Vector Calculus Theorems Green, Stokes, Divergence Cheat Sheet

A printable reference covering Green’s Theorem, Stokes’ Theorem, the Divergence Theorem, circulation, flux, curl, and divergence for college calculus.

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Vector calculus theorems connect integrals over curves, surfaces, and regions. This cheat sheet helps students recognize when to use Green’s Theorem, Stokes’ Theorem, or the Divergence Theorem. These results turn difficult integrals into simpler equivalent integrals when the orientation and hypotheses are correct. They are essential for multivariable calculus, electromagnetism, fluid flow, and advanced engineering mathematics. Green’s Theorem relates a line integral around a plane curve to a double integral over the region it encloses. Stokes’ Theorem relates circulation around a space curve to the flux of curl through a surface. The Divergence Theorem relates outward flux through a closed surface to a triple integral of divergence over the enclosed solid. The main skills are matching the theorem to the geometry, computing $\nabla \cdot \mathbf{F}$ or $\nabla \times \mathbf{F}$ , and using the correct orientation.

Key Facts

Green’s Theorem in circulation form is $\oint_C P\,dx+Q\,dy=\iint_D \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA$ , where $C$ is positively oriented around $D$ .
Green’s Theorem in flux form is $\oint_C P\,dy-Q\,dx=\iint_D \left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}\right)dA$ for outward flux across a plane curve.
Stokes’ Theorem is $\oint_C \mathbf{F}\cdot d\mathbf{r}=\iint_S \left(\nabla \times \mathbf{F}\right)\cdot \mathbf{n}\,dS$ , where $C=\partial S$ and the orientation follows the right-hand rule.
The Divergence Theorem is $\iint_S \mathbf{F}\cdot \mathbf{n}\,dS=\iiint_E \nabla \cdot \mathbf{F}\,dV$ , where $S$ is a closed surface bounding the solid $E$ .
For $\mathbf{F}=\langle P,Q,R\rangle$ , the divergence is $\nabla\cdot\mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$ .
For $\mathbf{F}=\langle P,Q,R\rangle$ , the curl is $\nabla\times\mathbf{F}=\left\langle \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right\rangle$ .
A positive orientation for Green’s Theorem means the region $D$ stays on the left as the curve $C$ is traversed.
The Divergence Theorem requires a closed surface, while Stokes’ Theorem applies to an oriented surface whose boundary is a closed curve.

Vocabulary

Circulation: Circulation measures the tendency of a vector field to flow along a closed curve, usually computed by $\oint_C \mathbf{F}\cdot d\mathbf{r}$ .
Flux: Flux measures how much of a vector field passes through a curve or surface, often computed by $\iint_S \mathbf{F}\cdot\mathbf{n}\,dS$ .
Divergence: Divergence is the scalar quantity $\nabla\cdot\mathbf{F}$ that measures the net source strength of a vector field at a point.
Curl: Curl is the vector quantity $\nabla\times\mathbf{F}$ that measures the local rotation of a vector field.
Orientation: Orientation is the chosen direction of a curve or normal direction of a surface that determines the sign of an integral.
Boundary: The boundary $\partial S$ or $\partial D$ is the curve or surface edge that encloses a region, surface, or solid.

Common Mistakes to Avoid

Using Green’s Theorem on a non-closed curve is wrong because the theorem requires a closed boundary curve $C=\partial D$ .
Forgetting orientation reverses the sign of the answer because clockwise orientation in Green’s Theorem gives the negative of the positive counterclockwise result.
Using the Divergence Theorem on an open surface is wrong because $S$ must be a closed surface that completely bounds a solid region $E$ .
Confusing curl and divergence leads to the wrong theorem because Stokes’ Theorem uses $\nabla\times\mathbf{F}$ while the Divergence Theorem uses $\nabla\cdot\mathbf{F}$ .
Ignoring singularities inside the region is wrong because the vector field must be sufficiently smooth on the region where the theorem is applied.

Practice Questions

1 Use Green’s Theorem to evaluate $\oint_C (x^2-y)\,dx+(x+y^2)\,dy$ , where $C$ is the positively oriented boundary of the rectangle $0\le x\le 2$ , $0\le y\le 3$ .
2 Use the Divergence Theorem to find the outward flux of $\mathbf{F}=\langle x,y,z\rangle$ through the sphere $x^2+y^2+z^2=4$ .
3 Use Stokes’ Theorem to rewrite $\oint_C \mathbf{F}\cdot d\mathbf{r}$ for $\mathbf{F}=\langle -y,x,0\rangle$ , where $C$ is the unit circle in the plane $z=0$ oriented counterclockwise as viewed from above.
4 A surface is a hemisphere without its circular base. Explain why the Divergence Theorem cannot be applied directly to only the curved hemisphere and what must be added.

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