Lessons — dy over dx

Part I

Describing Change

In progress

We begin by describing how quantities respond to change, then connect that idea to accumulation and differential equations. Limits arrive later to clarify and justify ideas that are already familiar.

1. Differentiation

1.1 Sensitivity to change
1.2 Differentials
1.3 The derivative
1.4 The Product Rule and the Chain Rule
1.5 Implicit differentiation

2. Applications of derivatives

2.1 Related rates
2.2 Optimization
2.3 Higher order derivatives
2.4 Linear approximation

3. Integration

3.1 The integral: accumulation of change
3.2 Properties of integrals
3.3 The Fundamental Theorem of Calculus
3.4 Average value and average rates

4. Differential equations

4.1 Differential equations
4.2 Exponential functions
4.3 Trigonometric functions
4.4 Logarithmic functions
4.5 Separation of variables

5. Limits

5.1 Limits and continuity
5.2 Algebraic techniques for limits
5.3 Limits and differentiation
5.4 Limits and integration
5.5 L’Hôpital’s Rule
5.6 Valuable theorems

Capstone: Proving the Fundamental Theorem

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Part II

Harnessing Infinity

Planned

This part develops the tools needed to work with curves, accumulation, and infinite processes. Series arise as useful approximations first, with questions of convergence following from their use.

6. Analytic geometry

6.1 Motion in the plane
6.2 Parametric curves
6.3 Polar coordinates
6.4 Arc length
6.5 Curvature

7. Area and volume

7.1 Area between curves
7.2 Volume by slicing
7.3 Volume by shells
7.4 Surfaces of revolution

8. Integration techniques

8.1 Integration by substitution
8.2 Integration by parts
8.3 Trigonometric substitution
8.4 Partial fractions
8.5 Advanced integration strategy

9. Special functions

9.1 Functions defined by integrals
9.2 Numerical integration
9.3 Improper integrals
9.4 Probability distributions

10. Infinite series

10.1 Taylor approximations and series
10.2 Error in Taylor series
10.3 Manipulating power series
10.4 Geometric series
10.5 Generating functions
10.6 Hyperbolic functions

Capstone: Fourier series

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Part III

New Dimensions

Planned

Calculus expands beyond a single input and output. Geometry, motion, approximation, and optimization are reconsidered in spaces where several quantities can change at once.

11. Higher dimensions

11.1 What are dimensions, really?
11.2 Extending to higher dimensions
11.3 Slicing through dimensions
11.4 Functions of multiple variables

12. Partial derivatives

12.1 Partial derivatives
12.2 Tangent plane approximation
12.3 Partial elasticity
12.4 Higher-order partial derivatives
12.5 Multivariable optimization

13. Vectors and motion

13.1 Vector-valued functions
13.2 Motion in space
13.3 The dot product and cross product
13.4 The Frenet-Serret frame

14. Gradients

14.1 The chain rule in higher dimensions
14.2 Directional derivatives and the gradient
14.3 Geometry of the gradient
14.4 Lagrange multipliers
14.5 Gradient descent

15. Multiple integrals

15.1 Double integrals and iterated integration
15.2 Double integrals over general regions
15.3 Surface area
15.4 Triple integrals
15.5 Center of mass
15.6 Multivariable probability

Capstone: Calculus of variations

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Part IV

Transforming Perspectives

Planned

Vector calculus and complex analysis develop alongside one another. Changing coordinates and representations reveals common structures that culminate in several far-reaching fundamental theorems.

16. Complex functions

16.1 The geometry of complex numbers
16.2 Euler’s formula and its consequences
16.3 Visualizing complex functions
16.4 Multivalued function behavior

17. Vector fields

17.1 Vector fields
17.2 Dynamical systems
17.3 New derivatives and the del operator
17.4 The Cauchy-Riemann equations
17.5 Potential functions

18. Coordinate transformations

18.1 Transformations in the plane
18.2 The Jacobian matrix and its determinant
18.3 Double integrals over polar regions
18.4 Cylindrical and spherical coordinates

19. Line and surface integrals

19.1 Line integrals
19.2 Work, circulation, and flux
19.3 Contour integrals
19.4 Surface integrals
19.5 Integrals over oriented surfaces

20. Fundamental theorems

20.1 Conservative vector fields
20.2 Green’s theorems
20.3 Fundamental theorems in higher dimensions
20.4 Integrals of holomorphic functions
20.5 Singularities and Laurent series
20.6 The residue theorem

Capstone: Differential forms and exterior calculus

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Part V

Firm Foundations

Planned

The final part asks why the familiar processes of calculus work, when they fail, and which properties of the real numbers make them possible. Real analysis brings these ideas into a precise framework and shows how their foundations fit together.

21. Properties of real numbers

21.1 The need for rigor
21.2 Ordered fields
21.3 The completeness property
21.4 Topology of the real numbers

22. Sequences and limits

22.1 Convergence of sequences
22.2 Proving limit laws
22.3 Limits of functions
22.4 Cauchy sequences
22.5 Constructing the real numbers

23. Differentiability

23.1 Differentiability classes
23.2 More on the Mean Value Theorem
23.3 Differentiability in higher dimensions
23.4 Inverse and implicit functions

24. Integrability

24.1 Compactness
24.2 Uniform continuity
24.3 Riemann and Darboux integrals
24.4 Conditions for and properties of integrability
24.5 Lebesgue measure

25. Series and limiting processes

25.1 Convergence of series
25.2 Absolute and conditional convergence
25.3 Pointwise and uniform convergence
25.4 Interchanging limit processes
25.5 Convergence of power series
25.6 Solutions to differential equations

Capstone: Non-standard analysis

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