Vector Fields A brief introduction

Functions---The Big Picture A set of inputs (domain!) A set of outputs (range!) A “rule” that takes any input and yields an output

Our Function “zoo” We have been concentrating mostly on functions with two or three input (real) variables and one output (real) variable. In Calc A and Calc B we study functions with one real input variable and one real output variable. We have been concentrating mostly on functions with two or three input (real) variables and one output (real) variable. In Calc A and Calc B we study functions with one real input variable and one real output variable. These are called “Scalar Fields.”

Scalar Fields A scalar field is one whose output values are real numbers

Scalar Fields in Higher Dimensions Harder to picture… Examples?

Vector Fields Functions in which the inputs are real numbers of vectors and the outputs are also vectors are called “vector fields.” Functions in which the inputs are real numbers of vectors and the outputs are also vectors are called “vector fields.” We will still concentrate on vectors with 2 or 3 coordinates to make these things easier to picture. But really we can work in any dimension. We will still concentrate on vectors with 2 or 3 coordinates to make these things easier to picture. But really we can work in any dimension.

Some special Cases Parametrically Defined Curves In the planeIn 3-dimensional space

An Example Trajectories in 3-dimensional space An Example Steel Dragon Nagashima Spaland, Japan 4 world records: Tallest, Longest, Fastest and Greatest Drop.

Vector Fields A vector field is one whose output values are vectors. Examples?

Vector Fields A vector field is one whose output values are vectors. Direction field---if we care only about direction and not magnitude