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Introduction to Vector Fields: Understanding Functions in Various Dimensions

Explore the concepts of vector fields in mathematics, distinguishing between scalar fields and vector fields. Learn about functions with vector inputs and outputs, as well as special cases of vector fields. Examples include parametric curves, trajectories in 3D space, and real-world applications like world records. Dive into the big picture of functions beyond traditional scalar fields.

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Introduction to Vector Fields: Understanding Functions in Various Dimensions

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  1. Vector Fields A brief introduction

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

  3. 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. These are called “Scalar Fields.”

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

  5. Scalar Fields in Higher Dimensions Harder to picture… Examples?

  6. 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.

  7. Some special CasesParametrically Defined Curves In the plane In 3-dimensional space

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

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

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

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