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Green's, Stokes' & Divergence Theorems cheat sheet - grade college

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Calculus Grade college

Green's, Stokes' & Divergence Theorems Cheat Sheet

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

TopicsCalculusIntegralsVectorsDerivativesStudy Guide
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This cheat sheet covers the three major integral theorems of vector calculus: Green's Theorem, Stokes' Theorem, and the Divergence Theorem. These theorems connect integrals over regions, surfaces, curves, and solids, making difficult calculations easier by changing the dimension of integration. Students need this reference to compare the hypotheses, orientations, and meanings of each theorem.

It is especially useful when deciding whether to compute a line integral, surface integral, or volume integral directly or by conversion.

Green's Theorem relates circulation or flux around a closed plane curve to a double integral over the enclosed region. Stokes' Theorem generalizes Green's circulation form by relating circulation around a space curve to the flux of curl through a surface. The Divergence Theorem relates outward flux through a closed surface to the triple integral of divergence over the solid inside.

The key ideas are orientation, boundary, curl measuring rotation, and divergence measuring net source strength.

Key Facts

  • Green's circulation form is ∮CP dx+Q dy=∬D(∂Q∂x−∂P∂y) dA\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\,dA, where CC is positively oriented around DD.
  • Green's flux form is ∮CF⋅n ds=∬D∇⋅F dA=∬D(∂P∂x+∂Q∂y) dA\oint_C \mathbf{F}\cdot \mathbf{n}\,ds = \iint_D \nabla \cdot \mathbf{F}\,dA = \iint_D \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}\right)\,dA for F=⟨P,Q⟩\mathbf{F}=\langle P,Q\rangle.
  • Stokes' Theorem is ∮CF⋅dr=∬S(∇×F)⋅n dS\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S \left(\nabla \times \mathbf{F}\right)\cdot \mathbf{n}\,dS, where C=∂SC=\partial S has the orientation induced by n\mathbf{n}.
  • The Divergence Theorem is ∬SF⋅n dS=∭E∇⋅F dV\iint_S \mathbf{F}\cdot \mathbf{n}\,dS = \iiint_E \nabla \cdot \mathbf{F}\,dV, where SS is a closed surface with outward orientation.
  • For F=⟨P,Q,R⟩\mathbf{F}=\langle P,Q,R\rangle, the divergence is ∇⋅F=∂P∂x+∂Q∂y+∂R∂z\nabla \cdot \mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}.
  • For F=⟨P,Q,R⟩\mathbf{F}=\langle P,Q,R\rangle, the curl is ∇×F=⟨∂R∂y−∂Q∂z,∂P∂z−∂R∂x,∂Q∂x−∂P∂y⟩\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 positively oriented plane curve in Green's Theorem travels counterclockwise so that the region DD stays on the left.
  • If ∇×F=0\nabla \times \mathbf{F}=\mathbf{0} on a simply connected region, then line integrals of F\mathbf{F} are path independent in that region.

Vocabulary

Circulation
Circulation is the line integral ∮CF⋅dr\oint_C \mathbf{F}\cdot d\mathbf{r} measuring how much a vector field flows along a closed curve.
Flux
Flux is the integral ∬SF⋅n dS\iint_S \mathbf{F}\cdot \mathbf{n}\,dS measuring how much a vector field passes through a surface.
Curl
Curl is the vector ∇×F\nabla \times \mathbf{F} that measures the local rotational tendency of a vector field.
Divergence
Divergence is the scalar ∇⋅F\nabla \cdot \mathbf{F} that measures the local net outward flow or source strength of a vector field.
Boundary
The boundary ∂S\partial S or ∂E\partial E is the curve or surface that encloses a higher-dimensional object.
Orientation
Orientation is the chosen direction of a curve or normal vector of a surface, and it determines the sign of the integral.

Common Mistakes to Avoid

  • Using the wrong orientation, which reverses the sign of the answer. For Green's Theorem the positive direction is counterclockwise, and for the Divergence Theorem the normal must point outward.
  • Applying the Divergence Theorem to an open surface, which is not allowed without closing the surface first. The theorem requires a closed surface S=∂ES=\partial E.
  • Forgetting that Stokes' Theorem needs a compatible boundary orientation, which can change the sign of ∮CF⋅dr\oint_C \mathbf{F}\cdot d\mathbf{r}. Use the right-hand rule to match the direction of CC with the chosen normal n\mathbf{n}.
  • Confusing curl and divergence, which leads to using the wrong derivative expression. Curl is used for circulation in Stokes' Theorem, while divergence is used for flux in the Divergence Theorem.
  • Ignoring smoothness and domain assumptions, which can make a theorem invalid. The vector field must be sufficiently differentiable on a region containing the curve, surface, or solid involved.

Practice Questions

  1. 1 Use Green's Theorem to evaluate ∮Cy dx+x2 dy\oint_C y\,dx + x^2\,dy, where CC is the counterclockwise boundary of the rectangle 0≤x≤20\le x\le 2, 0≤y≤30\le y\le 3.
  2. 2 Use the Divergence Theorem to find the outward flux of F=⟨x,y,z⟩\mathbf{F}=\langle x,y,z\rangle through the sphere x2+y2+z2=4x^2+y^2+z^2=4.
  3. 3 Use Stokes' Theorem to evaluate ∮CF⋅dr\oint_C \mathbf{F}\cdot d\mathbf{r} for F=⟨−y,x,0⟩\mathbf{F}=\langle -y,x,0\rangle, where CC is the counterclockwise circle x2+y2=9x^2+y^2=9 in the plane z=0z=0 viewed from above.
  4. 4 Explain how you decide whether Green's Theorem, Stokes' Theorem, or the Divergence Theorem is the best tool for a problem involving a vector field and an integral.

Understanding Green's, Stokes' & Divergence Theorems

The deepest idea is that these results are bookkeeping laws. Tiny effects inside a region add up to an effect measured around its edge. Imagine a fluid moving across a flat map.

Small local spinning tendencies throughout the map combine into one total tendency to circulate around the boundary. Small amounts of fluid created or removed throughout the map combine into the net flow crossing the boundary.

In three dimensions, the same pattern applies to fields around surfaces and inside solids. This is why the theorems appear in fluid flow, electromagnetism, heat transfer, and gravitational models.

Choosing the useful theorem depends on what is easier to describe. A boundary may be complicated while the interior has a simple derivative. In that case, convert the boundary calculation into an interior calculation.

The reverse can help too. A solid might be awkward to integrate through, while its outer surface is made of familiar pieces such as planes, cylinders, or spheres. Before converting anything, identify the geometric objects carefully.

Green's Theorem belongs to a flat region in the plane. Stokes' Theorem needs an open surface with a boundary curve. The Divergence Theorem needs a fully closed surface that encloses a volume.

Orientation is not a minor sign rule. It decides which direction counts as positive. For a plane boundary, the standard direction keeps the region on the left.

For a surface in space, use the right hand rule. Point your right thumb in the chosen normal direction, then curl your fingers. Your fingers show the required direction around the boundary.

For a closed surface, normals point outward. Reversing an orientation reverses the sign of the related circulation or flux. A quick sketch with arrows often prevents more mistakes than extra algebra does.

The theorems have conditions because derivatives must behave well across the region involved. Watch for points where a vector field is undefined, such as the center of a swirling field with division by distance. A hole can change the result and can destroy path independence.

This matters when deciding whether zero curl guarantees a potential function. It does only when the region has no hidden holes of the relevant kind. In practice, draw the domain, mark excluded points, and check whether the curve or surface encloses them.

Students should also separate circulation from flux. Circulation measures motion along a path, while flux measures motion through a boundary. The words along and through are a reliable guide when setting up a problem.