Wk1 reader math1011
1. Vector-Valued Functions and Functions of Several Variables
Scalar functions of one variable assign a real number to each real number in a subset of the real line.
Example: f(x) = x² where x is a real number.
In some fields like science, engineering, and economics, scalar functions are insufficient.
Example: Meteorology may require a function for air pressure that varies with latitude, longitude, and time.
Problems may involve:
One independent variable affecting multiple dependent variables (e.g., electromagnetic field components).
This can be represented as a vector with components correlating to changes in each variable.
Knowledge of how these functions change requires calculus, which can be extended from scalar to these new functions.
1.1 The Vector Space Rⁿ
Rⁿ consists of all n-tuples of real numbers, defined as Rⁿ = {(x₁, x₂, ..., xₙ) | x₁, x₂, ..., xₙ ∈ R}.
R² represents pairs of real numbers (x, y) visualized in the Cartesian plane.
R³ represents triples of real numbers identified with three-dimensional space.
1.2 Vector-Valued Functions
Vector-valued functions take a real number and return a vector in Rⁿ, typically R² or R³.
Formally:
r: D → R³, t ↦ r(t) = (f(t), g(t), h(t)).
Functions f(t), g(t), h(t) are coordinate functions.
Example: t could represent time with r(t) describing an object's position in space.
As t changes in D, r(t) traces a curve C in R³ representing the object's path.
Example function: r(t) = (sin(t), 2cos(t), t/10) defines a spatial curve.
C is given by parametric equations:
C: x=f(t), y=g(t), z=h(t) for t in D.
Remark: Any curve C has infinitely many parameterizations.
1.3 Functions of Two Variables
Two variables x and y can be represented as the vector (x, y) in R².
A function of two variables defines:
f: D → R, (x, y) ↦ f(x, y).
Example: Height above sea level can be determined by coordinates (x, y).
The graph of f maps to a surface in R³:
graph(f) = {(x, y, f(x, y)) | (x, y) ∈ D}.
1.4 Summary of Functions
Types of functions discussed:
Scalar-valued functions of one variable (f: D → R).
Vector-valued functions (f: D → Rⁿ, n ≥ 2).
Functions of several variables (f: D → R, n ≥ 2).
High school calculus primarily covers Type (1).
2. Limits and Continuity
2.1 Scalar-Valued Functions
Limits are crucial for understanding scalar-valued functions of one variable.
Definitions of limit include:
Open interval (a, b) includes real numbers x such that a < x < b.
Closed interval [a, b] includes real numbers x such that a ≤ x ≤ b.
Definition of limit:
f converges to L as x approaches a if f(x) can be made arbitrarily close to L as x approaches a.
Example of Limit: lim x→1 f(x) = f(1) for f(x) = x².
Limits can also be one-sided:
lim x→a⁻ f(x) and lim x→a⁺ f(x).
2.2 The Limit Laws
The limit laws allow combining limits to determine new limits from known functions:
lim x→a (f(x) ± g(x)) = lim x→a f(x) ± lim x→a g(x);
For constant c: lim x→a (c · f(x)) = c · lim x→a f(x);
lim x→a (f(x) · g(x)) = lim x→a f(x) · lim x→a g(x) if lim x→a g(x) ≠ 0.
2.3 Limits of Vector-Valued Functions
For a vector-valued function r(t) = (f(t), g(t), h(t)):
Definition: lim t→a r(t) = (lim t→a f(t), lim t→a g(t), lim t→a h(t)).
2.4 Continuity
Continuity means that small changes in input result in small changes in output.
A function is continuous at a if:
lim x→a f(x) = f(a).
A function is continuous throughout its domain if it is continuous at all points in its domain.