2nd derivative is curvature, rate of change of the rate: toward which direction is the rate changing
3rd derivative is "jerk", rate of change of curvature; it's the smoothness of the change
Think steering wheel and turning rate of a car
$\theta$ is the angle the car is pointing
$\phi$ is the angle of the steering wheel
which is proportional to $\mathrm d \theta \over \mathrm d t$
${\mathrm d ^2 \theta \over \mathrm d t^2} = {\mathrm d \phi \over\mathrm d t}$ is the acceleration of the direction the car is pointing -- how quickly the rate of turn is changing, and in which direction.
${\mathrm d ^3 \theta \over \mathrm d t^2} = {\mathrm d ^2 \phi \over\mathrm d t^2}$ is how quick the change is, of the acceleration of the direction the car is pointing -- if the jerk is large, it's like when you suddenly yank the steering wheel to one side.
The quick change from a steering wheel at rest to a moving steering wheel (large transient $\mathrm d^2 \phi \over \mathrm d t^2$) leads to people being slewed about in their seats.
The gradient points in the direction of fastest increase.
Based on the same principle, this little bookmarklet loads cancel in addition to mhchem:
javascript:(function(){if(window.MathJax===undefined){var%20script%20=%20document.createElement(%22script%22);script.type%20=%20%22text/javascript%22;script.src%20=%20%22http://cdn.mathjax.org/mathjax/latest...
The Laplacian, $\nabla\cdot\nabla\Theta \equiv\nabla^2\Theta$, is a measure of the local net curvature of a variable. At a local minimum $\nabla^2\Theta > 0$ strictly; at a local maximum $\nabla^2\Theta < 0$ strictly.
At other points, the value can be ~anything. One particular weird aspect is that $\nabla^2\Theta =0$ is possible even when a plot of the surface wouldn't look flat.
e.g., in two dimensions $\nabla^2\Theta = {\partial^2\Theta\over\partial x^2} + {\partial^2\Theta\over\partial y^2}$.
If ${\partial^2 \theta \over\partial x^2} = C$ and ${\partial^2 \theta \over\partial y^2} = -C$ at a certain point, then $\nabla^2\Theta = 0$ at that point, but the surface would look appreciably curved there.
The divergence, $\nabla \cdot \Theta$, is a measure of the "expandiness" or "contractiness" of a variable.
If the volume integral $\int{\int{\int{\nabla\cdot\Theta}}}$ over a region is greater than zero, then whatever $\Theta$ represents will have a tendency to "leak out" of that region.
Conversely, if the integral is less than zero, then there will be a "leaking in" tendency.
The curl, $\nabla\times\Theta$, gives a measure of the rotational character of the field. This is only defined in three dimensions, since the cross-product is only defined in three dimensions.
Yep, just create a new bookmark (I have mine in my bookmark bar) and copy the script text into the destination field.
The Hessian can be constructed from the outer product of $\nabla$ on $\Theta$, as $\nabla\nabla\Theta$. (Note this is not to my knowledge ever written as '$\nabla^2\Theta$'.)
The Periodic Table
The Periodic Table