# Scalar Field / Vector Field

Yao Yao on June 6, 2018

## Vector Field

A vector field in a plane is a map, which assigns each point $(x, y)$ to a vector $\vec F(x, y) = \langle P(x, y), Q(x, y) \rangle = \icol{P(x, y) \newline Q(x, y)}$.

A vector field in a 3D space is a map, which assigns each point $(x, y, z)$ to a vector $\vec F(x, y, z) = \langle P(x, y, z), Q(x, y, z), R(x, y, z) \rangle = \icol{P(x, y, z) \newline Q(x, y, z) \newline R(x, y, z)}$.

Quote from Wikipedia: Vector field:

If each component of $\vec F$ is continuous, then $\vec F$ is a continuous vector field, and more generally $\vec F$ is a $C^k$ vector field if each component of $\vec F$ is $k$ times continuously differentiable.

• $\vec F(1, 0) = \langle 0， -1 \rangle = \icol{0 \newline -1}$
• $\vec F(0, 1) = \langle 1， 0 \rangle = \icol{1 \newline 0}$
• $\vec F(-1, 0) = \langle 0， 1 \rangle = \icol{0 \newline 1}$
• $\vec F(0, -1) = \langle -1， 0 \rangle = \icol{-1 \newline 0}$

VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}]


## Scalar Field

A scalar field in a space is a map, which assigns each point $(x, y, z, \dots)$ to a scalar.

DensityPlot[x-y,{x,-10000,10000},{y,-10000,10000}]