also be a vector. For example, $\vec F(\vec r) = -G m_1 m_2 \vec r/||\vec r||^3$ also is a vector valued function. On the other hand, a function that maps from a vector space to a scalar space (e.g., potential energy as a function of position) is not a vector valued function.

How this kind of calculus is done?

With partial derivatives.