[proofplan] We prove equality of the two cochains by evaluating both sides on an arbitrary [smooth singular simplex](/page/Smooth%20Singular%20Simplex). The [cochain coboundary](/page/Cochain%20Coboundary) $\delta$ turns the evaluation of the [integration cochain](/page/Integration%20Cochain) $I_k(\omega)$ into the alternating sum of face integrals, written explicitly as Lebesgue integrals of the coefficient functions of the pulled-back forms in the standard oriented coordinates. [Stokes' theorem for manifolds with corners](/page/Stokes%20Theorem%20for%20Manifolds%20with%20Corners) converts this alternating face integral into the corresponding [Lebesgue integral](/page/Lebesgue%20Integral) of the coefficient of $d(\sigma^*\omega)$ over the simplex, and naturality of the [exterior derivative](/page/Exterior%20Derivative) identifies this with the coefficient of $\sigma^*(d\omega)$. [/proofplan] [step:Evaluate the coboundary on an arbitrary smooth singular simplex] Let \begin{align*} \sigma: \Delta^{k+1} &\to M \end{align*} be an arbitrary [smooth singular simplex](/page/Smooth%20Singular%20Simplex) of dimension $k+1$. For each index $i \in \{0,\dots,k+1\}$, let \begin{align*} \iota_i: \Delta^k &\to \Delta^{k+1} \end{align*} denote the standard smooth inclusion of the $i$-th oriented face. The [singular boundary](/page/Singular%20Boundary) of $\sigma$ is the smooth singular $k$-chain \begin{align*} \partial\sigma = \sum_{i=0}^{k+1} (-1)^i\, \sigma \circ \iota_i. \end{align*} For the standard oriented $m$-simplex $\Delta^m$, use the affine coordinates $(\lambda_1,\dots,\lambda_m)$ on the chart identifying $\Delta^m$ with $\{(\lambda_1,\dots,\lambda_m) \in \mathbb{R}^m : \lambda_j \ge 0,\ \sum_{j=1}^m \lambda_j \le 1\}$ and set $\lambda_0 = 1 - \sum_{j=1}^m \lambda_j$. For an oriented affine $m$-simplex $S$ equipped with such oriented coordinates and an $m$-form $\alpha \in \Omega^m(S)$, let $[\alpha]_S: S \to \mathbb{R}$ denote the coefficient function defined by writing $\alpha = [\alpha]_S\, d\lambda_1 \wedge \cdots \wedge d\lambda_m$ in those coordinates; integration over $S$ means the Lebesgue integral of this coefficient against $\mathcal{L}^m$. By the definition of the [cochain coboundary](/page/Cochain%20Coboundary) $\delta$, applied to the [integration cochain](/page/Integration%20Cochain) $I_k(\omega) \in C^k_{\mathrm{sm}}(M;\mathbb{R})$, we have \begin{align*} \delta(I_k(\omega))(\sigma) &= I_k(\omega)(\partial\sigma) \\ &= \sum_{i=0}^{k+1} (-1)^i\, I_k(\omega)(\sigma \circ \iota_i) \\ &= \sum_{i=0}^{k+1} (-1)^i \int_{\Delta^k} [(\sigma \circ \iota_i)^*\omega]_{\Delta^k}\, d\mathcal{L}^k. \end{align*} [/step] [step:Rewrite the alternating face sum as a boundary integral] For each $i \in \{0,\dots,k+1\}$, functoriality of pullback gives \begin{align*} (\sigma \circ \iota_i)^*\omega = \iota_i^*(\sigma^*\omega). \end{align*} The boundary orientation on $\partial\Delta^{k+1}$ is encoded by the alternating signs in \begin{align*} \partial\Delta^{k+1} = \sum_{i=0}^{k+1} (-1)^i\, \iota_i(\Delta^k). \end{align*} Therefore the preceding sum is the oriented boundary integral of the pulled-back $k$-form $\sigma^*\omega \in \Omega^k(\Delta^{k+1})$, written with explicit face measures as \begin{align*} \sum_{i=0}^{k+1} (-1)^i \int_{\Delta^k} [\iota_i^*(\sigma^*\omega)]_{\Delta^k}\, d\mathcal{L}^k. \end{align*} Consequently, \begin{align*} \delta(I_k(\omega))(\sigma) = \sum_{i=0}^{k+1} (-1)^i \int_{\Delta^k} [\iota_i^*(\sigma^*\omega)]_{\Delta^k}\, d\mathcal{L}^k. \end{align*} [guided] The alternating signs in the singular boundary are not extra bookkeeping; they are exactly the signs that define the [induced orientation on the boundary](/theorems/1528) faces of the standard simplex. For each face inclusion \begin{align*} \iota_i: \Delta^k &\to \Delta^{k+1}, \end{align*} the pullback of $\omega$ along the face simplex $\sigma \circ \iota_i$ satisfies \begin{align*} (\sigma \circ \iota_i)^*\omega = \iota_i^*(\sigma^*\omega) \end{align*} by functoriality of pullback. Thus \begin{align*} \delta(I_k(\omega))(\sigma) &= \sum_{i=0}^{k+1} (-1)^i \int_{\Delta^k} [(\sigma \circ \iota_i)^*\omega]_{\Delta^k}\, d\mathcal{L}^k \\ &= \sum_{i=0}^{k+1} (-1)^i \int_{\Delta^k} [\iota_i^*(\sigma^*\omega)]_{\Delta^k}\, d\mathcal{L}^k. \end{align*} The right-hand side is, by definition of integration over the oriented boundary of the simplex, the boundary integral written as an alternating sum of Lebesgue face integrals. Hence \begin{align*} \delta(I_k(\omega))(\sigma) = \sum_{i=0}^{k+1} (-1)^i \int_{\Delta^k} [\iota_i^*(\sigma^*\omega)]_{\Delta^k}\, d\mathcal{L}^k. \end{align*} This is the point where the sign convention for the singular boundary and the orientation convention for $\partial\Delta^{k+1}$ are matched. [/guided] [/step] [step:Apply Stokes theorem on the standard simplex with corners] The form $\sigma^*\omega$ is a smooth $k$-form on the standard simplex $\Delta^{k+1}$ in the sense that it extends smoothly to a neighbourhood of $\Delta^{k+1}$ in its affine span. The standard simplex is compact, oriented, and a smooth manifold with corners; its codimension-one faces, with the alternating orientations used above, form its oriented boundary. Therefore [Stokes' theorem for manifolds with corners](/page/Stokes%20Theorem%20for%20Manifolds%20with%20Corners) applies to $\sigma^*\omega$ on $\Delta^{k+1}$ and gives \begin{align*} \sum_{i=0}^{k+1} (-1)^i \int_{\Delta^k} [\iota_i^*(\sigma^*\omega)]_{\Delta^k}\, d\mathcal{L}^k = \int_{\Delta^{k+1}} [d(\sigma^*\omega)]_{\Delta^{k+1}}\, d\mathcal{L}^{k+1}. \end{align*} By naturality of the [exterior derivative](/page/Exterior%20Derivative) with respect to smooth pullback, \begin{align*} d(\sigma^*\omega) = \sigma^*(d\omega). \end{align*} Hence \begin{align*} \delta(I_k(\omega))(\sigma) = \int_{\Delta^{k+1}} [\sigma^*(d\omega)]_{\Delta^{k+1}}\, d\mathcal{L}^{k+1}. \end{align*} [guided] We now use the analytic content of the argument: [Stokes' theorem](/theorems/1530) on the domain simplex. The smooth singular simplex \begin{align*} \sigma: \Delta^{k+1} \to M \end{align*} pulls the smooth form $\omega \in \Omega^k(M)$ back to a smooth form \begin{align*} \sigma^*\omega \in \Omega^k(\Delta^{k+1}). \end{align*} The standard simplex $\Delta^{k+1}$ is compact, oriented, and a smooth manifold with corners. Its codimension-one faces, with the alternating signs from the preceding step, form the oriented boundary in the sense required by [Stokes' theorem for manifolds with corners](/page/Stokes%20Theorem%20for%20Manifolds%20with%20Corners). Thus that theorem applies to $\sigma^*\omega$ and gives \begin{align*} \sum_{i=0}^{k+1} (-1)^i \int_{\Delta^k} [\iota_i^*(\sigma^*\omega)]_{\Delta^k}\, d\mathcal{L}^k = \int_{\Delta^{k+1}} [d(\sigma^*\omega)]_{\Delta^{k+1}}\, d\mathcal{L}^{k+1}. \end{align*} The [exterior derivative](/page/Exterior%20Derivative) commutes with pullback by a smooth map, so \begin{align*} d(\sigma^*\omega) = \sigma^*(d\omega). \end{align*} Substituting this identity into the Stokes formula yields \begin{align*} \delta(I_k(\omega))(\sigma) = \int_{\Delta^{k+1}} [\sigma^*(d\omega)]_{\Delta^{k+1}}\, d\mathcal{L}^{k+1}. \end{align*} This is precisely the expression that appears when the integration map is applied to the [exterior derivative](/theorems/1525) $d\omega$. [/guided] [/step] [step:Identify the resulting integral with the integration cochain of $d\omega$] By the definition of the [integration cochain map](/page/Integration%20Cochain) \begin{align*} I_{k+1}: \Omega^{k+1}(M) &\to C^{k+1}_{\mathrm{sm}}(M;\mathbb{R}), \end{align*} we have \begin{align*} I_{k+1}(d\omega)(\sigma) = \int_{\Delta^{k+1}} [\sigma^*(d\omega)]_{\Delta^{k+1}}\, d\mathcal{L}^{k+1}. \end{align*} Combining this identity with the result of the preceding step gives \begin{align*} \delta(I_k(\omega))(\sigma) = I_{k+1}(d\omega)(\sigma). \end{align*} Since $\sigma: \Delta^{k+1} \to M$ was arbitrary, the two smooth singular $(k+1)$-cochains are equal: \begin{align*} \delta(I_k(\omega)) = I_{k+1}(d\omega). \end{align*} [/step]