#spaced # Vector Fields ### Intuitive Approach 1 A **vector field** (/posts/a type of [Field](Field.md)) is what you get if you associate each point in space with a *vector* (see [here](https://youtu.be/rB83DpBJQsE?t=7) for a clean, simple intro).  For example, these vectors could represent the velocities of particles of fluid at each point in space:  Or maybe they represent the force of gravity at each point in space:  ### Intuitive Approach 2 A vector field is a [way of visualizing](https://youtu.be/5FWAVmwMXWg?t=19) functions that have the same number of dimensions in their input as in their output. For instance, something such as: $f(x,y) = \begin{bmatrix} y^3 - 9y \\ x^3 - 9x \end{bmatrix}$ Imagine trying to visualize the above function with a graph. It would be *really hard*. This is because you have two dimensions in the input and two dimensions in the output. So, you would have to somehow visualize this in four dimensions. So, instead of using a graph, we look *only in the input space*, and for each input point, we look at the vector that it outputs, and attach that to the point:  We want to do the above for *every point in the input space*. Now, obviously as we plot more and more vectors things will become a huge mess, so we scale down their size to keep things tidy:  Now, yes, at this point we are kind of lying about what the vectors actually are, but that is just for visualization purposes. ### Technical Approach A vector field is the assignment of a vector to each point in a subset of space. Given a subset _S_ in $\mathbb{R}^n$, a **vector field** is represented by a [vector-valued function](https://en.wikipedia.org/wiki/Vector-valued_function) $V : S \rightarrow \mathbb{R}^n$ in standard [Cartesian coordinates](https://en.wikipedia.org/wiki/Cartesian_coordinates "Cartesian coordinates") $(x_1, …, x_n)$. --- References: * [Divergence and Curl - 3b1b](https://youtu.be/rB83DpBJQsE?t=7) * [Vector Field, introduction](https://www.youtube.com/watch?v=5FWAVmwMXWg) * [Wikipedia](https://en.wikipedia.org/wiki/Vector_field)