{
  "aliases": [
    "conservative-iff-path-independent"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0GRM3BS0DD"
    }
  ],
  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open and connected, and let $F \\colon U \\to \\mathbb{R}^n$ be a continuous vector field.\nThe following are equivalent.\n\n1. $F$ is [[conservative|05R9NF2J49]] on $U$.\n2. For all piecewise $C^1$ curves $\\gamma_1, \\gamma_2$ in $U$ with the same initial point and the same end point, $\\int_{\\gamma_1} F \\cdot d\\mathbf{r} = \\int_{\\gamma_2} F \\cdot d\\mathbf{r}$.\n3. For each closed piecewise $C^1$ curve $\\gamma$ in $U$, $\\int_\\gamma F \\cdot d\\mathbf{r} = 0$.\n\n## Remarks\n\nStatement 1 implies statement 2 by the [[gradient theorem|0SD237FEJD]].\nStatements 2 and 3 are equivalent: a closed curve splits into two curves with the same end points, and one uses [[reversal|0FSCXYG856]] and [[additivity|00ZEBSMNNV]].\nStatement 2 implies statement 1: fix $p \\in U$ and let $\\phi(x) = \\int_\\gamma F \\cdot d\\mathbf{r}$ for a path $\\gamma$ from $p$ to $x$. A short straight last segment in direction $e_i$ shows that $\\partial \\phi / \\partial x_i = F_i$.\nConnectedness is used to join $p$ to each point. On a disconnected set, apply the result on each component.\n\n## Explanations\n\n### L1 human\n\nFor a conservative field, the work from one point to another does not depend on the route.\nConversely, if the work never depends on the route, then the work from a fixed base point defines a potential for the field.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "conservative-iff-path-independent"
    ],
    "kind": "statement",
    "msc": [
      "26B20"
    ],
    "name": "Path independence of line integrals",
    "role": "theorem",
    "sources": [
      {
        "locator": "Path independence",
        "url": "https://en.wikipedia.org/wiki/Conservative_vector_field"
      },
      {
        "locator": "Path independence and conservative vector field",
        "url": "https://en.wikipedia.org/wiki/Conservative_vector_field"
      },
      {
        "locator": "Converse of the gradient theorem",
        "url": "https://en.wikipedia.org/wiki/Gradient_theorem"
      }
    ],
    "tag": "0BFN21TF5F",
    "uses": [
      "05R9NF2J49",
      "0YS8RX87F4",
      "0SD237FEJD",
      "0FSCXYG856",
      "00ZEBSMNNV",
      "05AXG3QRKR"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0BFN21TF5F/",
  "links": [
    {
      "field": "uses",
      "target": "05R9NF2J49"
    },
    {
      "field": "uses",
      "target": "0YS8RX87F4"
    },
    {
      "field": "uses",
      "target": "0SD237FEJD"
    },
    {
      "field": "uses",
      "target": "0FSCXYG856"
    },
    {
      "field": "uses",
      "target": "00ZEBSMNNV"
    },
    {
      "field": "uses",
      "target": "05AXG3QRKR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0BFN21TF5F.md",
  "msc": [
    "26B20"
  ],
  "name": "Path independence of line integrals",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0BF/0BFN21TF5F.md",
  "status": null,
  "tag": "0BFN21TF5F",
  "uri": "m2://0BFN21TF5F",
  "url": "https://math2.tamnd.com/wiki/path-independence-of-line-integrals/",
  "writing": {
    "tag": "0BFN21TF5F",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}