11 Calculus

Calculus is the mathematics of approximation. Here are the ideas that are fundamental to the study of calculus.

The limit. A limit is a value: the number a function approaches as its input gets arbitrarily close to a point.

The derivative. The derivative is a rate: the instantaneous rate of change of a function at a point. Geometrically it is the slope of the best straight-line approximation to the curve there.

The fundamental theorem of calculus. Differentiation and integration are inverse operations. Differentiation is the operation of finding a rate. Integration is the operation of accumulating a quantity. The fundamental theorem is the statement that these two operations undo each other.

11.1 Topics

1. Limit
2. Partial Derivative
3. Chain Rule
4. Jacobian
   1. Basic Derivation of Jacobian
   2. Derivation of Jacobian
5. Derivation of Taylor Expansion
6. Indefinite Integral
   1. Definite Integral
   2. Improper Integral
7. Real Parameter
8. Parameterization
9. Curve
   1. Smooth Curve
10. Envelope of Tangent Lines
11. Legendre Transform
    1. Legendre Transform Derivation
12. Vector Calculus
    1. Vector Field
    2. Vector Differential Operator
       1. Gradient
          1. Derivation of Grad
       2. Divergence
          1. Derivation of Divergence
       3. Curl
          1. Derivation of Curl
    3. Dot Product
    4. Cross Product
    5. Scalar Projection
    6. Vector Projection
    7. Line Integral
13. General Functions

11.2 References

1. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. — calculus as what becomes true in the limit, and the derivative as the best linear approximation.
2. Griffel, D. H. Applied Functional Analysis. — elementary calculus as the tangent line giving the best approximation to a curve at a point.
3. Stewart, J. Calculus: Early Transcendentals. — the limit as the value a function approaches.
4. Needham, T. Visual Differential Geometry and Forms. — the fundamental theorem as the inverse relation of differentiation and integration.

• Basic Derivation of Jacobian
• Chain Rule
• Cross Product
• Curl
• Curve
• Definite Integral
• Derivation of Curl
• Derivation of Divergence
• Derivation of Grad
• Derivation of Jacobian
• Derivation of Taylor Expansion
• Divergence
• Dot Product
• Envelope of Tangent Lines
• General Functions
• Gradient
• Improper Integral
• Indefinite Integral
• Jacobian
• Legendre Transform
• Legendre Transform Derivation
• Limit
• Line Integral
• Notes
• Parameterization
• Partial Derivative
• Real Parameter
• Scalar Projection
• Smooth Curve
• Vector Calculus
• Vector Differential Operator
• Vector Field
• Vector Projection