Mathematics MOC

Infinitesimal Calculus MOC

Calculus, specifically infinitesimal calculus, is the mathematical study of continuous change. Infinitesimal calculus is traditionally broken into two branches: Differential Calculus and Integral Calculus; and these are linked together by the Fundamental theorem of calculus. In education, Calculus forms the basic foundations of further analysis.

Foundations

• Limits (Calculus)
• Continuity

Differential Calculus

Differential calculus concerns instantaneous rates of change.

Techniques and identities

• Derivatives of inverse trigonometric functions
• Partial fraction decomposition
• Implicit differentiation

Function features

• Critical point
• Local extremum
• Concavity

Optimisation

• Mean value theorem
• Second derivative test for multivariable functions
• Lagrange multiplier
• Lagrangian function

Vector-valued

• Product rule
• Arc length of a vector-valued function

Multivariable rates of change

• Jacobian matrix
• Classical curvature
• Divergence
• Curl

Integral Calculus

Integral calculus concerns the accumulation of quantities.

Techniques and identities

See Integration techniques MOC

Types of integral

• Improper integral

Multivariable integral calculus

• Double integral
• Triple integral
• Area of a bounded region
• Change of basis in multiple integrals

Vector calculus

• Conservative vector field
• Circulation
• Flux

Series and sequences

The theory of series and sequences also plays an important role.

• Sequence
• Infinite series
• Taylor series
• Fourier series

Differential equations

See Differential equations MOC.

Complex calculus

Complex analysis MOC

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