{
  "aliases": [
    "line-integral-over-a-sum-of-paths"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0BFN21TF5F"
    }
  ],
  "body": "\n## Statement\n\nLet $\\gamma_1 \\colon [a, b] \\to \\mathbb{R}^n$ and $\\gamma_2 \\colon [b, c] \\to \\mathbb{R}^n$ be piecewise $C^1$ curves with $\\gamma_1(b) = \\gamma_2(b)$.\nLet $\\gamma \\colon [a, c] \\to \\mathbb{R}^n$ be their concatenation: $\\gamma = \\gamma_1$ on $[a, b]$ and $\\gamma = \\gamma_2$ on $[b, c]$.\nFor each continuous vector field $F$ and each continuous function $f$ on the image of $\\gamma$,\n$$\\int_\\gamma F \\cdot d\\mathbf{r} = \\int_{\\gamma_1} F \\cdot d\\mathbf{r} + \\int_{\\gamma_2} F \\cdot d\\mathbf{r}, \\qquad \\int_\\gamma f \\, ds = \\int_{\\gamma_1} f \\, ds + \\int_{\\gamma_2} f \\, ds.$$\n\n## Remarks\n\nProof: apply the [[parametric formula|0W3W1VYWBP]] on $[a, c]$ and split the integral at $t = b$.\nIf the parameter intervals do not match, first shift the parameter of $\\gamma_2$. A shift does not change line integrals.\nTogether with [[path reversal|0FSCXYG856]], this lets one compute line integrals over the boundary of a polygon, side by side.\nThe concatenation is often written $\\gamma_1 + \\gamma_2$ or $\\gamma_1 * \\gamma_2$.\n\n## Explanations\n\n### L1 human\n\nWalking from $A$ to $B$ and then from $B$ to $C$ is one walk from $A$ to $C$.\nThe line integral over the whole walk is the sum of the line integrals over the two parts.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "line-integral-over-a-sum-of-paths"
    ],
    "kind": "statement",
    "msc": [
      "26B12"
    ],
    "name": "Additivity of line integrals under concatenation",
    "role": "proposition",
    "sources": [
      {
        "locator": "Line integral of a vector field",
        "url": "https://en.wikipedia.org/wiki/Line_integral"
      }
    ],
    "tag": "00ZEBSMNNV",
    "uses": [
      "0YS8RX87F4",
      "0M50E4WYWJ",
      "0W3W1VYWBP"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/00ZEBSMNNV/",
  "links": [
    {
      "field": "uses",
      "target": "0YS8RX87F4"
    },
    {
      "field": "uses",
      "target": "0M50E4WYWJ"
    },
    {
      "field": "uses",
      "target": "0W3W1VYWBP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/00ZEBSMNNV.md",
  "msc": [
    "26B12"
  ],
  "name": "Additivity of line integrals under concatenation",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/00Z/00ZEBSMNNV.md",
  "status": null,
  "tag": "00ZEBSMNNV",
  "uri": "m2://00ZEBSMNNV",
  "url": "https://math2.tamnd.com/wiki/additivity-of-line-integrals-under-concatenation/",
  "writing": {
    "tag": "00ZEBSMNNV",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}