More vector stuff :-)

More vector stuff 🙂 Good to get familiar with this, as once you are, it really does simplify the description of so much physics. I’ve seen lots of students struggle with Electromagnetism and I suspect this is because they find the perceived complexity of the maths, disguises the simplicity of the physics. Familiarity with vectors and their operators really does make life so much easier. Like most (all?) of maths, it can seem impenetrable, even pointless when you first encounter it. As if some weird mathematician is just creating complexity for the sake of it, but give it time; practice, and soon you will see how everything is connected.

This time I’m going to try inserting the equations as images, as the HTML struggles at times, with the extended character sets. Actually… Scrub that! What a pain! It seems to be compound characters, the blog is struggling with, the ‘hats’ over the unit vectors. It must be related to the style sheet I’m using, so bear with. . . 😉

Spatial Derivatives

For a function f(x,y,z) a useful vector, the gradient can be derived:

grad f = (∂f/∂x, ∂f/∂y, ∂f/∂z) = x̂(∂f/∂x) + ŷ(∂f/∂y) + ẑ (∂f/∂z)

The gradient of f is the vector pointing in the direction in which f(x,y,z) changes most rapidly. Its magnitude is the directional derivative of f, the rate of increase of f in the direction of grad f

To make these types of equation easier to read, we define an operator ‘del’, ∇ = x̂(∂/∂x) + Å·(∂/∂y) + ẑ (∂/∂z).

This allows grad f to be written ∇f

The divergence of the vector v is, ∇ ∙ v = ∂vx/∂x + ∂vy/∂y +∂vz/∂z.

This is a scalar that describes the ‘source’ or ‘sink’ of a field or fluid.

The curl of vector v is, ∇ × v = (∂vz/∂y-∂vy/∂z) x̂+ (∂vx/∂z-∂vz/∂x) ŷ + (∂vy/∂x-∂vx/∂y)ẑ.

The physical meaning of this vector is a measure of how the vector circles around a point, e.g. the magnetic field circling around a current carrying wire.


~ by jamesdow2013 on March 19, 2013.

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