# Leibniz’ World of Calculus

* *Gottfried Wilhelm Leibniz (1646-1716).

Chapters: (Video Introduction to Fundamental Theorem of Calculus)

- Introduction: Symbolic vs Constructive Calculus
- Leibniz
- Real Numbers: Constructive Mathematics
- Differential Equations of Motion
- What Is a Function?
- Integral vs Derivative
- Fundamental Theorem of Calculus
- Integrals of Polynomials
- Exponential Function
- Logarithm
- Elementary Functions
- Trigonometric Functions
- Lipschitz Continuity
- Derivative with respect to Time
- Derivative with respect to Space
- Rules of Differentiation
- Proof of Fundamental Theorem
- Rules of Integration
- Solving f(u) = 0 by Time Stepping
- Solving f(u) = 0 by Newton’s Method
- From Residual to Root Error
- Contraction Mapping Theorem
- Generalized Fundamental Theorem
- The World as IVP
- Mathematics: Backward Magics or Forward Reason
- Standard Calculus as Ill-posed Unstable Backward Magic
- Constructive Calculus in Finite Precision
- Integration in Several Dimensions
- Gauss’, Green’s and Stokes’ Theorems
- Limits and Sequences
- Time Stepping Error Analysis
- Perspective on Reformation
- Who Invented Calculus?

*Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on in its conclusions, except that in the height the identity fades out of sight.*(*Goethe)*

*Making the simple complicated is commonplace; making the complicated simple, awesomely simple, that’s creativity.*(*Charlie Mingus)*

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