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  "body": "\n## Summary\n\nThe kernel of a linear [[integral equation|R3SG1SN7JQ]] is the function $K(x, s)$ of two variables that multiplies the unknown function inside the integral.\nIn $\\varphi(x) = f(x) + \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds$ it is $K$.\nThe properties of the kernel, such as continuity, symmetry or the form of its singularity, decide which theory applies to the equation.\n\n## Explanations\n\n### L1 human\n\nThe kernel says how strongly the value of the unknown at the point $s$ influences the equation at the point $x$.\nIt is like an infinite matrix whose rows are indexed by $x$ and whose columns are indexed by $s$.\nThe integral $\\int K(x, s) \\varphi(s) \\, ds$ is the continuous version of multiplying a matrix by a vector.\nFor example, in $\\varphi(x) = 1 + \\int_0^1 (x + s) \\varphi(s) \\, ds$ the kernel is $K(x, s) = x + s$.\n",
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