Problem 33 - Thomas Calculus

Question

Differentiable vector functions are continuous Show that if \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}\) is differentiable at \(t=t_{0},\) then it is continuous at \(t_{0}\) as well.

Step-by-Step Solution

Answer

If differentiable at \(t_0\), \(\mathbf{r}(t)\) is continuous at \(t_0\).

Step 1: Understand the Given Problem

We need to show that if a vector function \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}\) is differentiable at \(t=t_0\), it is also continuous at \(t_0\). We know the differentiation implies continuity in scalar functions, but we need to extend this idea to a vector function.

Step 2: Define Differentiability of a Vector Function

A vector function \(\mathbf{r}(t)\) is differentiable at \(t = t_0\) if the derivative \(\mathbf{r}'(t_0)\) exists. This is defined as \[\mathbf{r}'(t_0) = \lim_{\Delta t \to 0} \frac{\mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)}{\Delta t}\]. To be differentiable, this limit must exist.

Step 3: Express Continuity of the Vector Function

The vector function \(\mathbf{r}(t)\) is continuous at \(t = t_0\) if \[\lim_{t \to t_0} \mathbf{r}(t) = \mathbf{r}(t_0)\]. This means as \(t\) approaches \(t_0\), the values of \(\mathbf{r}(t)\) approach the value of \(\mathbf{r}(t_0)\).

Step 4: Use Differentiability to Show Continuity

Since \(\mathbf{r}(t)\) is differentiable at \(t_0\), we know \(\mathbf{r}'(t_0)\) exists. Consider the limit definition of the derivative: \[\lim_{\Delta t \to 0} \frac{\mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)}{\Delta t} = \mathbf{r}'(t_0)\]. Multiply both sides by \(\Delta t\) and take the limit as \(\Delta t \to 0\): \[\lim_{\Delta t \to 0} (\mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)) = 0\]. This implies \(\lim_{t \to t_0} \mathbf{r}(t) = \mathbf{r}(t_0)\), satisfying continuity.

Key Concepts

Continuity

In the field of mathematics, particularly calculus, the concept of continuity plays a fundamental role in understanding how functions behave. A function is said to be continuous at a point if there is no interruption or gap in its value as you approach that point. For a vector function, such as \( \mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k} \), we say it is continuous at \( t_0 \) if:

• \( \lim_{t \to t_0} \mathbf{r}(t) = \mathbf{r}(t_0) \)

This means that as \( t \) gets closer to \( t_0 \), the vector function approaches the vector value at \( t_0 \) itself.
The importance of continuity is that it ensures smoothness in the transition of function values; there are no breaks or jumps. Understanding this in the context of vector functions extends the basic notion of continuity from simple scalar functions (like \( f(x) \)) to more complex, multidimensional objects.

Differentiability

Differentiability is closely linked to the idea of continuity, but it goes a step further. For a vector function to be differentiable at a point \( t_0 \), it must have a well-defined tangent vector at that point. This is akin to saying the slope, or rate of change, of the vector function exists. For \( \mathbf{r}(t) \), differentiability at \( t_0 \) means:

• \( \mathbf{r}'(t_0) = \lim_{\Delta t \to 0} \frac{\mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)}{\Delta t} \)

This limit must exist and be finite, meaning no abrupt changes. If a function is differentiable at a point, it implies it is also continuous there.
Think of differentiability as an assurance that the function not only has no gaps or jumps but also that it transitions smoothly enough that we can calculate rates of change.

Vector Calculus

Vector calculus expands the ideas of calculus into higher dimensions, dealing with vector fields and spaces. In vector calculus, functions like \( \mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k} \) are common, representing a vector's path as a function of an independent variable \( t \).

• These functions can describe physical phenomena, such as motion along a path in space, where each component of the vector may represent different dimensions like x, y, and z.

Understanding continuity and differentiability in vector calculus is crucial for analyzing motion, fluxes, and other multidimensional changes.
Diving into vector calculus allows us to explore how such vector functions change and behave, gaining insights into their continuity (smooth, gap-free behavior) and differentiability (smooth rate of change). These concepts enable us to analyze and predict complex systems modeled by vector fields.