Sign in to save

Bookmark this page so you can find it later.

Sign in to save

Bookmark this page so you can find it later.

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

Click image to open full size

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
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

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.