Charles队长 · Bilibili · 1:28
This video provides a visual proof of Green's Theorem using the "small rectangle method." The proof follows four logical steps: partitioning, approximation, summation, and taking limits. First, a planar region D is divided into a fine grid of infinitesimal rectangles. Next, the line integral around a single small rectangle is approximated using first-order Taylor expansions of partial derivatives, yielding an area integral form. Then, summing the integrals over all sub-rectangles reveals that internal shared boundaries cancel out due to opposite traversal directions, leaving only a staircase-like outer boundary. Finally, as the grid size approaches zero, this staircase boundary converges to the original smooth curve L, and the discrete Riemann sum transforms into a double integral, rigorously deriving Green's Theorem.
Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.
Generated from the video's visuals and explanation; not verbatim speech.
Welcome to the geometric intuition behind Green's Theorem via the small rectangle method. Our core proof strategy involves four steps: partitioning, approximation, summation, and taking limits. Step one is partitioning. We divide the closed planar region D into countless tiny rectangular grids denoted as . The goal is to transform a complex curvilinear domain problem into a collection of simple, straight-edged rectangular problems.
Step two is approximation. We isolate a single small rectangle and calculate the circulation of the vector field along its counterclockwise boundary. For the horizontal bottom and top edges, the difference in y-coordinates is Δy. Using a first-order Taylor approximation, their combined integral simplifies to - · Δx Δy. Similarly, the vertical right and left edges yield · Δx Δy. Summing these four sides, the line integral of the individual rectangle is approximately (∂Q/∂x - ∂P/∂y) Δσ, where Δσ represents the area of the small rectangle.
Step three is summation. We add up the contour integrals of all the small rectangles within region D. Observe any two adjacent rectangles; they share a common edge. Because each rectangle is traversed counterclockwise, this shared boundary is traveled in exactly opposite directions by the neighboring cells. Consequently, the integrals along all internal grid lines perfectly cancel each other out. After summation, only the outermost unshared edges remain, forming a stepped boundary that closely hugs the original curve L.
Step four is taking the limit. As the mesh becomes infinitely dense—meaning Δx and Δy approach zero—the stepped outer boundary on the left side converges uniformly to the original smooth curve L, making its line integral converge to the integral along L. Simultaneously, the summation expression on the right becomes the exact Riemann sum for the function (∂Q/∂x - ∂P/∂y) over region D, which naturally converts into a double integral. Thus, we have derived Green's Theorem: the line integral along the closed curve L equals the double integral of the curl over the enclosed region D. This is a geometric proof outline. A rigorous limit also requires continuous first partial derivatives of P,Q near the region, a piecewise smooth boundary and control of boundary approximation errors; proximity of curves alone does not imply convergence of their line integrals.
A classic elementary proof technique for Green's Theorem based on fundamental calculus principles. It partitions a complex region into simple small rectangles, applies local differential approximations, performs global summation leveraging the cancellation of internal boundaries, and finally takes the continuous limit to bridge line integrals and double integrals.
When evaluating the line integral on a micro-rectangle, first-order Taylor expansions convert finite differences of the functions P and Q into their respective partial derivatives. The horizontal edges generate the - term, while the vertical edges produce the term. Multiplying by the area element Δσ facilitates the local transition from a 1D line integral to a 2D area integral.
This is the most crucial topological property in the small rectangle method. When summing the counterclockwise boundary integrals of all adjacent sub-rectangles, any common edge between two rectangles is traversed once in each direction. Since line integrals are path-dependent and orientation-sensitive, reversing the direction negates the value, causing all internal mesh contributions to vanish and isolating the true outer boundary.
Under the stated smoothness and piecewise-smooth boundary hypotheses, the mesh-boundary approximation can be controlled so its line integrals approach the boundary integral; geometric closeness alone is insufficient. Algebraically, the displayed sum is a Riemann sum for over .
The small-rectangle argument adds oriented boundary integrals. Common internal edges are traversed in opposite directions and cancel, leaving the outer boundary. Refinement connects the sum with an area integral in Green’s theorem. This applies integral additivity and orientation, under the usual differentiability and boundary hypotheses, rather than establishing a theorem for arbitrary regions or nonsmooth fields.
A rigorous limit requires continuous first partial derivatives of P and Q near the region, a piecewise smooth boundary, and control of boundary approximation errors. Proximity of curves alone does not imply convergence of their line integrals.
Conditions: The proof is being taken to a rigorous limit.; P and Q are defined near the region D.; The boundary L is considered.
The line integral is approximated using first-order Taylor expansions of the partial derivatives. The horizontal edges yield a term involving , and the vertical edges yield a term involving .
Conditions: The rectangle is infinitesimally small.; P and Q have continuous first partial derivatives.
The proof proceeds in four steps: partitioning the region D into small rectangles, approximating the line integral on a single rectangle using Taylor expansions, summing these integrals so that internal boundaries cancel out, and taking the limit as the grid size approaches zero to derive the double integral of the curl.
Conditions: The region D is a closed planar region.; The functions P and Q have continuous first partial derivatives near D.; The boundary L is piecewise smooth.
Internal boundaries cancel out because adjacent rectangles share a common edge that is traversed in exactly opposite directions. Since line integrals are orientation-sensitive, the integral along the shared edge for one rectangle negates the integral for the neighboring rectangle, leaving only the outer boundary.
Conditions: Each rectangle is traversed counterclockwise.; The rectangles are adjacent and share a common edge.
As the grid size approaches zero, the stepped outer boundary converges uniformly to the original smooth curve L, and the summation expression becomes the exact Riemann sum for the function , which converts into a double integral over region D.
Conditions: The mesh becomes infinitely dense ().; P and Q have continuous first partial derivatives.; The boundary L is piecewise smooth.
YouTube2:26Prerequisite in Calculus: Limits → Definite integrals.
Limits · View learning sequence
YouTube3:52Next in Calculus: Definite integrals → Integration by parts.
Integration by parts · View learning sequence
YouTube2:38Prerequisite in Calculus: Limits → Definite integrals.
Limits · View learning sequence
YouTube2:28English video covering Definite integrals; the current video is in Chinese.
Definite integrals
YouTube4:16English video covering Definite integrals; the current video is in Chinese.
Definite integrals
YouTube4:26English video covering Definite integrals; the current video is in Chinese.
Definite integralsYouTube2:28Explanation of Definite integrals, complementing the current application.
Definite integralsYouTube4:16Explanation of Definite integrals, complementing the current application.
Definite integralsYouTube4:26Explanation of Definite integrals, complementing the current application.
Definite integrals