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  "body": "\n## Summary\n\nA smooth manifold with corners is a space modelled near each point on an open subset of the closed orthant $[0, \\infty)^n$.\nIts changes of chart are smooth.\nIt allows boundary points where several boundary faces meet at an angle, like the vertices of a square or a cube.\nProducts of manifolds with boundary are manifolds with corners.\nThis is one reason the notion is needed, for example in cobordism theory and in Stokes' theorem on cubes.\n\n## Explanations\n\n### L1 human\n\nA closed square has an edge, like a closed disk, but its edge has four sharp corners.\nAs a topological space the square is just a disk, so the corners only matter when one wants to do calculus.\nA manifold with corners keeps track of them: near a corner it looks like the corner of the quarter-plane $\\set{x \\ge 0, y \\ge 0}$.\nThe solid cube is a $3$-dimensional example, with faces, edges and vertices as boundary strata.\n",
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