Problem 44 - Thomas Calculus

Question

Prove the Sum and Difference Rules for vector functions.

Step-by-Step Solution

Answer

The derivative of the sum is the sum of their derivatives, and similarly, the derivative of the difference is the difference of their derivatives.

Step 1: Understanding Vector Functions

A vector function is a function that takes one or more variables, like time, and outputs a vector. The output vector depends continuously on the input variables. For two vector functions \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \), the components of these vectors can be treated with calculus rules similar to scalar functions.

Step 2: State the Sum Rule

The Sum Rule for vector functions states that the derivative of the sum of two vector functions is equal to the sum of the derivatives of the individual functions. Mathematically, it is expressed as: \[ \frac{d}{dt} ( \mathbf{u}(t) + \mathbf{v}(t) ) = \frac{d}{dt} \mathbf{u}(t) + \frac{d}{dt} \mathbf{v}(t) \].

Step 3: Prove the Sum Rule

Assume \( \mathbf{u}(t) = [u_1(t), u_2(t), ..., u_n(t)] \) and \( \mathbf{v}(t) = [v_1(t), v_2(t), ..., v_n(t)] \). Then \( \mathbf{u}(t) + \mathbf{v}(t) = [u_1(t)+v_1(t), u_2(t)+v_2(t), ..., u_n(t)+v_n(t)] \). Derivative of each component gives: \( \frac{d}{dt} (u_i(t) + v_i(t)) = \frac{d}{dt} u_i(t) + \frac{d}{dt} v_i(t) \) by the sum rule of derivatives for scalars. Apply it to all components, confirming the vector form.

Step 4: State the Difference Rule

The Difference Rule for vector functions states that the derivative of the difference of two vector functions is equal to the difference of the derivatives of the individual functions. Mathematically, it is expressed as: \[ \frac{d}{dt} ( \mathbf{u}(t) - \mathbf{v}(t) ) = \frac{d}{dt} \mathbf{u}(t) - \frac{d}{dt} \mathbf{v}(t) \].

Step 5: Prove the Difference Rule

Using the same vector functions \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \) as before, \( \mathbf{u}(t) - \mathbf{v}(t) = [u_1(t)-v_1(t), u_2(t)-v_2(t), ..., u_n(t)-v_n(t)] \). The derivative of each component is \( \frac{d}{dt} (u_i(t) - v_i(t)) = \frac{d}{dt} u_i(t) - \frac{d}{dt} v_i(t) \), again using the differentiation rule for scalars component-wise.

Key Concepts

Sum Rule

The Sum Rule in vector calculus is a fundamental principle that simplifies the differentiation of sums of vector functions. Imagine you have two vector functions, \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \), each carrying their own set of components. The Sum Rule states that the derivative of their sum is simply the sum of their derivatives.

This means instead of dealing with the entire vector function at once, we can break it down into manageable pieces. Each component of the vector is differentiated separately, and we add these results together.

• If \( \mathbf{u}(t) \) is a vector with components \([u_1(t), u_2(t), ..., u_n(t)]\), and \( \mathbf{v}(t) \) has \([v_1(t), v_2(t), ..., v_n(t)]\), the sum is \( \mathbf{u}(t) + \mathbf{v}(t) = [u_1(t)+v_1(t), u_2(t)+v_2(t), ..., u_n(t)+v_n(t)] \).
• Derivatively speaking, it's expressed as: \[ \frac{d}{dt} ( \mathbf{u}(t) + \mathbf{v}(t) ) = \frac{d}{dt} \mathbf{u}(t) + \frac{d}{dt} \mathbf{v}(t) \].

Using this rule simplifies complex tasks in vector calculus, making it crucial for calculations involving multiple vector fields.

Difference Rule

The Difference Rule complements the Sum Rule by addressing the differentiation of vector function differences. For two vector functions \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \), the Difference Rule states that the derivative of their difference is simply the difference of their derivatives.

Just as with the Sum Rule, this rule allows us to separate concerns. Each component of the vector difference is treated individually, making calculus with vectors a lot like handling numbers.

• This is mathematically represented as: \[ \frac{d}{dt} ( \mathbf{u}(t) - \mathbf{v}(t) ) = \frac{d}{dt} \mathbf{u}(t) - \frac{d}{dt} \mathbf{v}(t) \].
• For \( \mathbf{u}(t) = [u_1(t), u_2(t), ..., u_n(t)] \) and \( \mathbf{v}(t) = [v_1(t), v_2(t), ..., v_n(t)] \), the difference becomes \([u_1(t)-v_1(t), u_2(t)-v_2(t), ..., u_n(t)-v_n(t)]\).

The Difference Rule is particularly useful in areas like physics and engineering, where subtracting vector quantities is common.

Vector Functions

Vector functions play a critical role in calculus, especially when dealing with multiple dimensions or directions. Unlike scalar functions, which output a single value, vector functions output a vector. This vector consists of several components, each possibly dependent on variables like time.

Understanding vector functions is essential for interpreting physical phenomena, such as:

• Describing the trajectory of a moving object, where position could be given as a function of time.
• Representing the electric or magnetic field in a region, where each field component depends on spatial coordinates.

Consider a vector function \( \mathbf{r}(t) = [x(t), y(t), z(t)] \), which might describe the position of a particle in space. Here, \( x(t), y(t), \) and \( z(t) \) are the scalar functions of time, providing coordinates in a three-dimensional space.

Derivative

In calculus, derivatives are all about change. They measure how a function changes as its input changes. For vector functions, derivatives are a bit more complex than for scalars, because we deal with changes in magnitude and direction.

Think of a vector function \( \mathbf{r}(t) = [x(t), y(t), z(t)] \). The derivative \( \frac{d}{dt} \mathbf{r}(t) \) provides a new vector that indicates the rate of change of the position, showing not just how fast something moves but also where it's heading.

The derivative of a vector function encompasses:

• Direction: Indicating where the values of the function are heading.
• Magnitude: Showing the speed of the change.

This makes derivatives crucial in modeling motion for real-world applications, like physics and engineering, where understanding both the direction and velocity is vital.

Calculus Rules

In calculus, there are several fundamental rules that facilitate the differentiation and integration of functions. For vector functions, these rules provide the foundation for manipulating multiple variables.

Some basic rules to keep in mind include:

• Sum and Difference Rules: For adding and subtracting derivatives.
• Product Rule: For differentiating products of two functions.
• Quotient Rule: For differentiating ratios of functions.

These rules are not just random formulas—they are logical consequences of how we have defined derivatives. When dealing with vector functions, knowing and applying these rules helps solve complex problems efficiently. With these tools, students can tackle challenging calculus problems with greater ease.