Mathematics & Science
Exploring Calculus, Multivariable Calculus, and Mathematical Concepts through Interactive Visualizations
Calculus Concepts
Differential Calculus
Differential calculus focuses on the concept of the derivative, which measures how a function changes as its input changes. The derivative represents the instantaneous rate of change or the slope of the tangent line at a point.
- •Limits and Continuity
- •Derivatives and Differentiation Rules
- •Chain Rule and Implicit Differentiation
- •Applications: Optimization and Related Rates
Integral Calculus
Integral calculus deals with the accumulation of quantities and the area under curves. The fundamental theorem of calculus connects differentiation and integration, showing they are inverse operations.
- •Riemann Sums and Definite Integrals
- •Fundamental Theorem of Calculus
- •Integration Techniques (Substitution, Parts, etc.)
- •Applications: Area, Volume, and Arc Length
Multivariable Calculus
Partial Derivatives
In multivariable calculus, we extend the concept of derivatives to functions of multiple variables. Partial derivatives measure how a function changes with respect to one variable while keeping others constant.
- •Partial Derivatives and Gradient
- •Directional Derivatives
- •Chain Rule for Multiple Variables
- •Optimization: Critical Points and Lagrange Multipliers
Multiple Integrals
Multiple integrals extend integration to functions of multiple variables, allowing us to compute volumes, masses, and other quantities in higher dimensions.
- •Double and Triple Integrals
- •Change of Variables (Jacobian)
- •Line and Surface Integrals
- •Green's, Stokes', and Divergence Theorems
Pi Raptor Computation Works
Detailed mathematical computations showcasing problem-solving approaches and analytical techniques
Computation Works
Showcasing detailed mathematical computations and problem-solving approaches
Interactive Visualizations
Explore mathematical concepts through interactive 3D animations and dynamic graphs
Gabriel's Horn
A fascinating mathematical paradox: a shape with infinite surface area but finite volume. Formed by rotating the curve y = 1/x (for x ≥ 1) around the x-axis.
Mathematical Properties:
- • Surface Area: A = 2π ∫₁^∞ (1/x)√(1 + 1/x⁴) dx = ∞
- • Volume: V = π ∫₁^∞ (1/x²) dx = π
- • This paradox demonstrates that infinite surface area doesn't imply infinite volume
Riemann Sums - Integral Approximation
Visualize how discrete rectangular sections approximate the area under a curve. As the number of rectangles increases, the approximation converges to the definite integral.
Approximated Area:
21.280000
As n → ∞, this approximation approaches the exact value of the definite integral ∫[a to b] f(x) dx
Derivatives: Secant to Tangent
Watch as the secant line approaches the tangent line as h approaches zero. The derivative is the limit of the difference quotient as h → 0.
Mathematical Definition:
f'(x) = limh→0 [f(x+h) - f(x)] / h
As h approaches zero, the secant line (blue, dashed) converges to the tangent line (green, solid) at the point of tangency (red dot). The slope of the tangent line is the derivative at that point.