What are we What are we Studying?Studying?

The “Zoo” of Vector AnalysisThe “Zoo” of Vector Analysis

Functions---The Big PictureFunctions---The Big Picture

A set of inputs(domain!)

A set of outputs(range!)

A “rule” that takes any input and yields an output

Our “zoo”Our “zoo”

Calculus C---concentrates (with some Calculus C---concentrates (with some exceptions) on exceptions) on

Vector Analysis studies Vector Analysis studies

(Though our pictures and many applications are (Though our pictures and many applications are still in the plane or in 3-space.)still in the plane or in 3-space.)

2:f R R

: n mf R R

Some special CasesSome special CasesParametrically Defined CurvesParametrically Defined Curves

In the plane In 3-dimensional space

: mf R R

Trajectories in SpaceTrajectories in Space

Trajectories in SpaceTrajectories in Space

Trajectories in SpaceTrajectories in Space

Trajectories in 3-dimensional space

An ExampleAn Example

Steel DragonNagashima Spaland, Japan

4 world records: Tallest, Longest, Fastest and Greatest Drop.

Scalar FieldsScalar FieldsA scalar field is one whose output values are real numbers

2:f R R:f R R

Scalar Fields in Higher Scalar Fields in Higher DimensionsDimensions

Examples?

Vector FieldsVector FieldsA vector field is one whose output values are vectors.

2 2:f R R

Examples?

Vector FieldsVector FieldsA vector field is one whose output values are vectors.

2 2:f R R Direction field---if we care only about direction and not magnitude