Question

Use the definition of differentiability to prove that the following functions are differentiable at \((0,0) .\) You must produce functions \(\varepsilon_{1}\) and \(\varepsilon_{2}\) with the required properties. $$f(x, y)=x y$$

Short answer

To prove that the function \(f(x, y) = xy\) is differentiable at the point (0, 0), we used the definition of differentiability in multiple variables and found functions \(\varepsilon_1(x, y) = y\) and \(\varepsilon_2(x, y) = x\). We then showed that the limits of \(\varepsilon_1(x, y)\) and \(\varepsilon_2(x, y)\) as \((x, y) \to (0, 0)\) are both equal to 0. Therefore, the function \(f(x, y) = xy\) is differentiable at the point (0, 0).

Step-by-step solution

Evaluate the function at \((0, 0)\)

First, we need to determine the value of the function \(f(x, y) = xy\) at the point \((0, 0)\): $$f(0, 0) = 0 \times 0 = 0$$

Find functions \(\varepsilon_1(x, y)\) and \(\varepsilon_2(x, y)\)

Now, we need to write \(f(x, y)\) as \(f(0, 0) + \varepsilon_1(x, y)x + \varepsilon_2(x, y)y\). Since \(f(x, y) = xy\) and \(f(0, 0) = 0\), we have: $$f(x, y) = \varepsilon_1(x, y)x + \varepsilon_2(x, y)y = xy$$ Comparing the terms, we may choose \(\varepsilon_1(x, y) = y\) and \(\varepsilon_2(x, y) = x\).

Show that the limits of \(\varepsilon_1(x, y)\) and \(\varepsilon_2(x, y)\) as \((x, y) \to (0, 0)\) are equal to 0

It remains to show that the limits of \(\varepsilon_1(x, y)\) and \(\varepsilon_2(x, y)\) as \((x, y) \to (0, 0)\) are equal to 0: $$\lim_{(x, y) \to (0, 0)}\varepsilon_1(x, y) = \lim_{(x, y) \to (0, 0)}y = 0$$ $$\lim_{(x, y) \to (0, 0)}\varepsilon_2(x, y) = \lim_{(x, y) \to (0, 0)}x = 0$$ Since both limits are equal to 0, we can conclude that the function \(f(x, y) = xy\) is differentiable at the point \((0, 0)\).

Key concepts

Multivariable Calculus

Grasping the essence of multivariable calculus is essential for understanding functions of more than one variable and how they change. It extends the concepts of single-variable calculus—such as limits, derivatives, and integrals—to functions of multiple variables.

Unlike single-variable calculus where the graph of a function is a curve, in multivariable calculus, we deal with functions whose graphs can be surfaces or even hypersurfaces in higher dimensions. This adds complexity, as each input point can be described by two or more numbers, representing coordinates in the plane or space.

For instance, when given a function like \(f(x, y) = xy\), it's not just about how the function changes with respect to \(x\) alone or \(y\) alone, but how the function changes as both \(x\) and \(y\) vary simultaneously. Analyzing the differentiability of such a function requires an understanding of how increment changes in inputs affect the output, which leads to the necessity of partial derivatives and limits as foundational tools.

Limit of a Function

Approaching Limits in Multiple Dimensions

The notion of the limit is a cornerstone in calculus. It provides a way to describe the behavior of functions as inputs approach a certain point or infinity. In the context of multivariable functions, this concept becomes subtler since we must consider all possible paths by which we can approach a given point.

Considering the function \(f(x, y) = xy\) from the exercise, we examine the limit as the point \( (x, y) \) gets infinitely close to \( (0, 0) \). In two dimensions, this means simultaneously shrinking the distances from \(x\) to 0 and from \(y\) to 0. A function is said to have a limit at a point if no matter the path taken to approach that point, the limit is the same.

In practice, showing that limits of error functions \(\varepsilon_1(x, y)\) and \(\varepsilon_2(x, y)\) approach zero is fundamental to proving differentiability at that point. The limit processes for \(\varepsilon_1\) and \(\varepsilon_2\) suggest that the change in the function's value will become negligibly small as \( (x, y)\) gets closer to \( (0, 0)\).

Partial Derivatives

Dissecting Changes with Partial Derivatives

Understanding partial derivatives is critical in multivariable calculus. While a regular derivative looks at how a function changes as its input changes, partial derivatives tell us how a multivariable function changes as one of the variables changes, holding the other variables constant. To put it simply, it isolates the effect of one variable at a time.

In the given exercise, the function \(f(x, y) = xy\) involves both \(x\) and \(y\). Determining how \(f\) changes as only \(x\) varies (with \(y\) kept fixed) or as only \(y\) varies (with \(x\) kept fixed) leads us to the concept of partial derivatives, denoted as \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) respectively.

In the context of differentiability, finding and analyzing partial derivatives is a fundamental step. Functions that have all partial derivatives are good candidates for being differentiable, but just having partial derivatives isn't enough. Those derivatives also need to be continuous and satisfy certain limit conditions as illustrated by the error functions in our exercise. If every point around \( (0,0) \) has partial derivatives that don’t vary wildly, then the function is smooth around that point, building a case for its differentiability at that point.