Author(s): Joel Feldman, Andrew Rechnitzer and Elyse Yeager
Series: CLP Calculus 03
Edition: Text
Publisher: University of British Columbia
Year: 2021
Language: English
Tags: mathematics; maths; math; calc; calculus; single variable; limits; differentiation; differential; integration; integral; continuity; differentiability; analysis; real; complex; multiple variables; multivariable; multivariate; several variables; many variables
Vectors and Geometry in Two and Three Dimensions
Points
Vectors
Addition of Vectors and Multiplication of a Vector by a Scalar
The Dot Product
(Optional) Using Dot Products to Resolve Forces — The Pendulum
(Optional) Areas of Parallelograms
The Cross Product
(Optional) Some Vector Identities
(Optional) Application of Cross Products to Rotational Motion
(Optional) Application of Cross Products to Rotating Reference Frames
Equations of Lines in 2d
Equations of Planes in 3d
Equations of Lines in 3d
Curves and their Tangent Vectors
Derivatives and Tangent Vectors
Sketching Surfaces in 3d
Level Curves and Surfaces
Cylinders
Quadric Surfaces
Partial Derivatives
Limits
Optional — A Nasty Limit That Doesn't Exist
Partial Derivatives
Higher Order Derivatives
Optional — The Proof of Theorem 2.3.4
Optional — An Example of 2 fxy(x0,y0) 2 fyx(x0,y0)
The Chain Rule
Memory Aids for the Chain Rule
Chain Rule Examples
Review of the Proof of ddtf(to.x(t))to. = dfdx(to.x(t))to. dxdt(t)
Proof of Theorem 2.4.1
Tangent Planes and Normal Lines
Surfaces of the Form z=f(x,y).
Surfaces of the Form G(x,y,z)=0.
Linear Approximations and Error
Quadratic Approximation and Error Bounds
Optional — Taylor Polynomials
Directional Derivatives and the Gradient
A First Look at Partial Differential Equations
Optional — Solving the Advection and Wave Equations
Really Optional — Derivation of the Wave Equation
Maximum and Minimum Values
Absolute Minima and Maxima
Lagrange Multipliers
(Optional) An Example with Two Lagrange Multipliers
Multiple Integrals
Double Integrals
Vertical Slices
Horizontal Slices
Volumes
Examples
Optional — More about the Definition of Rf(x,y) dxdy
Even and Odd Functions
Double Integrals in Polar Coordinates
Polar Coordinates
Polar Curves
Integrals in Polar Coordinates
Optional— Error Control for the Polar Area Formula
Applications of Double Integrals
Surface Area
Triple Integrals
Triple Integrals in Cylindrical Coordinates
Cylindrical Coordinates
The Volume Element in Cylindrical Coordinates
Sample Integrals in Cylindrical Coordinates
Triple Integrals in Spherical Coordinates
Spherical Coordinates
The Volume Element in Spherical Coordinates
Sample Integrals in Spherical Coordinates
Optional— Integrals in General Coordinates
Optional — Dropping Higher Order Terms in du,dv
Trigonometry
Trigonometry — Graphs
Trigonometry — Special Triangles
Trigonometry — Simple Identities
Trigonometry — Add and Subtract Angles
Inverse Trigonometric Functions
Powers and Logarithms
Powers
Logarithms
Table of Derivatives
Table of Integrals
Table of Taylor Expansions
3d Coordinate Systems
Cartesian Coordinates
Cylindrical Coordinates
Spherical Coordinates
ISO Coordinate System Notation
Polar Coordinates
Cylindrical Coordinates
Spherical Coordinates
Conic Sections and Quadric Surfaces
