Differentiability

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We introduce differentiability for functions of several variables and find tangent planes.

Differentiability

Previously in your calculus experience, we often asked the question, “Is this function differentiable?” When looking at $f: \R \to \R$ , the answer to that question was the same as the answer to,“Does the derivitive exist?” Now that we are working with multivariable functions, we’ve already seen that there is more than one derivative we can define at any given point, giving us different information about the surface. Therefore, we need a different strategy. Another way we could have answered whether a function $f: \R \to \R$ was differentiable was to ask whether there exists a tangent line (that is, a linear approximation) to the function at that point. This idea we can generalize to higher dimensions.

Recall, for $f: \R \to \R$ , we say a function is differentiable if the limit $f'(a)=\lim _{x \to a} \frac {f(x)-f(a)}{x-a}$ exists. This is equivalent to saying that

$\begin{align*} \lim_{x \to a} \frac{f(x)-f(a)}{x-a} -f'(a)=0 \\ \lim_{x \to a} \frac{f(x)-f(a) -f'(a)(x-a)}{x-a}=0\\ \lim_{x \to a} \frac{f(x)-(f(a) +f'(a)(x-a))}{x-a}=0\\ \lim_{x \to a} \frac{f(x)-\l(x)}{x-a}=0 \end{align*}$ where $\l (x)$ is the equation of the tangent line to $f$ at $a$ .

Let’s use this same idea to define differentiability in higher dimensions.

Let $F: \R ^n\to \R$ and let $\vec {x}=\vector {x_1, x_2, \dots , x_n}$ and $\vec {a} = \vector {a_1, a_2, \dots , a_n}$ be vectors in the interior of the domain of $F$ . We say $F$ is differentiable at a vector $\vec {a}$ if there exists $L(\vec {x})=F(\vec {0})+\sum _{i=1}^n c_i (x_i-a_i)$ for some real constants $c_i$ such that $\lim _{\vec {x} \to \vec {a}} \frac {F(\vec {x})-L(\vec {x})}{|\vec {x}-\vec {a}|} = 0$ A function is differentiable on an open set if it is differentiable at every point in that set.

In $\R ^2$ , we can interpret this definition as saying that a function $F$ is differentiable at a vector $\vec {a}$ if there is a plane $L(\vec {x})$ at that point such that $F$ approaches $L(\vec {x})$ faster than $\vec {x}$ approaches $\vec {a}$ . In this case, we call this plane the tangent plane. We interpret this differentiability as, if one “zooms in” on the graph of $F$ at $(\vec {a}, F(\vec {a}))$ sufficiently, it looks more and more like the tangent plane.

You can use this interactive to visualize a tangent plane.

How do we interpret this definition in higher dimensions? If we zoom in on a differentiable curve $f:\R \to \R$ , we will see $f$ approach its tangent line. If we zoom in on a differentiable surface $F:\R ^2 \to \R$ , we will see $F$ approach its tangent plane. Similarly, if we zoom in on a differentiable surface $F:\R ^n \to \R$ , we should expect to see $F$ approach a flat surface $L(x)$ which looks like $\R ^n$ .

Keep in mind that this statement of differentiability is much stronger than just saying the partial derivatives of $F$ exist at $\vec {a}$ . Because this is a limit in a higher dimension, we are saying this limit must exist and be equal for every path by which $\vec {x}$ could approach $\vec {a}$ , not just the paths along the axes. Because of this, it is very difficult to use this definition to show that a function is differentiable. Thankfully we have the following theorem.

A Sufficient Criterion for Differentiability Let $F:\R ^n\to \R$ be defined on an open set $S$ containing $\vec {a}$ . If $\pp [F]{x_i}(x_1, x_2, \dots , x_n)$ for $i$ from $1$ to $n$ are all continuous on $S$ , then $F$ is differentiable on $S$ .

In words, this theorem is saying that if all of the partial derivatives are all continuous on all of $S$ , then $F$ is differentiable on $S$ . Notice how much easier this is to use than the definition! This theorem assures us that essentially all functions that we see in the course of our studies here are differentiable on their natural domains.

Remember, we were trying to generalize the concept of differentiability for $f:\R \to \R$ ? Our concept of differentiability for $f:\R \to \R$ had a lot of nice properties. Many of these properties also hold for differentiability in higher dimensions. In particular, here are a few examples.

Differentiability Implies Partial Derivatives Exist Let $F:\R ^n\to \R$ be defined on an open set $S$ containing $\vec {a}$ . If $F$ is differentiable at $\vec {a}$ , then all the partial derivatives of $F$ exist at $\vec {a}$ .

Differentiability Implies Continuity Let $F:\R ^n\to \R$ be defined on an open set $S$ containing $\vec {a}$ . If $F$ is differentiable at $\vec {a}$ , then $F$ is continuous at $\vec {a}$ .

Since most of the functions we see in this course are differentiable on their domains, they are also continuous.

A note of caution: differentiability in higher dimensions is a much more subtle notion. For instance, it is possible for a function $F$ to be differentiable yet some partials may not be continuous. It is also possible for all the partial derivatives to exist at a point and the function still fail to be continuous, let alone differentiable, at that point. Such strange behavior of functions is a source of delight for many mathematicians.