🧪Integral Theorems: Connecting Derivatives to Boundaries

Integral theorems connect derivatives to boundaries by converting integrals over a region of derivative quantities (such as gradients, partial derivatives, or divergences) into expressions evaluated at or involving the region's boundary. This interplay is critical for solving and understanding physical problems where flux, circulation, or accumulated quantities inside a volume or along a curve are related to conditions on the boundary.

This section emphasizes a comprehensive understanding of fundamental calculus theorems (Divergence, Green's, Stokes') through interactive exploration and analytical application, demonstrating their power in relating seemingly disparate concepts like volume and surface integrals, understanding physical phenomena (e.g., mass-space interaction, electric flux), and deriving key mathematical and physical laws.

🎬Animated result and interactive web

how a volume integral of a divergence can be related to the flux of a vector field through the boundary surface

how mass influences the space around it-how forces arise from potential energy landscapes and work done by conservative force

how the sum of the divergences within a volume equates to the net flux passing through the outer boundary of that volume

how the Divergence Theorem proves the formulas for the volume and surface area of a sphere

Gauss Law in both its integral and differential forms and how it's derived using the Divergence Theorem

Greens Theorem for line integrals and area integrals

The flux of the curl depends only on the boundary curve rather than the specific shape of the surface spanning that boundary