Hot Spots on a Circuit Board

Two chips heat a board unevenly. A partial derivative asks one simple question: slide a sensor a little east, or a little north, and how fast does the temperature change? Each direction gets its own rate.

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

How it works

Partial derivatives

In one sentence

For $T(x, y)$ , the partial derivative $\partial T/\partial x$ is the ordinary derivative in $x$ with $y$ held constant (and likewise for $y$ ); together they form the gradient $\nabla T$ , which points in the direction of fastest increase.

General formula

{\htmlData{sym=0}{\frac{\partial T}{\partial x}}} = \lim_{{\htmlData{sym=4}{h}}\to 0}\frac{{\htmlData{sym=3}{T}}(x + {\htmlData{sym=4}{h}},\,y) - {\htmlData{sym=3}{T}}(x,\,y)}{\htmlData{sym=4}{{h}}}, \qquad {\htmlData{sym=2}{\nabla T}} = \left({\htmlData{sym=0}{\frac{\partial T}{\partial x}}},\ {\htmlData{sym=1}{\frac{\partial T}{\partial y}}}\right)

Freeze one coordinate and the surface becomes an ordinary curve. The limit is just the slope of that curve, measured in degrees per centimeter.

What the symbols mean

$\frac{\partial T}{\partial x}$
Rate going east
How many degrees the temperature changes per centimeter moved in the x direction, with y held fixed.
$\frac{\partial T}{\partial y}$
Rate going north
The same rate in the y direction, with x held fixed.
$\nabla T$
Gradient
Both rates as one arrow. It points toward the fastest warming, and its length is how fast.
$T$
Temperature
Degrees Celsius at each point (x, y) of the board.
$h$
Small step
A tiny distance moved east; the limit makes it shrink to zero.

Hold $y$ fixed and the board becomes one line of temperatures; $\partial T/\partial x$ is that line's slope.
Hold $x$ fixed instead and you get $\partial T/\partial y$ , the slope from south to north.
Together they make the gradient $\nabla T$ , which points toward the fastest warming. Its length is how fast that is.
The tangent plane at the point is built from the two slopes: $T \approx T_0 + T_x\,\Delta x + T_y\,\Delta y$ .

Learn the math

Partial DerivativesCalc III · Lesson 19, Partial Derivatives Interpreting Partial DerivativesCalc III · Lesson 20, Partial Derivatives