Problem 54 - Thomas Calculus

Question

Establish the following properties of integrable vector functions. a. The Constant Scalar Multiple Rule: $$ \int_{a}^{b} k \mathbf{r}(t) d t=k \int_{a}^{b} \mathbf{r}(t) d t \quad \text { (any scalar } k ) $$ The Rule for Negatives, $$ \int_{a}^{b}(-\mathbf{r}(t)) d t=-\int_{a}^{b} \mathbf{r}(t) d t $$ \(\quad\) is obtained by taking \(k=-1\) b. The Sum and Difference Rules: $$ \int_{a}^{b}\left(\mathbf{r}_{1}(t) \pm \mathbf{r}_{2}(t)\right) d t=\int_{a}^{b} \mathbf{r}_{1}(t) d t \pm \int_{a}^{b} \mathbf{r}_{2}(t) d t $$ c. The Constant Vector Multiple Rules: $$ \begin{array}{l}{\int_{a}^{b} \mathbf{C} \cdot \mathbf{r}(t) d t=\mathbf{C} \cdot \int_{a}^{b} \mathbf{r}(t) d t \quad \text { (any constant vector } \mathbf{C} )} \\ {\text { and }} \\ {\int_{a}^{b} \mathbf{C} \times \mathbf{r}(t) d t=\mathbf{C} \times \int_{a}^{b} \mathbf{r}(t) d t \quad \text { (any constant vector } \mathbf{C} )}\end{array} $$

Step-by-Step Solution

Answer

The rules demonstrate linearity and effectiveness in simplifying vector function integrals.

Step 1: Understanding Constant Scalar Multiple Rule

The Constant Scalar Multiple Rule states that when a vector function \( \mathbf{r}(t) \) is multiplied by a constant scalar \( k \) and integrated over an interval, the result is the same as multiplying \( k \) by the integral of \( \mathbf{r}(t) \). This can be expressed as: \( \int_{a}^{b} k \mathbf{r}(t) \, dt = k \int_{a}^{b} \mathbf{r}(t) \, dt \). The Rule for Negatives is a specific case where \( k = -1 \), leading to: \( \int_{a}^{b} (-\mathbf{r}(t)) \, dt = -\int_{a}^{b} \mathbf{r}(t) \, dt \). This reflects that negating the function is equivalent to negating the integral.

Step 2: Analyzing Sum and Difference Rules

The Sum and Difference Rules state that integrating a sum (or difference) of two vector functions \( \mathbf{r}_1(t) \) and \( \mathbf{r}_2(t) \) over an interval is equivalent to the sum (or difference) of their integrals. This is written as: \( \int_{a}^{b} \left( \mathbf{r}_1(t) \pm \mathbf{r}_2(t) \right) \, dt = \int_{a}^{b} \mathbf{r}_1(t) \, dt \pm \int_{a}^{b} \mathbf{r}_2(t) \, dt \). This property simplifies the integration process of combined vector functions.

Step 3: Exploring Constant Vector Multiple Rules

The Constant Vector Multiple Rules involve the dot product and cross product with constant vectors. For a dot product, the rule states: \( \int_{a}^{b} \mathbf{C} \cdot \mathbf{r}(t) \, dt = \mathbf{C} \cdot \int_{a}^{b} \mathbf{r}(t) \, dt \). For a cross product, the rule states: \( \int_{a}^{b} \mathbf{C} \times \mathbf{r}(t) \, dt = \mathbf{C} \times \int_{a}^{b} \mathbf{r}(t) \, dt \). These rules show that the integral of a vector function involved in a product with a constant vector can be computed by taking the product with the integral of the vector function.

Key Concepts

Constant Scalar Multiple Rule

The Constant Scalar Multiple Rule is fundamental in vector integration. This rule applies when you integrate a vector function \( \mathbf{r}(t) \) that is multiplied by a constant scalar \( k \) over an interval \([a, b]\). Instead of directly integrating \( k \mathbf{r}(t) \), you can simplify the process by taking the integral of \( \mathbf{r}(t) \) first, and then multiplying the result by \( k \). Essentially, this can be expressed as: \[\int_{a}^{b} k \mathbf{r}(t) \, dt = k \int_{a}^{b} \mathbf{r}(t) \, dt\] This formula shows us the convenience of separating constants from the integral. It is particularly helpful because it allows you to avoid directly dealing with the constant during integration, which can save you time and simplify calculations.

An important application of this rule is the Rule for Negatives, which is just a specific case where \( k = -1 \). This results in the following property: \[\int_{a}^{b} (-\mathbf{r}(t)) \, dt = -\int_{a}^{b} \mathbf{r}(t) \, dt\] This rule tells us that the integral of the negative of a vector function is simply the negative of the integral of that function.

Sum and Difference Rules

The Sum and Difference Rules are very useful for integrating combinations of vector functions. When you are working with two vector functions, \( \mathbf{r}_1(t) \) and \( \mathbf{r}_2(t) \), these rules allow you to break down the integration process. They state that the integral of the sum (or difference) of these functions is equivalent to the sum (or difference) of their individual integrals. Mathematically, this can be expressed as: \[\int_{a}^{b} \left( \mathbf{r}_1(t) \pm \mathbf{r}_2(t) \right) \, dt = \int_{a}^{b} \mathbf{r}_1(t) \, dt \pm \int_{a}^{b} \mathbf{r}_2(t) \, dt\] These rules simplify the process significantly. Instead of integrating a more complex function all at once, you can handle each component separately.

This aspect becomes particularly advantageous when dealing with intricate functions in applications of physics and engineering, where each vector might represent a different dimension or physical quantity.

Constant Vector Multiple Rules

The Constant Vector Multiple Rules deal with vector functions involving dot and cross products with a constant vector. When you have a constant vector \( \mathbf{C} \), these rules establish a straightforward method for computing integrals involving these vector operations.

• For the dot product, the rule states: \[\int_{a}^{b} \mathbf{C} \cdot \mathbf{r}(t) \, dt = \mathbf{C} \cdot \int_{a}^{b} \mathbf{r}(t) \, dt\]
• For the cross product, it states: \[\int_{a}^{b} \mathbf{C} \times \mathbf{r}(t) \, dt = \mathbf{C} \times \int_{a}^{b} \mathbf{r}(t) \, dt\]

If you’re performing these types of integrations, the rules reveal that you can compute the integral of the vector function first, and then perform the dot or cross product with the constant vector. This approach is efficient as it reduces the complexity of calculations, especially when vector operations are involved in the analysis of forces, fields, and other vector quantities in engineering and physics.