The coordinates are the current location, measured on the $x,y,z$ axes. With me so far? We type in any coordinate, and the microwave spits out the gradient at that location.īe careful not to confuse the coordinates and the gradient. But this was well worth it: we really wanted that clock. Unfortunately, the clock comes at a price - the temperature inside the microwave varies drastically from location to location. The microwave also comes with a convenient clock. We can type any 3 coordinates (like “3,5,2″) and the display shows us the gradient of the temperature at that point. Suppose we have a magical oven, with coordinates written on it and a special display screen: I’m a big fan of examples to help solidify an explanation. Likewise, with 3 variables, the gradient can specify and direction in 3D space to move to increase our function. If we have two variables, then our 2-component gradient can specify any direction on a plane. However, now that we have multiple directions to consider ($x$, $y$ and $z$), the direction of greatest increase is no longer simply “forward” or “backward” along the $x$-axis, like it is with functions of a single variable. The gradient of a multi-variable function has a component for each direction.Īnd just like the regular derivative, the gradient points in the direction of greatest increase ( here's why: we trade motion in each direction enough to maximize the payoff). The regular, plain-old derivative gives us the rate of change of a single variable, usually $x$. Now that we know the gradient is the derivative of a multi-variable function, let’s derive some properties. “Gradient” can refer to gradual changes of color, but we’ll stick to the math definition if that’s ok with you. Yes, you can say a line has a gradient (its slope), but using 'gradient' for single-variable functions is unnecessarily confusing. The term 'gradient' is typically used for functions with several inputs and a single output (a scalar field). Is zero at a local maximum or local minimum (because there is no single direction of increase).Points in the direction of greatest increase of a function ( intuition on why).The gradient is a fancy word for derivative, or the rate of change of a function.

Gradient symbol