Problem 48 - Thomas Calculus

Question

Constant Function Rule Prove that if \(u\) is the vector function with the constant value \(\mathbf{C},\) then \(d \mathbf{u} / d t=0 .\)

Step-by-Step Solution

Answer

If \( \mathbf{u}(t) = \mathbf{C} \), then \( \frac{d \mathbf{u}}{dt} = 0 \) because the derivative of a constant is 0.

Step 1: Understanding the Problem

We need to show that if a vector function, \( \mathbf{u} \), is constant, meaning it has the value \( \mathbf{C} \) at all times, then its derivative with respect to time, \( \frac{d \mathbf{u}}{dt} \), is equal to 0.

Step 2: Defining the Function

Define the vector function \( \mathbf{u}(t) = \mathbf{C} \), where \( \mathbf{C} \) is a constant vector. This means that at any time \( t \), \( \mathbf{u}(t) \) does not change; it remains \( \mathbf{C} \).

Step 3: Taking the Derivative

To find the rate of change of \( \mathbf{u}(t) \) with respect to \( t \), calculate the derivative \( \frac{d \mathbf{u}}{dt} \). Since \( \mathbf{C} \) is a constant, its rate of change with respect to \( t \) is 0.

Step 4: Result Conclusion

Conclusively, since the derivative of a constant is 0, we have \( \frac{d \mathbf{u}}{dt} = 0 \). Thus, the proof is complete.

Key Concepts

Vector Calculus

Vector calculus is a field of mathematics that deals with vector fields and differential operators. Vectors are quantities that have both magnitude and direction. These can represent physical quantities like velocity or force. In vector calculus, we often work with vector functions, which assign a vector to each point in space.

• Vector fields: These are assignments of a vector to each point in a region of space.
• Vector functions: These describe how vectors change over time or space, such as a moving particle's velocity.

A critical part of vector calculus involves using operations like differentiation to analyze these changes. Understanding the behavior of vector functions is crucial in physics and engineering, as it helps in modeling the dynamics of systems.

Derivative

The derivative is a fundamental concept in calculus that represents the rate of change of a quantity. In simple terms, it allows us to see how a function changes as its input changes. For vector functions, taking a derivative helps us understand how the vector's components change.

• Constant function derivative: If a function is constant, its derivative is zero because there is no change to measure.
• Mathematical notation: The derivative of a vector function \( \mathbf{u}(t) = \mathbf{C} \) is represented as \( \frac{d \mathbf{u}}{dt} \).

In the context of the exercise, we are dealing with a constant vector function, meaning it does not "move" or change as time progresses. Therefore, its derivative is zero, reaffirming the key property that a constant does not change.

Rate of Change

The rate of change is a way to describe how one quantity changes relative to another. It's a crucial idea in mathematics especially when dealing with functions and their derivatives. Through the concept of a derivative, the rate of change can be quantified.

• Constant rate: For a constant function, the rate of change over time is zero.
• Practical understanding: Think of driving at a constant speed. The speed doesn't change, indicating a constant rate of movement, so your acceleration, or rate of change of speed, is zero.
• Application to vectors: If a vector function like \( \mathbf{u}(t) \) is constant, its components don't vary over time, leading to a derivative of zero.

This means when you hear "constant function," you can immediately think of zero change over time, which translates mathematically to a zero derivative. Understanding this helps simplify problems involving constant functions across different scientific domains.