vector fields

What does a solenoidal / divergence free vector field mean?

It has zero flux in/out of any choice of region in the x-y plane.

It has zero divergence everywhere.

What is the divergence of a vector field?

The divergence is the dot product of the gradient operator and the vector field.

\text{Let } \mathbf{v}(x,y)=U(x,y)\mathbf{i}+V(x,y)\mathbf{j}

\operatorname{div}\mathbf{v}=\nabla\cdot\mathbf{v}

\operatorname{div}\mathbf{v}=\frac{\partial U}{\partial x}+\frac{\partial V}{\partial y}

\text{In } \mathbb{R}^3,\ \mathbf{v}(x,y,z)=U\mathbf{i}+V\mathbf{j}+W\mathbf{k}

\operatorname{div}\mathbf{v}=\frac{\partial U}{\partial x}+\frac{\partial V}{\partial y}+\frac{\partial W}{\partial z}

\operatorname{div}\mathbf{v}=0 \Rightarrow \mathbf{v}\text{ is divergence-free}

What is the divergence theorem in 2d and 3d?

For an area A and a differentiable vector \vec{V} the divergence theorem states

\int\int_{A}div\left(\underline{}\underline{v}\right)dA=\int_{C}\underline{_{}v}\cdot\underline{n}ds where C is closed perimeter of A and n is an outwards unit normal from A.

in 3d

\int\int\int_{V}div\left(\underline{v}\right)dV=\iint_{S}\underline{v}\cdot\underline{n}dS where S is the closed surface of V and n is an outward pointing unit vector normal to S.

what is the circulation of a vector field?

Circulation measures the tendency to ‘rotate’ around a closed path.

\int_{C}\mathbf{v}(x,y)\cdot d\mathbf{r}=\int_{C}U(x,y)\,dx+V(x,y)\,dy

with

\mathbf{v}(x,y)=\begin{pmatrix}U(x,y)\\ V(x,y)\end{pmatrix},\mathbf{dr}=\begin{pmatrix}dx\\ dy\end{pmatrix}\qquad

What does it mean for a vector field to be irrotational?

It has zero circulation for all parts.

What is the ‘curl’ of a vector?

\operatorname{curl}\mathbf{v}=\begin{pmatrix} W_y - V_z \\ U_z - W_x \\ V_x - U_y \end{pmatrix}

The curl is equivalent to the cross product of the gradient operator and the vector field

\operatorname{curl}\mathbf{v}=\nabla\times\mathbf{v}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ U&V&W\end{vmatrix}

for vector v=(U(x,y,z)+V(x,y,z)+W(x,y,z))

What is stokes’ theorem?

For a surface S in (x,y,z) bounded by a closed path C, and a differentiable vector field V(x,y,z), Stokes’ theorem says

\iint_{S}\operatorname{curl}\underline{\mathbf{v}}\cdot\underline{\mathbf{n}}\,dS=\oint_{C}\underline{\mathbf{v}}\cdot\underline{d\mathbf{r}}

It says the flux of curlV across the surface = the circulation around C.

n is the unit normal to S

What is Greens theorem

\iint_S \left(V_x - U_y\right)\,dx\,dy=\oint_C U\,dx+V\,dy

S is now a region of the (x, y) plane. In this case we have chosen +k as the unit normal to S, so the path integral must be performed in an anti-clockwise sense in the plane.