@math.visualizations: Which row is the fundamental theorem of calculus? The single-variable (Riemann) integral of f over an interval is the signed area between the graph of f and the x-axis. The Lebesgue integral measures the same quantity by partitioning the range instead of the domain: it integrates the measure of the set where f exceeds each level y. For the function shown, both equal 13.72. The line integral of a vector field F along a curve C adds up the tangential component F.T over the length of the curve; for a force field it is the work done along the path. The double integral adds up a function over a plane region D. Green's theorem states that the line integral of F around a closed curve equals the double integral of (curl F).k over the region it encloses. Here both equal 6.22. The flux (surface) integral of F over a surface S measures the rate at which the field passes through S, from the normal component F.n. The triple integral adds up a function over a solid region E, such as a density to give a mass. The divergence theorem states that the outward flux through a closed surface equals the triple integral of div F over the solid inside. Here both equal 7.54. #math #calculus #mathstudent #mathtok #manim

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Friday 18 September 2026 06:57:49 GMT