Topics

Selected topics from:

Semester 1 Topics

OSC3: OpenStax Calculus Volume 3
OSC2: OpenStax Calculus Volume 2
OSP: OpenStax Precalculus

ABD: Calculus (Anton, Bivens, Davis, 10th ed)

Unit	Topic	Old Chapters	Open Chapters
1	3D Space	ABD Ch 11	OSC3 Ch 2
2	Vector-valued functions	ABD Ch 12	OSC3 Ch 3
3	Functions of Several Variables	ABD Ch 13	OSC3 Ch 4

Unit	Topic	Old Chapters	Open Chapters
4	Optimization	ABD Ch 13.8-13.9	OSC3 4.7-4.8
5	Multiple Integrals	ABD Ch 14	OSC3 Ch 5
6	Differential Equations		OSC2 Ch 4
6 (alt)	Geometric Algebra

1.1 3D space
- 3D coordinate systems
- distance formula
- equations of simple planes, spheres
- reference: ABD 11.1, OSC3 2.2
1.2 Vectors
- vector operations
- reference: ABD 11.2, OSC3 2.1-2.2
1.3 Dot Product
- dot product
- projections
- reference: ABD 13.3, OSC3 2.3
1.4 Matrices and determinants
- matrix operations
- column vectors
- linear transformations and matrix multiplication
- determinant and area
- 2D transformations: scale, reflection, rotation, projection
- reference: OSP 9.5, 9.7, 9.8
1.5 Cross Product
- linear transformations in $\mathbb{R}^3$
- orthogonality
- cross product, area of parallelogram
- scalar triple product
- reference: ABD 11.4, OSC3 2.4
1.6 Lines and Planes
- parametric equations of lines
- normal vectors
- equations of planes
- calculate distances from points to lines and planes
- reference: ABD 11.5-11.6, OSC3 2.5
1.7 Cylindrical and spherical coordinates
- cylindrical coordinates
- spherical coordinates
- figure out conversion formulas from picture, not memorization
- reference: ABD 11.8, OSC3 2.7
1.8 Projection and linear regression
- bonus lesson on statistics viewed from the point of view of linear algebra

2.1 Vector-valued functions
- parametric curves
- vector valued functions (curves)
- reference: ABD 12.1, OSC3 3.1
2.2 Calculus of vector-valued functions
- derivatives of parametric curves
- tangent vectors of curves
- reference: ABD 12.2, OSC3 3.2
2.3 Arc length
- arc length
- reparametrization
- reference: ABD 12.3, OSC3 3.3
2.4 Tangent, normal, binormal
- tangent, normal, binormal
- reference: ABD 12.4, OSC3 3.3
2.5 Motion
- position, velocity, acceleration
- projectile motion
- reference: ABD 12.6, OSC3 3.4
2.6 Polar conics
- recall conics topics from precalculus: focus, directrix, eccentricity
- geometric definitions of conic sections
- polar equations for conic sections
- reference: ABD 10.6, OSP 10.5
2.7 Kepler’s Laws
- Newton’s Laws -> Kepler’s Laws
- reference: ABD 12.7, OSC3 3.4

3.1 Functions of two or more variables
- functions of several variables
- level sets
- origami hyperbolic paraboloids
- reference: ABD 13.1, OSC3 4.1
3.2 Limits and continuity
- limits and continuity (multivariable)
- reference: ABD 13.2, OSC3 4.2
3.3 Partial derivatives
- partial derivatives
- reference: ABD 13.3, OSC3 4.3
3.4 The derivative
- derivative as linear transformation
- differentiability
- tangent planes
- reference: ABD 13.4, OSC3 4.4
3.5 Linear transformations and approximation
- derivative, tangent plane, linear approximation
- composition of linear transformations
- differentiability and linear transformations
- reference: OSC3 4.4
3.6 Chain rule
- composition of linear transformations
- multivariable chain rule
- reference: ABD 13.5, OSC3 4.5
3.7 Directional derivatives and gradient
- directional derivatives
- gradient
- reference: ABD 13.6, OSC3 4.6
3.8 Level surfaces and tangent planes
- level surfaces
- tangent planes of level surfaces
- reference: ABD 13.7, OSC3 4.6

4.1 Rotation of axes
- conic sections
- removing $xy$ term in general quadratics
- reference: ABD 10.5, OSP 10.4
4.2 Maxima and minima
- 2nd derivative test
- Extreme value theorem
- maxima, minima, saddle points
- reference: ABD 13.8, OSC3 4.7
4.3 Constrained optimization
- handling constraints by substitution
- reference: ABD 13.8, OSC3 4.7
4.4 Lagrange multipliers
- handling constraints by using Lagrange multipliers
- reference: ABD 13.9, OSC3 4.8

5.1 Double integrals
- double integrals
- reference: ABD 14.1, OSC3 5.1
5.2 More double integrals
- integration over non-rectangular regions
- reference: ABD 14.2, OSC3 5.2
5.3 More more double integrals
- volumes
- reference: ABD 14.2, OSC3 5.2
5.4 Double integrals in polar coordinates
- double integrals in polar coordinates
- reference: ABD 14.3, OSC3 5.3
5.5 Parametric surfaces and surface area
- cylinder, cone, sphere
- parametric equations for surfaces
- surface area
- reference: ABD 14.4, OSC3 6.6
5.6 More surfaces
- parametrization of sphere
- torus surface area and volume
- reference: ABD 14.4, OSC3 6.6
5.7 Triple integrals
- triple integrals
- average value of a function
- reference: ABD 14.5, OSC3 5.4
5.8 Cylindrical / spherical coordinates
- integration in polar, cylindrical, spherical
- volume elements
- reference: ABD 14.6, OSC3 5.5
5.9 Change of variables
- Jacobians
- change of variable formula derivations: polar, cylindrical, spherical
- reference: ABD 14.7, OSC3 5.7
5.10 Centers of Mass
- centroids of curves and regions
- reference: ABD 14.8, OSC3 5.6
5.11 Disks and Shells
- volumes and surfaces of revolution
- computing volumes using disks and shells
- disks and shells as triple integrals
5.12 Pappus’ Centroid Theorems
- surface area
- volume

6.1 Separable Equations
- separable differential equations
- reference: OSC2 4.1, 4.3
6.2 Integrating factors
- first order linear equations
- reference: OSC2 4.5
6.3 Population growth
- exponential and logistic population models
- reference: OSC2 4.4

6.1 Wedge Product
- wedge product in $\mathbb{R}^2$
6.2 Geometric Product
- geometric product in $\mathbb{R}^2$
6.3 Wedge Product in 3D
- wedge product in $\mathbb{R}^3$
- comparison with cross product
- equation of plane
- distance to plane
6.4 Geometric Product in 3D
- geometric product in $\mathbb{R}^3$
- reflections and rotations
6.5 Cramer’s Rule
- Cramer’s rule
6.6 Rotations
- more on composing rotations in 3D