Tangent Planes and Linear Approximations: Learn It 4

Differentials

You first encountered differentials when studying linear approximations in single-variable calculus. The differential of [latex]y[/latex], written [latex]dy[/latex], is defined as [latex]f'(x)dx[/latex]. We use differentials to approximate [latex]\Delta y = f(x+\Delta x)-f(x)[/latex], where [latex]\Delta x=dx[/latex].

We can extend this idea to the linear approximation of a function of two variables at the point[latex](x_0, y_0)[/latex] to create the formula for the total differential for a function of two variables.

total differential

Let [latex]z=f(x,y)[/latex] be a function of two variables with [latex](x_0, y_0)[/latex] in the domain of [latex]f[/latex] and let [latex]\Delta x[/latex] and [latex]\Delta y[/latex] be chosen so that [latex](x_0+\Delta x, y_0+\Delta y)[/latex] is also in the domain of [latex]f[/latex]. If [latex]f[/latex] is differentiable at the point [latex](x_0, y_0)[/latex], then the differentials [latex]dx[/latex] and [latex]dy[/latex] are defined as:

[latex]dx= \Delta x[/latex] and [latex]dy= \Delta y[/latex]

The differential [latex]dz[/latex] also called the total differential of [latex]z=f(x,y)[/latex] at [latex](x_0,y_0)[/latex], is defined as:

[latex]dz=f_x(x_0,y_0)dx+f_y(x_0,y_0)dy[/latex].

Notice that we use [latex]d[/latex] (not the partial derivative symbol [latex]\partial[/latex]) for the total differential.

Now, let’s define [latex]\Delta z=f(x+\Delta x, y+\Delta y) - f(x,y)[/latex]. We use [latex]dz[/latex] to approximate [latex]\Delta z[/latex]:

[latex]\Delta z\approx dz=f_x(x_0,y_0)dx+f_y(x_0,y_0)dy[/latex]

The differential approximates the change in [latex]z=f(x_0, y_0)[/latex] at point [latex](x_0, y_0)[/latex] for given values of [latex]\Delta x[/latex] and [latex]\Delta y[/latex]. Since [latex]\Delta z=f(x+ \Delta x, y+ \Delta y)-f(x, y)[/latex], we can use this to approximate [latex]f(x+ \Delta x, y+ \Delta y)[/latex]:

[latex]\hspace{8cm}\begin{alignat}{2} f(x+\Delta{x},y+\Delta{y})&=f(x,y)+\Delta{z}\\ &\approx{f}(x,y)+f_x(x_0,y_0)\Delta{x}+f_y(x_0,y_0)\Delta{y}.\\ \end{alignat}[/latex]

See the following figure.

A surface f in the xyz plane, with a tangent plane at the point (x, y, f(x, y)). On the (x, y) plane, there is a point marked (x + Δx, y + Δy). There is a dashed line to the corresponding point on the graph of f and the line then continues to the tangent plane; the distance to the graph of f is marked f(x + + Δx, y + Δy), and the distance to the tangent plane is marked as the linear approximation.
Figure 2. The linear approximation is calculated via the formula [latex]\small{f(x+\Delta{x},y+\Delta{y}) \approx f(x,y)+f_{x}(x_{0},y_{0})\Delta{x}+f_{y}(x_{0},y_{0})\Delta{y}}[/latex].

Applications to Error Propagation

One practical application of differentials is determining error propagation. For example, if you’re manufacturing a gadget and your measurements are off by a small amount, you can use the differential to estimate the error in the total volume of the gadget.

Find the differential [latex]dz[/latex] of the function [latex]f(x,y)=3x^{2}-2xy+y^{2}[/latex] and use it to approximate [latex]\Delta z[/latex] at point [latex](2, -3)[/latex]. Use [latex]\Delta x=0.1[/latex] and [latex]\Delta y=-0.05[/latex] What is the exact value of [latex]\Delta z[/latex]?

Differentiability of a Function of Three Variables

All of the results for differentiability of functions of two variables generalize naturally to functions of three variables.

differentiable function of three variables

A function [latex]f(x, y, z)[/latex] is differentiable at a point [latex]P(x_0, y_0, z_0)[/latex] if for all points [latex](x, y, z)[/latex] in a [latex]\delta[/latex] disk around [latex]P[/latex] we can write:

[latex]\begin{alignat}{2}\hspace{6cm}f(x, y, z)&=f_x(x_0, y_0, z_0)(x-x_0)+f_y(x_0, y_0, z_0)(y-y_0)\\ & \;\; +f_z(x_0, y_0, z_0)(z-z_0)+E(x,y,z),\\ \end{alignat}[/latex]

where the error term [latex]E[/latex] satisfies:

[latex]\displaystyle\lim_{(x,y,z)\to(x_0,y_0,z_0)}=\dfrac{E(x,y,x)}{\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}}=0.[/latex]

If a function of three variables is differentiable at [latex](x_0, y_0, z_0)[/latex], then it is continuous there. Furthermore, continuity of first partial derivatives at that point guarantees differentiability.