Problem 20 - Calculus Volume 3

Question

Find the domain of \(\mathbf{r}(t)=2 e^{-t} \mathbf{i}+e^{-t} \mathbf{j}+\ln (t-1) \mathbf{k}\) .

Step-by-Step Solution

Answer

The domain is \( (1, \infty) \).

Step 1: Identify Function Components

The given vector function is \[ \mathbf{r}(t) = 2e^{-t} \mathbf{i} + e^{-t} \mathbf{j} + \ln(t-1) \mathbf{k} \].This function is expressed as a linear combination of functions of time \( t \) along unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).

Step 2: Analyze Each Component's Domain

The unit vector components have the following functions:- \( 2e^{-t} \) and \( e^{-t} \) are defined for all real numbers \( t \) since the exponential function is always defined.- \( \ln(t-1) \) is defined only when \( t-1 > 0 \). Therefore, its domain is \( t > 1 \).

Step 3: Determine the Overall Domain

Since \( 2e^{-t} \) and \( e^{-t} \) are defined for all \( t \), but \( \ln(t-1) \) restricts us to \( t > 1 \), the domain of the entire function \( \mathbf{r}(t) \) is constrained by \( \ln(t-1) \).

Step 4: Conclude the Domain

The domain of the vector function \( \mathbf{r}(t) \) is the set of real numbers for which each component function is defined. Therefore, the domain of \( \mathbf{r}(t) \) is \( (1, \infty) \).

Key Concepts

Exponential Functions

Exponential functions are one of the fundamental concepts in mathematics, represented in the general form as \( a^t \), often seen as \( e^t \) when the base is the mathematical constant \( e \). These functions are unique because they show constant growth rates and appear in many natural phenomena, like population growth or radioactive decay.
In the context of our vector function \( \mathbf{r}(t) = 2e^{-t} \mathbf{i} + e^{-t} \mathbf{j} + \ln(t-1) \mathbf{k} \), the exponential components \( 2e^{-t} \) and \( e^{-t} \) are defined for all real values of \( t \). This is because exponential functions do not inherently have any restrictions, meaning they exist smoothly and continuously over the entire real number line.

• The function \( 2e^{-t} \) starts at 2 when \( t = 0 \) and approaches 0 as \( t \) increases towards infinity.
• The component \( e^{-t} \) decreases similarly, representing exponential decay.

Because these components are always valid, they do not restrict the domain of the given vector function. However, they contribute to the dynamic behavior of the vector in space, showing exponential decay along the \( \mathbf{i} \) and \( \mathbf{j} \) axes as \( t \) increases.

Logarithmic Functions

Logarithmic functions are the inverse operations of exponential functions, commonly expressed as \( \log_b(x) \), where \( b \) is the base of the logarithm. In this exercise, we see the natural logarithm \( \ln(t-1) \), which uses the base \( e \).
Logarithmic functions have specific requirements for their domain, primarily concerning the argument inside the logarithm.

• For \( \ln(t-1) \) to be defined, the expression inside the logarithm, \( t-1 \), must be greater than zero.
• This condition gives us \( t > 1 \), which is the critical domain restriction for the vector function \( \mathbf{r}(t) \).

The reason for this restriction is because taking the logarithm of zero or a negative number is undefined in the real number system. Hence, understanding this limitation is crucial when determining the domain of any function involving logarithms. The restriction ensures that the overall expression remains mathematically valid and provides realistic outputs.

Vector Calculus

Vector calculus is a branch of mathematics focused on vector fields and differentiable functions. It provides the tools for describing and analyzing vector quantities in multiple dimensions, essential in physics and engineering.
The vector function \( \mathbf{r}(t) = 2e^{-t} \mathbf{i} + e^{-t} \mathbf{j} + \ln(t-1) \mathbf{k} \) is expressed as a combination of unit vectors, \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), representing directions in three-dimensional space. Each part functions independently along its vector component.

• \( 2e^{-t} \mathbf{i} \) and \( e^{-t} \mathbf{j} \) indicate how the vector decays exponentially along the \( x \)- and \( y \)-axes.
• \( \ln(t-1) \mathbf{k} \) introduces a logarithmic growth along the \( z \)-axis, shaped by the additional domain constraint \( t > 1 \).

This vector representation illustrates how time \( t \) affects each component differently, showing the importance of understanding each function's domain. These concepts form the basis of many applications in fields such as electromagnetism, fluid dynamics, and navigation, making vector calculus an indispensable part of modern science and engineering.