## Integration over Vector Fields

I mentioned vector fields in a previous post in the context of differential equations and over the last week or so I have been looking at them in a bit more detail. Vector fields sound quite complicated but they can be very simple. A vector field can be presented visually as a vector attached to each point in space. The space may be the $x$-$y$ plane, three-dimensional space, it could be a region of the $x$-$y$ plane or even a manifold. A typical vector field in 2-dimensions might look as follows.

An image of a simple vector field drawn in SAGE Math

The arrows represent the vector that is attached to that particular point – the direction of the arrow gives the direction of the vector and the size of the arrow gives an idea of its relative magnitude. Graphical representations of vector fields can be a little misleading as it is tempting to think that only certain points have vectors attached to them – this is not the case; every point has a vector attached to it but if we were to try to show all of them the diagram would be too cluttered.

Vector fields are useful to model flows of liquids or gases; for example in weather prediction a vector field that changes over time could be used to model wind patterns. The vector attached to each point would tell you the direction and strength of the wind at that point and the vector field would evolve from one moment to the next. If you define a surface in a vector field then you can use integration to measure the flux across the surface – the physical interpretation of flux is as a measure of the amount of substance flowing across a surface. I came across this question in the book Advanced Engineering Mathematics Fifth Edition by Stroud and Booth and decided to give it a go.

Evaluate $\int_{S}\mathbf{F}.\mathrm{d}\mathbf{S}$ over the surface $S$ defined by $x^{2}+y^{2}+z^{2}=4$ for $z\geq0$ and bounded by $x=0$, $y=0$, $z=0$ and $\mathbf{F}=x\mathbf{i}+2z\mathbf{j}+y\mathbf{k}$.

The pictures below give an idea of what the vector field and surface both look like from a few different angles. This problem is asking to integrate this vector field over the surface to find the flux across the surface.

For this problem, since the surface that we are integrating over is part of a sphere, it is convenient to change to spherical polar co-ordinates given by$$x=r\mathrm{sin}\theta\mathrm{cos}\phi$$ $$y=r\mathrm{sin}\theta\mathrm{sin}\phi$$ $$z=r\mathrm{cos}\theta$$.The integration itself is quite straightforward although some of the integrands look a bit of a pain at first glance, but some techniques from A-Level Further Maths courses should clear things up. I used some reduction formulae to deal with some of the integrals that I ended up with which really simplified things (which is always good). You can download and view my full solution here – Integrating Vector Fields.