CALCULUS · FIELDS & FLUX

Vector calculus

Vector calculus connects local information — curl and divergence — to global circulation and flux. Green's, Stokes' and the divergence theorem are different-dimensional versions of one central idea.

local curl ↔ circulation around the boundary

1. Vector fields, divergence and curl

A vector field assigns a vector to each point. In three dimensions, write \(\mathbf F=(P,Q,R)\).

Divergence

\(\nabla\cdot\mathbf F=P_x+Q_y+R_z\) measures net outward source strength.

Curl

\(\nabla\times\mathbf F\) measures local rotational tendency.

Potential

If \(\mathbf F=\nabla\phi\), the field is conservative and path-independent on suitable domains.

2. Line integrals and circulation

For a smooth curve \(\mathbf r(t)\), the work done by a vector field along the curve is

\[\int_C \mathbf F\cdot d\mathbf r=\int_a^b \mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt.\]

If \(\mathbf F=\nabla\phi\), the fundamental theorem for line integrals gives

\[\int_C\mathbf F\cdot d\mathbf r=\phi(B)-\phi(A).\]
Path independence requires more than zero curl in general. On simply connected domains, continuously differentiable fields with zero curl are conservative.

3. Green's theorem

For a positively oriented simple closed curve \(C=\partial R\) in the plane,

\[\oint_C P\,dx+Q\,dy=\iint_R\left(Q_x-P_y\right)dA.\]

Green's theorem converts circulation around a boundary into accumulated scalar curl over the enclosed region.

Worked example

Let \(\mathbf F=(-y,x)\) and let \(C\) be the unit circle counterclockwise. Since \(Q_x-P_y=1-(-1)=2\),

\[\oint_C -y\,dx+x\,dy=\iint_R2\,dA=2\pi.\]

4. Stokes' theorem

For an oriented smooth surface \(S\) with boundary curve \(\partial S\),

\[\oint_{\partial S}\mathbf F\cdot d\mathbf r=\iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS.\]

Stokes' theorem is the three-dimensional circulation theorem: the circulation around the edge equals the flux of curl through any compatible spanning surface.

Orientation matters. The boundary direction must follow the right-hand rule relative to the chosen normal.

5. Divergence theorem

For a solid region \(V\) with closed outward-oriented boundary \(\partial V\),

\[\iint_{\partial V}\mathbf F\cdot\mathbf n\,dS=\iiint_V\nabla\cdot\mathbf F\,dV.\]

The total outward flux through a closed surface equals the total source strength inside the volume.

Worked example

For \(\mathbf F=(x,y,z)\) on the ball of radius \(R\), \(\nabla\cdot\mathbf F=3\). Hence

\[\iint_{\partial V}\mathbf F\cdot\mathbf n\,dS=3\operatorname{Vol}(V)=3\cdot\frac{4}{3}\pi R^3=4\pi R^3.\]

6. One theorem pattern

TheoremBoundary integralInterior derivative
Fundamental theorem of calculusendpoint differenceordinary derivative
Greenclosed planar line integralscalar curl over area
Stokesclosed spatial line integralcurl flux over surface
Divergenceflux through closed surfacedivergence over volume

The recurring structure is: integral over the boundary = integral of a derivative over the interior.

7. Interactive linear-field diagnostic

For \(\mathbf F(x,y)=(ax+by,\;cx+dy)\), compute planar divergence and scalar curl.

8. Common mistakes

  • Using the wrong orientation on a curve or surface.
  • Applying the divergence theorem to an open surface without first closing it.
  • Confusing circulation with flux; they pair with different geometric components of a field.
  • Assuming zero curl implies a global potential without checking the domain topology.
  • Forgetting that Stokes' theorem uses the curl dotted with the oriented normal.

9. Practice

  1. For \(\mathbf F=(x^2,-2xy)\), compute planar divergence and scalar curl.
  2. Use Green's theorem to evaluate \(\oint_C -y\,dx+x\,dy\) around the circle of radius \(R\).
  3. For \(\mathbf F=(x,y,z)\), find the outward flux through the unit sphere.
Solutions

1) divergence \(=2x-2x=0\), scalar curl \(=Q_x-P_y=-2y-0=-2y\). 2) \(2\pi R^2\). 3) \(4\pi\).