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S = contains supplemental resources
Course: Calculus III
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. Startign 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 r(t) and know that r'(t) is the tangent vector (a.k.a. velocity vector)
• 14.2.4 Find the unit tangent vector T(t) = r'(t)/||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 r(0) = i)

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.