# Divergence Theorem (Gauss's Theorem): Statement and Examples > The divergence theorem: the outward flux of a vector field through a closed surface equals the triple integral of its divergence. Statement and examples. Source: https://practicecalculus.com/theorems/divergence-theorem Updated: 2026-11-03 $$ \iint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_E \nabla\cdot\mathbf{F}\,dV $$ ## At a glance - **Also called:** Gauss's theorem, Ostrogradsky's theorem - **Hypotheses:** $E$ is a solid region whose boundary $S$ is a closed, piecewise smooth surface oriented outward, and $\mathbf{F}$ has continuous partial derivatives on an open region containing $E$ - **Divergence:** $\nabla\cdot\mathbf{F} = P_x + Q_y + R_z$ for $\mathbf{F} = \langle P, Q, R \rangle$ - **In words:** Total outflow through the surface equals the total source strength inside - **2D version:** The flux form of Green's theorem - **Course:** Calculus 3 ## Key takeaways - The outward flux through a closed surface equals the triple integral of the divergence over the solid inside. - The surface must be closed and oriented outward. For an open surface, close it up first and subtract the added piece. - $\mathbf{F}$ must be smooth throughout the solid. A point source inside, like an inverse square field, breaks the theorem. - Use it when the divergence is simpler than the field, or when the surface has many pieces. The **divergence theorem**, also called Gauss's theorem, says that the total flux of a vector field out of a closed surface equals the total divergence of the field inside. What flows out through the skin must be produced by sources in the interior. ## Statement of the divergence theorem Let 1. $E$ be a solid region in space whose boundary $S$ is a **closed, piecewise smooth surface**, 2. $S$ be oriented with the **outward** normal $\mathbf{n}$, and 3. $\mathbf{F} = \langle P, Q, R \rangle$ have **continuous partial derivatives** on an open region containing $E$. Then $$\iint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_E \nabla\cdot\mathbf{F}\,dV, \qquad \nabla\cdot\mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$$ Here $\iint_S \mathbf{F}\cdot d\mathbf{S} = \iint_S \mathbf{F}\cdot\mathbf{n}\,dS$ is the outward flux. ## What the divergence theorem means Think of $\mathbf{F}$ as the velocity of a fluid. The divergence at a point is the rate at which fluid is created there per unit volume: positive at a source, negative at a sink. Adding up all the creation inside the solid gives the total amount leaving through its surface. As in [Green's theorem](/theorems/greens-theorem), flux across every internal wall between tiny boxes cancels, and only the outer boundary is left. ## Why each hypothesis matters **Smoothness inside.** The inverse square field $\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^3}$, where $\mathbf{r} = \langle x, y, z \rangle$, has $\nabla\cdot\mathbf{F} = 0$ everywhere except the origin, where it is undefined. Yet on the unit sphere $\mathbf{F}\cdot\mathbf{n} = 1$, so the flux is the surface area, $4\pi$, not $0$. The theorem fails because $E$ contains the singular point. This is exactly Gauss's law: the flux measures the point charge inside. **A closed surface.** Flux through an open surface such as a hemisphere is not determined by the divergence alone. Close the surface first. **Outward orientation.** Using inward normals flips the sign of the flux. ## Worked examples **Example 1: $\mathbf{F} = \langle x, y, z \rangle$ through a sphere of radius $R$.** **Step 1.** $\nabla\cdot\mathbf{F} = 1 + 1 + 1 = 3$. **Step 2.** $\iiint_E 3\,dV = 3\cdot\frac{4}{3}\pi R^3 = 4\pi R^3$. **Check directly.** On the sphere, $\mathbf{n} = \frac{\mathbf{r}}{R}$, so $\mathbf{F}\cdot\mathbf{n} = \frac{x^2 + y^2 + z^2}{R} = R$. The flux is $R$ times the area $4\pi R^2$, which is $4\pi R^3$. **Example 2: $\mathbf{F} = \langle x^3, y^3, z^3 \rangle$ through the unit sphere.** $\nabla\cdot\mathbf{F} = 3x^2 + 3y^2 + 3z^2 = 3\rho^2$ in spherical coordinates. Then $$\iiint_E 3\rho^2\,dV = \int_0^{2\pi}\int_0^{\pi}\int_0^1 3\rho^2\cdot\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta = 3\cdot\frac{1}{5}\cdot 2\cdot 2\pi = \frac{12\pi}{5}$$ Computing this flux as a surface integral would take far more work. **Example 3: a cube with six faces.** Find the outward flux of $\mathbf{F} = \langle x^2, yz, z \rangle$ through the surface of the unit cube $[0, 1]^3$. **Step 1.** $\nabla\cdot\mathbf{F} = 2x + z + 1$. **Step 2.** Integrate over the cube, term by term (each other variable integrates to $1$): $$\iiint_E (2x + z + 1)\,dV = 1 + \frac{1}{2} + 1 = \frac{5}{2}$$ One triple integral replaces six separate surface integrals. ## Common mistakes 1. **Applying it to an open surface.** The theorem needs a closed boundary of a solid. 2. **Missing a singularity.** Check that $\mathbf{F}$ is defined and smooth at every point inside, especially the origin. 3. **Confusing divergence with curl.** Divergence is a scalar, $P_x + Q_y + R_z$; curl is a vector and belongs to [Stokes' theorem](/theorems/stokes-theorem). ## Related theorems and guides The divergence theorem, [Stokes' theorem](/theorems/stokes-theorem) and [Green's theorem](/theorems/greens-theorem) are all versions of one idea: integrating a derivative over a region equals integrating the original field over the boundary, just like the [fundamental theorem of calculus](/theorems/fundamental-theorem-of-calculus-part-2). For the triple integral side, start with [double integrals](/blog/double-integrals) and for vector operations see [vectors, dot and cross product](/blog/vectors-dot-and-cross-product). ## Frequently asked questions ### What does the divergence theorem say? If $E$ is a solid region with closed boundary surface $S$ oriented outward, and $\mathbf{F}$ has continuous partial derivatives on a region containing $E$, then $\iint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_E \nabla\cdot\mathbf{F}\,dV$. ### When do you use the divergence theorem? Use it to compute flux through a closed surface such as a sphere, cube or cylinder with caps, especially when the divergence is a simple expression. One triple integral replaces several surface integrals. ### Can you use the divergence theorem on an open surface? Not directly. Add a surface that closes it, such as a flat disk on a hemisphere, apply the theorem to the closed surface, then subtract the flux through the added piece. ### What is the difference between the divergence theorem and Stokes' theorem? The divergence theorem relates flux through a closed surface to divergence in the volume inside. Stokes' theorem relates circulation around a closed curve to the curl flux through a surface bounded by that curve.