Differentials

If we look at the linear approximation to a function in the infinitesimal limit (``as $\Delta \rightarrow d$'') then we get the following statement relating the differentials of a function of two variables:

df = f_x dx + f_y dy.

If we rewrite this in partial derivative notation,

$$df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy,$$

then we sort of see how the differential of a function is constructed. If we think of cancelling the $\partial x$ and dx in the first term we are left with $\partial f$. Similarly cancelling in the second term leaves us with $\partial f$.

Sometimes, the differential id referred to as the total derivative of a function since it includes information on how the function changes based on all of the variables. Thus, the differential of a function h(x,y,z,t) is

dh = h_x dx + h_y dy + h_z dz + h_t dt.

Differentials are often used to calculate the error bars in a function. For example suppose you measure the length, width, and depth of a rectangular prism to be x, y, and z. To calculate the volume of the object, you use the function V(x,y,z) = xyz. What happens to the volume if you make small errors in the measurements of the object's dimensions? We can calculate this effect by using the differential of V. If the errors in the measured values are dx, dy, and dz, respectively, then the error in the volume is

$$dV = \frac{\partial V}{\partial x} dx + \frac{\partial V}{\p... ... + \frac{\partial V}{\partial z} dz = (yz)dx + (xz)dy + (xy)dz.$$

One way that you may see a problem like this stated is:

You have measured the length, width, and depth of a rectangular object to be $x_0 \pm dx$, $y_0 \pm dy,$ and $z_0 \pm dz$, respectively. What is the error in the volume measurement? Thus, if the measurements are $4 \pm 0.5, 3 \pm 0.5$ and $6 \pm 0.5$ we have the volume as $V = (4)(3)(6) \pm dV = 72 \pm dV$ where dV = (3)(6)(0.5) + (4)(6)(0.5) + (4)(3)(0.5) = 27.

Any time a function is used to calculate a quantity based on quantities that have errors, the differential can be used to get the appropriate error in the calculation of the function.