Temperature Felt by a Drone

A drone flies an oval loop over a field that is warm in one corner and cool in another. The temperature it feels changes for two reasons at once: the field changes from place to place, and the drone is moving. The chain rule adds the two effects.

Drag on the surface to move the point, or click a red marker to jump to it.

How it works

The multivariable chain rule

In one sentence

If $T(x, y)$ is differentiable and $x(t)$ , $y(t)$ are differentiable, then $\frac{d}{dt}T(x(t), y(t)) = T_x\,x'(t) + T_y\,y'(t) = \nabla T\cdot\mathbf r'(t)$ .

General formula

{\htmlData{sym=0}{\frac{dT}{dt}}} = {\htmlData{sym=1}{\frac{\partial T}{\partial x}}}\,{\htmlData{sym=3}{\frac{dx}{dt}}} + {\htmlData{sym=2}{\frac{\partial T}{\partial y}}}\,{\htmlData{sym=4}{\frac{dy}{dt}}}

Each term is a rate times a rate: degrees per kilometer east times kilometers east per second, and the same for north. Adding them gives degrees per second.

What the symbols mean

$\frac{dT}{dt}$
Rate the drone feels
How fast the temperature reading changes per second of flight.
$\frac{\partial T}{\partial x}$
East-west temperature slope
Degrees per kilometer going east.
$\frac{\partial T}{\partial y}$
North-south temperature slope
Degrees per kilometer going north.
$\frac{dx}{dt}$
East-west speed
How fast the drone moves east.
$\frac{dy}{dt}$
North-south speed
How fast the drone moves north.

The drone's position is $(x(t), y(t)) = (R\cos t,\ 0.75R\sin t)$ ; the temperature it reads is $T(x(t), y(t))$ .
$T_x\,\frac{dx}{dt}$ is how fast temperature changes because of the drone's east-west motion; $T_y\,\frac{dy}{dt}$ is the north-south part.
Their sum is $\frac{dT}{dt}$ : equivalently, $\nabla T\cdot\mathbf v$ , the gradient dotted with the velocity.
When the velocity runs along a contour line, $\frac{dT}{dt} = 0$ even though the drone is moving.

Learn the math

The Chain RuleCalc III · Lesson 23, Partial Derivatives