Problem 47 - 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

Differentiability at \( t_0 \) implies \( \mathbf{r}(t) \) is continuous at \( t_0 \).

Step 1: Define Differentiability

A vector function \( \mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k} \) is differentiable at \( t = t_0 \) if the limit \[ \lim_{\Delta t \to 0} \frac{\mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)}{\Delta t} \] exists. Differentiability implies the existence of this limit.

Step 2: Express Continuity

A vector function \( \mathbf{r}(t) \) is continuous at \( t=t_0 \) if \[ \lim_{t \to t_0} \mathbf{r}(t) = \mathbf{r}(t_0) \]. We need to show that if \( \mathbf{r}(t) \) is differentiable at \( t_0 \), this condition holds.

Step 3: Analyze the Difference Quotient

Since \( \mathbf{r}(t) \) is differentiable at \( t = t_0 \), \[ \lim_{\Delta t \to 0} \frac{\mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)}{\Delta t} \] exists. This implies that \( \lim_{\Delta t \to 0} (\mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)) = 0 \).

Step 4: Show the Limit Condition

From Step 3, since \( \lim_{\Delta t \to 0} (\mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)) = 0 \), it follows that \( \lim_{\Delta t \to 0} \mathbf{r}(t_0 + \Delta t) = \mathbf{r}(t_0) \). Thus, \( \mathbf{r}(t) \) is continuous at \( t = t_0 \).

Key Concepts

Continuity

Continuity in vector functions means that the function behaves predictably without any abrupt changes at a given point. For a vector function \( \mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k} \), continuity at a point \( t_0 \) implies that \\[ \lim_{t \to t_0} \mathbf{r}(t) = \mathbf{r}(t_0) \]This means as \( t \) gets closer to \( t_0 \), the output of the function approaches \( \mathbf{r}(t_0) \) closely. This predictable behavior ensures there are no jumps or gaps. Continuity is crucial because it ensures the function values change smoothly as the input values shift.
For a function to be continuous:

• The limit as \( t \to t_0 \) must exist.
• The value of the function at \( t_0 \) must be the same as this limit.
• The function behavior should exhibit no jumps, dips, or gaps at that point.

Existence of Limits

The existence of limits is a fundamental concept in calculus, crucial for understanding both differentiability and continuity. For a function to be differentiable at \( t_0 \), it must admit a limit as the interval approaches zero.
This is often expressed through:\[ \lim_{\Delta t \to 0} \frac{\mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)}{\Delta t} \]The existence of this limit confirms that the function's output behaves consistently and predictably as changes to the input become infinitesimally small.
If this limit exists, it also implies that:

• Any small change in \( t \) around \( t_0 \) leads to a consistent small change in \( \mathbf{r}(t) \).
• The values of the function around \( t_0 \) do not deviate wildly but instead conform to a predictable pattern.

This reliable behavior is what makes differentiability leading to continuity since differentiability inherently assumes the light existence condition of limits.

Difference Quotient

The difference quotient is an important tool used to compute the derivative of a vector function, acting as a bridge between basic algebra and calculus concepts. The difference quotient for a vector function \( \mathbf{r}(t) \) is expressed as:\[ \frac{\mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)}{\Delta t} \]This quotient gives an average rate of change over the interval \( \Delta t \).
When the interval \( \Delta t \) approaches zero, the difference quotient morphs into the derivative, provided the limit exists.
This transition from average to instantaneous change highlights the essence of calculus:

• Calculating the velocity of a particle when \( \mathbf{r}(t) \) represents position over time.
• Predicting the behavior of physical systems in engineering and physics.
• Smoothing out real-life applications where tracking changes becomes essential.

Grasping the concept of the difference quotient leads to deeper insights into both calculus and real-world applications, laying down a path from discrete computations towards continuous processes.

Vector Calculus

Vector calculus extends traditional calculus into the realm of multi-dimensional processes, utilizing vectors to explore complex systems. Unlike scalar calculus, which deals with quantities described by single values, vector calculus takes into account both magnitude and direction.
In the context of differentiable vector functions, we express these functions in vector form as:\[ \mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k} \]These functions harness the power of vector fields, helpful in fields such as physics (e.g., electromagnetism), engineering, and computer graphics. By understanding and implementing vector calculus:

• We can formulate methods for describing and predicting the behavior of 3D objects.
• We can effectively manage systems where directionality and magnitude are core factors.
• It provides tools to navigate and solve complex problems requiring multidimensional reasoning.

Vector calculus serves as a vital element in interpreting and shaping the natural world's mathematical models, allowing for profound interactions with multi-dimensional phenomena.