{
  "aliases": [
    "orientation-reversal-of-line-integrals"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0BFN21TF5F"
    }
  ],
  "body": "\n## Statement\n\nLet $\\gamma \\colon [a, b] \\to \\mathbb{R}^n$ be a piecewise $C^1$ curve with image $C$, and let $\\gamma^- \\colon [a, b] \\to \\mathbb{R}^n$, $\\gamma^-(t) = \\gamma(a + b - t)$, be the reversed curve.\nFor each continuous vector field $F$ on $C$ and each continuous function $f$ on $C$,\n$$\\int_{\\gamma^-} F \\cdot d\\mathbf{r} = -\\int_\\gamma F \\cdot d\\mathbf{r}, \\qquad \\int_{\\gamma^-} f \\, ds = \\int_\\gamma f \\, ds.$$\n\n## Remarks\n\nProof: by the [[parametric formula|0W3W1VYWBP]], $(\\gamma^-)'(t) = -\\gamma'(a + b - t)$. The substitution $u = a + b - t$ then gives both identities.\nThe [[line integral of a vector field|0YS8RX87F4]] depends on the orientation of the curve. The [[line integral of a scalar field|0M50E4WYWJ]] does not, because it uses only $\\|\\gamma'\\|$.\nMore generally, under a $C^1$ change of parameter $\\varphi$ with $\\varphi' \u003e 0$, the vector line integral does not change. With $\\varphi' \u003c 0$ it changes sign.\n\n## Explanations\n\n### L1 human\n\nIf a force helps you along a path, it works against you on the way back.\nSo the work, a line integral of a vector field, changes sign when you reverse the path. The length of the path and the mass of a wire do not.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "orientation-reversal-of-line-integrals"
    ],
    "kind": "statement",
    "msc": [
      "26B12"
    ],
    "name": "Reversing a path changes the sign of a line integral",
    "role": "proposition",
    "sources": [
      {
        "locator": "Line integral of a vector field",
        "url": "https://en.wikipedia.org/wiki/Line_integral"
      },
      {
        "locator": "Line integral of a scalar field",
        "url": "https://en.wikipedia.org/wiki/Line_integral"
      }
    ],
    "tag": "0FSCXYG856",
    "uses": [
      "0YS8RX87F4",
      "0M50E4WYWJ",
      "0W3W1VYWBP"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0FSCXYG856/",
  "links": [
    {
      "field": "uses",
      "target": "0YS8RX87F4"
    },
    {
      "field": "uses",
      "target": "0M50E4WYWJ"
    },
    {
      "field": "uses",
      "target": "0W3W1VYWBP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0FSCXYG856.md",
  "msc": [
    "26B12"
  ],
  "name": "Reversing a path changes the sign of a line integral",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0FS/0FSCXYG856.md",
  "status": null,
  "tag": "0FSCXYG856",
  "uri": "m2://0FSCXYG856",
  "url": "https://math2.tamnd.com/wiki/reversing-a-path-changes-the-sign-of-a-line-integral/",
  "writing": {
    "tag": "0FSCXYG856",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}