28 May 2025

Gauss, Green, and Stokes Theorems

Integral theorems connecting local derivatives with flux and circulation on boundaries.

bsc semester-ii mathematical-physics divergence-theorem greens-theorem stokes-theorem

The integral theorems express one geometrical principle: an accumulated local derivative inside a region equals a field integral over its boundary.

Gauss divergence theorem

Let $V$ have closed boundary $S=\partial V$ with outward orientation. For a small cell,

\[\oint_{\partial(\Delta V)}\mathbf A\cdot d\mathbf S =(\nabla\cdot\mathbf A)\Delta V+o(\Delta V).\]

On adding all cells, fluxes through shared interior faces cancel because neighbouring outward normals are opposite. In the limit,

\[\boxed{ \iiint_V(\nabla\cdot\mathbf A)\,dV =\iint_{\partial V}\mathbf A\cdot d\mathbf S}.\]

For $\mathbf A=x\hat{\mathbf x}+y\hat{\mathbf y}+z\hat{\mathbf z}$ inside $r\leq R$,

\[\iiint_V\nabla\cdot\mathbf A\,dV =3\frac{4\pi R^3}{3}=4\pi R^3.\]

On the sphere, $\mathbf A=R\hat{\mathbf r}$ and $d\mathbf S=\hat{\mathbf r}R^2\sin\theta\,d\theta d\varphi$, so

\[\iint_S\mathbf A\cdot d\mathbf S =R^3\int_0^{2\pi}d\varphi\int_0^\pi\sin\theta\,d\theta =4\pi R^3.\]

Stokes theorem

Let an oriented surface $S$ have boundary $C=\partial S$. The right-hand rule fixes the positive direction around $C$. Circulations on shared edges of small surface patches cancel, leaving

\[\boxed{ \iint_S(\nabla\times\mathbf A)\cdot d\mathbf S =\oint_{\partial S}\mathbf A\cdot d\boldsymbol\ell}.\]

For $\mathbf A=-y\hat{\mathbf x}+x\hat{\mathbf y}$ on $x^2+y^2\leq R^2$,

\[\nabla\times\mathbf A=2\hat{\mathbf z}, \qquad \iint_S2\,dS=2\pi R^2.\]

On $C$, $\mathbf r=R(\cos\varphi,\sin\varphi)$ gives

\[\mathbf A=R(-\sin\varphi,\cos\varphi), \quad d\boldsymbol\ell=R(-\sin\varphi,\cos\varphi)d\varphi,\]

and therefore

\[\oint_C\mathbf A\cdot d\boldsymbol\ell =R^2\int_0^{2\pi}d\varphi=2\pi R^2.\]

Green’s theorems

For a positively oriented curve $C$ bounding a plane region $D$, Stokes theorem applied to $\mathbf A=P\hat{\mathbf x}+Q\hat{\mathbf y}$ gives

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

The planar flux form is

\[\boxed{ \oint_C(P\,dy-Q\,dx) =\iint_D\left(P_x+Q_y\right)dA}.\]

From $\nabla\cdot(u\nabla v)=\nabla u\cdot\nabla v+u\nabla^2v$ and Gauss’ theorem,

\[\boxed{ \iiint_V(\nabla u\cdot\nabla v+u\nabla^2v)dV =\iint_Su\frac{\partial v}{\partial n}dS}.\]

Subtracting the equation with $u,v$ exchanged gives

\[\boxed{ \iiint_V(u\nabla^2v-v\nabla^2u)dV =\iint_S\left(u\frac{\partial v}{\partial n} -v\frac{\partial u}{\partial n}\right)dS}.\]

Gauss relates a volume to its closed surface; Stokes relates an oriented surface to its edge; Green is the planar case with counterclockwise positive orientation.