Problem 56 - Thomas Calculus

Question

Antiderivatives of vector functions a. Use Corollary 2 of the Mean Value Theorem for scalar functions to show that if two vector functions \(\mathbf{R}_{1}(t)\) and \(\mathbf{R}_{2}(t)\) have identical derivatives on an interval \(I\) , then the functions differ by a constant vector value throughout \(I .\) b. Use the result in part (a) to show that if \(\mathbf{R}(t)\) is any anti- derivative of \(\mathbf{r}(t)\) on \(I,\) then any other antiderivative of \(\mathbf{r}\) on \(I\) equals \(\mathbf{R}(t)+\mathbf{C}\) for some constant vector \(\mathbf{C}\) .

Step-by-Step Solution

Answer

Two vector functions with identical derivatives differ by a constant vector. Any antiderivative differs by a constant vector.

Step 1: Understand the Statement of the Mean Value Theorem for Scalar Functions

Corollary 2 of the Mean Value Theorem for scalar functions states that if two functions have the same derivative on an interval, then they differ by a constant value on that interval. Mathematically, if two functions \( f'(t) = g'(t) \) for all \( t \) in \( I \), then \( f(t) = g(t) + C \), where \( C \) is a constant.

Step 2: Apply to Vector Functions

Considering vector functions \( \mathbf{R}_1(t) \) and \( \mathbf{R}_2(t) \), if they have identical derivatives over an interval \( I \), i.e., \( \mathbf{R}_1'(t) = \mathbf{R}_2'(t) \) for all \( t \) in \( I \), then each component of these vector functions must have equal derivatives. According to the scalar version of the Mean Value Theorem, each component must therefore differ by a constant.

Step 3: Conclude the Difference is a Constant Vector

Since each component of \( \mathbf{R}_1 \) and \( \mathbf{R}_2 \) differ by a constant, the whole vector functions differ by a vector constant. Therefore, \( \mathbf{R}_2(t) = \mathbf{R}_1(t) + \mathbf{C} \), where \( \mathbf{C} \) is a constant vector.

Step 4: Show Any Two Antiderivatives Differ by a Constant Vector

Given \( \mathbf{R}(t) \) is an antiderivative of \( \mathbf{r}(t) \), any other antiderivative \( \mathbf{S}(t) \) of \( \mathbf{r}(t) \) will satisfy \( \mathbf{S}'(t) = \mathbf{R}'(t) = \mathbf{r}(t) \). By the result in Step 3, \( \mathbf{S}(t) = \mathbf{R}(t) + \mathbf{C} \), where \( \mathbf{C} \) is a constant vector.

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental concept in calculus, important for both scalar and vector functions. Corollary 2 of the MVT for scalar functions tells us that if two functions have the same derivative on a specific interval, they actually differ by a constant over that interval. This means that if you have two functions, say \( f(t) \) and \( g(t) \), and their derivatives \( f'(t) \) and \( g'(t) \) are equal on an interval \( I \), then there exists some constant \( C \) such that \( f(t) = g(t) + C \).
In the context of vector calculus, this theorem can be extended to vector functions. If two vector functions, \( \mathbf{R}_1(t) \) and \( \mathbf{R}_2(t) \), have the same derivative on an interval, this means their components differ only by constant vectors. Since each individual component of these vector functions behaves according to the MVT, it implies that the entire functions differ by a constant vector. This is crucial for understanding antiderivatives of vector functions.

Vector Calculus

Vector calculus is a branch of mathematics focused on vector fields, which are functions that assign a vector to every point in space. In the scenario we are considering, we deal with vector functions, \( \mathbf{R}_1(t) \) and \( \mathbf{R}_2(t) \), each being a collection of scalar functions representing different components, such as \( (x(t), y(t), z(t)) \).
When dealing with derivatives of these vector functions, the derivative of the entire vector function is simply the vector of the derivatives of each component. For example, if \( \mathbf{R}'(t) = (x'(t), y'(t), z'(t)) \), then each component derivative is treated using regular derivative rules for scalars.

• Identical derivatives: When vector functions have identical derivatives, it essentially means that every part or dimension represented by the vector has the same rate of change.
• Antiderivatives: The antiderivative of a vector function can be visualized as the reverse process of finding a derivative, similar to what is done for scalar functions.

Understanding these principles is fundamental to tackling problems involving differentiation and integration of vector fields.

Constant Vector Difference

A constant vector difference between two vector functions means that, although the functions themselves might look different, their change or variation is completely described by a single vector constant. To visualize this, imagine two parallel paths traced in space. They may start at different points but maintain the same trajectory.
Given two vector functions \( \mathbf{R}_1(t) \) and \( \mathbf{R}_2(t) \) such that \( \mathbf{R}_1'(t) = \mathbf{R}_2'(t) \), the functions only differ by a constant vector \( \mathbf{C} \), i.e., \( \mathbf{R}_2(t) = \mathbf{R}_1(t) + \mathbf{C} \). This constant vector can be thought of as a static adjustment or shift applied to \( \mathbf{R}_1(t) \) to get \( \mathbf{R}_2(t) \).
The concept is crucial when considering antiderivatives, where any two antiderivative functions \( \mathbf{R}(t) \) and \( \mathbf{S}(t) \) of a given vector function \( \mathbf{r}(t) \) only differ by that constant vector. Hence, once you have one antiderivative, you can derive others by simply adding a constant vector.