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  "body": "\n## Proof\n\nWe reduce to $C^1$ pieces and apply the fundamental theorem of calculus to $\\phi \\circ \\gamma$.\n\n1. Since $\\gamma$ is piecewise $C^1$, there is a [[partition|069P20KYQT]] $a = p_0 \u003c \\dots \u003c p_m = b$ such that the restriction $\\gamma^{(l)}$ of $\\gamma$ to $[p_{l-1}, p_l]$ is $C^1$ for each $l$.\n2. Since $\\phi$ is of [[class $C^1$|0TCBZMEYGN]], its partial derivatives are continuous on $U$. So the [[gradient|0NS2QYVC4Y]] field $\\nabla \\phi$ is continuous, and its [[line integrals|0YS8RX87F4]] over $\\gamma$ and over each $\\gamma^{(l)}$ are defined.\n3. By [[additivity under concatenation|00ZEBSMNNV]], applied $m - 1$ times, $\\int_\\gamma \\nabla \\phi \\cdot d\\mathbf{r} = \\sum_{l=1}^{m} \\int_{\\gamma^{(l)}} \\nabla \\phi \\cdot d\\mathbf{r}$.\n4. By the [[parametric formula|0W3W1VYWBP]], $\\int_{\\gamma^{(l)}} \\nabla \\phi \\cdot d\\mathbf{r} = \\int_{p_{l-1}}^{p_l} \\nabla \\phi(\\gamma(t)) \\cdot \\gamma'(t) \\, dt$.\n5. Since its partial derivatives are continuous, $\\phi$ is differentiable at each point of $U$, by [[continuous partial derivatives imply differentiability|0KXSK8TEEB]].\n6. Let $t \\in (p_{l-1}, p_l)$. By the [[chain rule for a composition with a curve|0A8TJ0VHC7]] on the open interval $(p_{l-1}, p_l)$, $(\\phi \\circ \\gamma)'(t) = D\\phi(\\gamma(t))(\\gamma'(t))$.\n7. By the definition of the gradient, $D\\phi(\\gamma(t))(\\gamma'(t)) = \\nabla \\phi(\\gamma(t)) \\cdot \\gamma'(t)$.\n8. The function $\\phi \\circ \\gamma$ is continuous on $[p_{l-1}, p_l]$. The function $t \\mapsto \\nabla \\phi(\\gamma(t)) \\cdot \\gamma'(t)$ is continuous there, and by steps 6 and 7 it is the derivative of $\\phi \\circ \\gamma$ on the open interval.\n9. By the [[fundamental theorem of calculus|0NHGY0N0JB]], $\\int_{p_{l-1}}^{p_l} \\nabla \\phi(\\gamma(t)) \\cdot \\gamma'(t) \\, dt = \\phi(\\gamma(p_l)) - \\phi(\\gamma(p_{l-1}))$.\n10. Summing over $l$ with steps 3 and 4, the sum telescopes: $\\int_\\gamma \\nabla \\phi \\cdot d\\mathbf{r} = \\phi(\\gamma(p_m)) - \\phi(\\gamma(p_0)) = \\phi(\\gamma(b)) - \\phi(\\gamma(a))$.\n\n## Idea\n\nAlong the curve, the integrand $\\nabla \\phi(\\gamma(t)) \\cdot \\gamma'(t)$ is the derivative of $\\phi(\\gamma(t))$, by the chain rule. So the line integral is the integral of a derivative, and the fundamental theorem of calculus leaves only the values of $\\phi$ at the ends.\n",
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