Problem 41 - 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 antiderivative 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

If vector functions have identical derivatives, they differ by a constant vector; any antiderivative differs by a constant vector.

Step 1: Understanding Corollary 2 of the Mean Value Theorem

Corollary 2 of the Mean Value Theorem for scalar functions states that if two differentiable functions have the same derivative on an interval, then they differ by a constant. This applies to vector functions as well.

Step 2: Applying Corollary 2 to Vector Functions

For two vector functions \( \mathbf{R}_{1}(t) \) and \( \mathbf{R}_{2}(t) \), if their derivatives are identical for all \( t \) in some interval \( I \) — that is, \( \frac{d}{dt} \mathbf{R}_{1}(t) = \frac{d}{dt} \mathbf{R}_{2}(t) \) — then \( \mathbf{R}_{1}(t) - \mathbf{R}_{2}(t) \) must be constant on \( I \). The components of these vector functions each satisfy the conditions of the corollary, so they differ by a constant vector \( \mathbf{C} \).

Step 3: Analyzing Part b with Antiderivatives

If \( \mathbf{R}(t) \) is an antiderivative of \( \mathbf{r}(t) \) on \( I \), any other antiderivative can be represented by \( \mathbf{R}_{2}(t)\). Since both \( \mathbf{R}(t) \) and \( \mathbf{R}_{2}(t) \) have the same derivative \( \mathbf{r}(t) \), they must differ by a constant vector \( \mathbf{C} \). Thus, \( \mathbf{R}_{2}(t) = \mathbf{R}(t) + \mathbf{C} \).

Key Concepts

Vector Functions

A vector function is a function that takes one or more variables and outputs a vector. These functions are essential in engineering, physics, and computer graphics, because they can describe quantities with both direction and magnitude.
Commonly, vector functions are denoted using bold letters, such as \( \mathbf{R}(t) \). Here, \( t \) is usually a parameter such as time, and \( \mathbf{R} \) represents a vector consisting of component functions.

• For example, \( \mathbf{R}(t) = \langle f(t), g(t), h(t) \rangle \), where \( f(t), g(t), h(t) \) are real-valued functions.
• Each component is a function on its own, which collectively determines the path or position described by the vector.

Understanding vector functions is crucial because they allow us to model real-world phenomena like motion paths, forces acting on objects, and more.

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental theorem of calculus that relates values of a differentiable function at two points to its derivative in between those points. While often considered for scalar functions, it holds special significance for vector functions as well.
For scalar functions, the MVT states that if a function \( f \) is continuous over \([ a, b ]\) and differentiable over \( (a, b) \), there exists some point \( c \) in \( (a, b) \) such that:

• \( f'(c) = \frac{f(b) - f(a)}{b - a} \)

In vector functions, each component satisfies MVT separately, ensuring that if two vector functions have the same derivative over an entire interval, they differ only by a constant vector. This is because the difference in vector functions' rate of change is zero, leaving just a constant difference.

Constant Vector

A constant vector is a vector that does not change in direction or magnitude over time or any parameter. Within the context of antiderivatives or the Mean Value Theorem, a constant vector serves as an adjustment or offset between two similar functional forms.
For instance, if two vector functions \( \mathbf{R}_1(t) \) and \( \mathbf{R}_2(t) \) differ by a constant vector \( \mathbf{C} \), this means:

• Their derivatives are identical over a specific interval.
• \( \mathbf{R}_2(t) = \mathbf{R}_1(t) + \mathbf{C} \) describes how each function traces the same path but is shifted by the constant vector \( \mathbf{C} \).

This vector can be thought of as a fixed translation, indicating the uniformity between antiderivatives of a function.

Differentiable Functions

A differentiable function is one that has a derivative at every point within its domain. This means the function is "smooth" with no sharp corners or discontinuities, allowing the calculation of its rate of change.
For a function to be differentiable, it must be continuous. In mathematical terms, this involves the limit:

• The function \( f \) is differentiable at a point \( a \) if the limit \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \) exists.
• For vector functions, the differentiability of each component must be established independently and collectively ensures the vector function's differentiability.

Differentiable functions are pivotal in calculus and real analysis, facilitating the exploration of function behavior and the existence of antiderivatives.