26 May 2025

Divergence, Curl, and Vector Integrals

Local sources and circulation, followed by line, surface, and volume integrals.

bsc semester-ii mathematical-physics divergence curl vector-integrals

The derivatives of a vector field separate into two local quantities: divergence measures source strength, while curl measures circulation density.

Divergence from flux

For $\mathbf A=A_x\hat{\mathbf x}+A_y\hat{\mathbf y}+A_z\hat{\mathbf z}$, the outward flux through the two faces of a box normal to $x$ is

\[\begin{aligned} d\Phi_x &=[A_x(x+dx)-A_x(x)]dy\,dz\\ &=\frac{\partial A_x}{\partial x}dx\,dy\,dz+O(dx^2)\,dy\,dz. \end{aligned}\]

Adding the $y$ and $z$ face pairs gives

\[d\Phi= \left(\frac{\partial A_x}{\partial x} +\frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z}\right)dV.\]

Thus

\[\boxed{\nabla\cdot\mathbf A =\lim_{\Delta V\to0}\frac1{\Delta V} \oint_{\partial(\Delta V)}\mathbf A\cdot d\mathbf S}.\]

Positive divergence means net outward flux; negative divergence means net inward flux. If $[\mathbf A]=Q$, then $[\nabla\cdot\mathbf A]=Q\,{\rm m^{-1}}$.

Curl from circulation

Traverse a rectangle $dx\times dy$ counterclockwise as seen from $+z$. Its circulation is

\[\begin{aligned} \oint\mathbf A\cdot d\boldsymbol\ell ={}&A_x(x,y)dx+A_y(x+dx,y)dy\\ &-A_x(x,y+dy)dx-A_y(x,y)dy\\ ={}&\left(\frac{\partial A_y}{\partial x} -\frac{\partial A_x}{\partial y}\right)dx\,dy. \end{aligned}\]

This coefficient is $(\nabla\times\mathbf A)_z$. Repeating in the other planes,

\[\boxed{ \nabla\times\mathbf A =\begin{vmatrix} \hat{\mathbf x}&\hat{\mathbf y}&\hat{\mathbf z}\\ \partial_x&\partial_y&\partial_z\\ A_x&A_y&A_z \end{vmatrix}}.\]

For a unit normal $\hat{\mathbf n}$,

\[\boxed{ \hat{\mathbf n}\cdot(\nabla\times\mathbf A) =\lim_{\Delta S\to0}\frac1{\Delta S} \oint_{\partial(\Delta S)}\mathbf A\cdot d\boldsymbol\ell}.\]

The right-hand rule fixes the sign between the circulation and $\hat{\mathbf n}$.

\(\mathbf A_s=(x,y)\) has divergence 2 and zero curl; \(\mathbf A_r=(-y,x)\) has zero divergence and curl \(2\hat{\mathbf z}\).

Line, surface, and volume integrals

For a curve $C$ parametrized by $\mathbf r(t)$,

\[\boxed{\int_C\mathbf A\cdot d\boldsymbol\ell =\int_a^b\mathbf A(\mathbf r(t))\cdot\frac{d\mathbf r}{dt}dt}.\]

Reversing the path reverses the sign. For a gradient field,

\[\int_C\nabla\phi\cdot d\boldsymbol\ell =\int_a^b\frac{d\phi}{dt}dt =\phi(\mathbf r_b)-\phi(\mathbf r_a),\]

so the integral depends only on the endpoints.

If $\mathbf r(u,v)$ parametrizes an oriented surface,

\[d\mathbf S=\left(\frac{\partial\mathbf r}{\partial u} \times\frac{\partial\mathbf r}{\partial v}\right)du\,dv, \qquad \boxed{\Phi=\iint_S\mathbf A\cdot d\mathbf S}.\]

Changing the orientation changes the sign. On a closed surface, $d\mathbf S$ points outward.

For a scalar volume density $\rho$,

\[\boxed{Q=\iiint_V\rho\,dV}.\]

Because $[\rho]=[Q]/{\rm m^3}$, the integral has units $[Q]$. A vector volume integral is evaluated component by component.

Equality of mixed partial derivatives gives

\[\boxed{\nabla\times(\nabla\phi)=\mathbf0}, \qquad \boxed{\nabla\cdot(\nabla\times\mathbf A)=0}.\]