Topic: Vector-Valued Functions

Subtopic: Calculus of VV Functions

**Overview**

All calculus operations such as differentiating and integrating can be applied to vector valued (VV) functions as they are to singel-variabled functions. Starting with the definition of functions as a ratio of the change in the vector-valued function over a change in time, we work thorugh various calculus operations in 3D. Fun stuff!

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 14.2.1 Understand and apply the limit definition of derivative of a vv function
- 14.2.2 Find points at which a vv function is discontinuous
- 14.2.3 Derive a vv function `vecr(t)` and know that `vec(r')(t)` is the tangent vector (a.k.a. velocity vector)
- 14.2.4 Find the unit tangent vector `vecT(t) = (vec(r')(t))/||vec(r')(t)||`
- 14.2.5 Know and apply derivative rules for vv functions including product rule, chain rule, dot product rule, cross product rule
- 14.2.6 Find higher order derivatives of vv functions
- 14.2.7 Find antiderivative of vv function
- 14.2.8 Find indefinite and definite integral of vv function
- 14.2.9 Find specific antiderivative of vv function given initial condition (such as `vecr(0) = hati`)

**Terminology**

Define: tangent vector, unit tangent vector, velocity vector

**Supplemental Resources (optional)**

Dale Hoffman's Contemporary Calculus III: Derivatives & Antiderivatives of Vector-Valued Functions (study the calculus processes of VV functions and "Tangent Vectors" sections; save "Velocity" and "Acceleration" sections for a later lesson)

Paul's OL Notes - Calc III: Calculus with Vector Functions

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down right column to "Vector Valued Functions". There are several related titles in the first half of that list.