Link to Current (updated) notes

Gradients and Directional Derivatives

Scalar functions $F(x,y,z) = F(\vec{x})$ have a natural vector field associated with them--at each point $\vec{x}$ there is a direction $\hat{n}(\vec{x})$ pointing in the direction of the most rapid increase of $F$ . Associating the magnitude of a vector in the direction of steepest increase with the rate of increase of $F$ defines the gradient.

The gradient is therefore a vector function with a vector argument ($\vec{x}$ in this case) and it is commonly written as $\ensuremath{\nabla}F$ .

Here are some natural examples:

Meteorology

The ``high pressure regions'' are commonly displayed with weather reports--as are the "isobars" or curves of constant barometric pressure. Thus displayed, pressure is a scalar function of latitude and longitude.

At any point on the map, there is a direction that points to local high pressure center--this is the direction of the gradient. The rate at which the pressure is increasing gives the magnitude of the gradient.

The gradient of pressure should be a vector that is related to the direction and the speed of wind.

Mosquitoes

It is known that hungry mosquitoes tend to fly towards sources (or local maxima) of carbon dioxide. Therefore, it can be hypothesized that mosquitoes are able to determine the gradient in the concentration of carbon dioxide.

Heat

In an isolated system, heat flows from high-temperature ( $T(\vec{x})$ ) regions to low-temperature regions.

The Fourier empirical law of heat flow states that the rate of heat flows is proportional to the local decrease in temperature.

Therefore, the local rate of heat flow should be a proportional to the vector which is minus the gradient of $T(\vec{x})$ : $-\ensuremath{\nabla}T$

Finding the Gradient

Potentials and Force Fields

Force is a vector. Force projected onto a displacement vector $\vec{dx}$ is the rate at which work, $dW$ , is done on an object $dW = -\vec{F} \cdot \vec{dx}$ .

If the work is being supplied by an external agent (e.g., a charged sphere, a black hole, a magnet, etc.), then it may be possible to ascribe a potential energy ( $E(\vec{x})$ , a scalar function with vector argument) to the agent associated with the position at which the force is being applied.¹This $E(\vec{x})$ is the potential for the agent and the force field due to the agent is $\vec{F}(\vec{x}) = - \ensuremath{\nabla}E(\vec{x})$ .

Sometimes the force (and therefore the energy) scale with the ``size'' of the object (i.e., the object's total charge in an electric potential due to a fixed set of charges, the mass of an object in the gravitational potential of a black hole, the magnetization of the object in a magnetic potential, etc.). In these cases, the potential field can be defined in terms of a unit size (per unit charge, per unit mass, etc.). One can determine whether such a scaling is applied by checking the units.