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  "body": "\n## Definition\n\nLet $\\Omega$ be an open subset of $\\mathbb{R}^2$ and let $f : \\Omega \\to \\mathbb{R}$ be a function.\nThe slope field of the first-order equation $y' = f(x, y)$ is the map that assigns to each point $(x, y) \\in \\Omega$ the line through $(x, y)$ with slope $f(x, y)$.\nIn a picture, one draws a short segment of this line at each point of a grid in $\\Omega$.\n\n## Remarks\n\nLet $u$ be a differentiable function on an interval. It is a [[solution|04SW7WECS8]] of $y' = f(x, y)$ if and only if its graph is tangent to the slope field at each of its points.\nFor $y' = y$, the segments are horizontal on the $x$-axis and get steeper as $|y|$ grows.\nFor a system $x' = F(x)$ in the plane, the same picture with arrows is a picture of the vector field $F$.\n\n## Explanations\n\n### L1 human\n\nA slope field shows the direction that a solution must take at each point.\nYou can sketch a solution without a formula: start at a point and follow the small segments.\nThis gives a quick picture of how all the solutions behave.\n",
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