Problem 69 - Essential Calculus Early Transcendentals

Question

Suppose \(\mathbf{u}\) and \(\mathbf{v}\) are vector functions that possess limits as \(t \rightarrow a\) and let \(c\) be a constant. Prove the following properties of limits. (a) $$\lim _{t \rightarrow a}[\mathbf{u}(t)+\mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t)+\lim _{t \rightarrow a} \mathbf{v}(t)$$ (b) $$\lim _{r \rightarrow a} c \mathbf{u}(t)=c \lim _{t \rightarrow a} \mathbf{u}(t)$$ (c) $$\lim _{t \rightarrow a}[\mathbf{u}(t) \cdot \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \cdot \lim _{t \rightarrow a} \mathbf{v}(t)$$ (d) $$\lim _{t \rightarrow a}[\mathbf{u}(t) \times \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \times \lim _{t \rightarrow a} \mathbf{v}(t)$$

Step-by-Step Solution

Answer

Limits of vector sum, scalar multiplication, dot, and cross product align with their algebraic counterparts.

Step 1: Vector Addition Limit Property

Given vectors \(\mathbf{u}(t)\) and \(\mathbf{v}(t)\), both with limits as \(t \to a\), we aim to prove that \(\lim_{t \to a} [\mathbf{u}(t) + \mathbf{v}(t)] = \lim_{t \to a} \mathbf{u}(t) + \lim_{t \to a} \mathbf{v}(t)\). By definition, \[\lim_{t \to a} [\mathbf{u}(t) + \mathbf{v}(t)] = \lim_{t \to a} \mathbf{u}(t) + \lim_{t \to a} \mathbf{v}(t)\]. The properties of limits for real-valued functions apply here for vector components. Since limits apply component-wise, for vectors \(\mathbf{u}(t) = [u_1(t), u_2(t), u_3(t)]\) and \(\mathbf{v}(t) = [v_1(t), v_2(t), v_3(t)]\), the limit sought is simply the addition of individual limits of each corresponding component. Hence, the limit of the sum is equal to the sum of the limits.

Step 2: Scalar Multiplication Limit Property

For a constant \(c\), we need to show \(\lim_{t \to a} c \mathbf{u}(t) = c \lim_{t \to a} \mathbf{u}(t)\).Scalar multiplication by \(c\) is linear, implying that limits interact with scalar multiplication directly. Thus, \[\lim_{t \to a} c \mathbf{u}(t) = [\lim_{t \to a} (c u_1(t)), \lim_{t \to a} (c u_2(t)), \lim_{t \to a} (c u_3(t))]\], which simplifies by factorization, employing limit properties, to \[c [\lim_{t \to a} u_1(t), \lim_{t \to a} u_2(t), \lim_{t \to a} u_3(t)] = c \lim_{t \to a} \mathbf{u}(t)\].

Key Concepts

Vector Functions

Vector functions are mathematical descriptions used to represent quantities that have both magnitude and direction. Unlike scalar functions, which deal with only one-dimensional quantities, vector functions take a point in one space and map it to a vector in another. These are often denoted in terms of a parameter, typically expressed as a function of time, such as \( \mathbf{u}(t) = [u_1(t), u_2(t), u_3(t)] \). Here, each component \( u_i(t) \) is a function of \( t \), capturing the behavior of the respective dimension over this parameter.

• Vector functions can describe physical quantities like velocity, force, and displacement that naturally require direction and magnitude.
• The dimensionality of the vector depends on the number of components it possesses, which is usually pertinent to the spatial dimensions involved.

Understanding vector functions is crucial because it allows us to model real-world phenomena accurately, using mathematics to predict changes and interactions in a given system.

Limits of Vector Functions

The concept of limits in calculus can be extended beyond scalar functions to vector functions as well. When describing the limit of a vector function \( \mathbf{u}(t) \) as \( t \to a \), we mean that each component of the vector approaches a specific value. Essentially, as \( t \) approaches \( a \), the vector \( \mathbf{u}(t) \) closely approaches the vector \( \mathbf{L} \), written as \( \lim_{t \to a} \mathbf{u}(t) = \mathbf{L} \).

• This concept is crucial as it helps ensure continuity and predictability of vector functions in applied problems.
• Each component follows traditional limit rules, allowing comprehensive analysis using known calculus techniques for each coordinate value.

The properties of limits allow further operations, such as adding vector limits or multiplying them by scalars. Mastery of vector limits enables one to extend calculus results to numerous scientific fields such as physics and engineering.

Limit Properties for Vectors

The limit properties for vector functions are analogous to those for scalar limits. They provide essential tools for calculating limits involving vector operations. These properties make it easier to manage complex vector limits by reducing them to the limits of their components.

• Sum of Limits: If \( \lim_{t \to a} \mathbf{u}(t) \) and \( \lim_{t \to a} \mathbf{v}(t) \) exist, then \( \lim_{t \to a} [\mathbf{u}(t) + \mathbf{v}(t)] = \lim_{t \to a} \mathbf{u}(t) + \lim_{t \to a} \mathbf{v}(t) \).
• Product with Scalars: For a constant \( c \), \( \lim_{t \to a} [c \mathbf{u}(t)] = c \lim_{t \to a} \mathbf{u}(t) \).
• These properties hold true because limits distribute over addition and scalar multiplication.

Efficient use of these properties simplifies complex calculations by handling higher-dimensional analysis in a straightforward manner through the component-wise evaluation.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar. When applied to vector functions, this concept means multiplying every component function by the same constant. For instance, given a vector function \( \mathbf{u}(t) \), and a scalar \( c \), the multiplication results in \( c \mathbf{u}(t) = [c u_1(t), c u_2(t), c u_3(t)] \).

• This process scales the vector's magnitude without changing its direction if \( c > 0 \), or it reverses the direction if \( c < 0 \).
• Scalar multiplication is both distributed over vector addition and associative.

Understanding scalar multiplication within vector calculus is fundamental, as it allows manipulation of vectors in a simple and predictable manner, helping in modeling and solving real-world problems such as scaling forces, velocities, and more.