Polar Curves, Arc Length, Areas, and Functions of Several Variables — Lecture Notes
Polar Curves: Rose Curves, Cardioids, and Limacons
Rose curves have equations of the form r = a\cos(n\theta) or r = a\sin(n\theta).
If n is odd, there are n petals.
If n is even, there are 2n petals.
Intuition: cos or sin traces a circle, but the factor n compresses the angular travel, producing multiple petals instead of a full circle.
Cardioids are a family given by r = a \pm b\cos\theta\quad\text{or}\quad r = a \pm b\sin\theta.
When the ratio \frac{a}{b} = 1, the curve is a cardioid.
They have a notch at the origin because there exists a theta with r(\theta) = 0.
Symmetry: typically symmetric about an axis (depending on cosine vs sine forms).
Limacons generalize cardioids: r = a \pm b\cos\theta\quad\text{or}\quad r = a \pm b\sin\theta.
Shape determined by the ratio \frac{a}{b}:
\frac{a}{b} < 1: inner loop present.
1 \le \frac{a}{b} < 2: dimple.
\frac{a}{b} \ge 2: convex (no loop or dimple).
Symmetry and plotting notes:
Graphs often exhibit axis symmetry; checking symmetry helps sketch them more efficiently.
Polars curves can be easier to plot by considering symmetries than by brute plotting.
Why these shapes matter: polar curves appear naturally where there is radial or spherical symmetry, or near-symmetric systems with perturbations (e.g., perturbed orbital trajectories).
Applications: orbital mechanics, acoustics, optics, electric and gravitational fields.
Connection to other math concepts:
Polar curves can be treated as special cases of parametric curves, with the parameter being (\theta).
This view lets us compute areas and arc lengths using polar-specific formulas.
Arc Length in Polar Coordinates: setup and example
View a polar curve as a parametric curve with parameter (\theta):
x(\theta) = r(\theta)\cos\theta,
\quad y(\theta) = r(\theta)\sin\theta,
with (t = \theta).
Parametric arc length formula:
L = \int_a^b \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2}\,d\theta.
Derivatives:
\frac{dx}{d\theta} = \frac{dr}{d\theta}\cos\theta - r\sin\theta,
\frac{dy}{d\theta} = \frac{dr}{d\theta}\sin\theta + r\cos\theta.
After simplification (using (\cos^2\theta + \sin^2\theta = 1)), the velocity magnitude becomes
\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2 = \left(\frac{dr}{d\theta}\right)^2 + r^2.
Polar arc length formula (special case):
L = \int_a^b \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r^2 }\, d\theta.
Example: cardioid (r(\theta) = 1 + \sin\theta), full arc (\theta in [0,2\pi])
( \frac{dr}{d\theta} = \cos\theta ) and (r^2 = (1+\sin\theta)^2).
Integrand becomes (\sqrt{\cos^2\theta + (1+\sin\theta)^2} = \sqrt{2 + 2\sin\theta}).
To evaluate, multiply by the conjugate inside the square root to use a difference-of-squares:
Consider (\sqrt{2+2\sin\theta}\cdot\sqrt{2-2\sin\theta} = \sqrt{(2+2\sin\theta)(2-2\sin\theta)} = \sqrt{4 - 4\sin^2\theta} = 2|\cos\theta|.
Thus, treat the integral piecewise on intervals where (\cos\theta) is nonnegative vs negative.
After evaluation, the length is
L = 8.
Practical note: the absolute value from (|\cos\theta|) requires splitting the integral where cosine changes sign; many problems are solved with a similar conjugate-trick to simplify the square root expression.
Areas in Polar Coordinates: theory and how to compute
Start from the Riemann-sum idea: approximate area by summing small pieces.
In polar coordinates, instead of rectangles we use polar rectangles (wedges):
Each wedge has angular width (\Delta\theta) and radial extent from 0 to (r(\theta^)) for some sample angle (\theta^).
The area of a polar rectangle is roughly the area of a sector of radius (r) with angle (\Delta\theta):
\Delta A \approx \frac{1}{2} [r(\theta^*)]^2 \Delta\theta.
Summing wedges and taking the limit yields the polar area formula:
A = \int_a^b \frac{1}{2} r(\theta)^2 \; d\theta.
Check with a circle: if (r(\theta) = R) is constant and we integrate from (0) to (2\pi), we get
A = \int_0^{2\pi} \frac{1}{2} R^2 \; d\theta = \pi R^2,
which matches the standard circle area.
Example: area between two polar curves (purple region in the lecture): circle (r = 3\sin\theta) and cardioid (r = 1 + \sin\theta)
First find intersections: solve (3\sin\theta = 1 + \sin\theta) → (\sin\theta = \tfrac{1}{2}) → (\theta = \tfrac{\pi}{6},\; \tfrac{5\pi}{6}).
On ([\pi/6, 5\pi/6]), the circle is the outer boundary and the cardioid is the inner boundary for the region considered.
Set up the area as the difference of the two polar-area integrals:
A = \int_{\pi/6}^{5\pi/6} \frac{1}{2} \left[ (3\sin\theta)^2 - (1+\sin\theta)^2 \right] d\theta.Simplify the integrand:
\frac{1}{2} \left[ 9\sin^2\theta - (1 + 2\sin\theta + \sin^2\theta) \right] = \frac{1}{2} \left[ 8\sin^2\theta - 2\sin\theta - 1 \right].Use the identity (\sin^2\theta = \frac{1 - \cos(2\theta)}{2}) to integrate and obtain the final area
A = \pi.
Important caution: when computing areas between curves, ensure you are integrating the correct region and handle any sign changes or multiple intersections carefully.
Functions of Several Variables: Domain, Range, and Graphs
From one-variable functions to two-variable functions:
A function of two variables is usually written as f: \mathbb{R}^2 \to \mathbb{R}. Inputs are pairs ((x,y) \in \mathbb{R}^2).
The domain is the set of admissible input pairs; the range is the set of possible outputs in (\mathbb{R}).
Examples of explicit two-variable functions and their domains:
Example 1: f(x,y) = \frac{\sqrt{x+y+1}}{x-1}.
Domain constraints:
Denominator cannot be zero: (x \neq 1).
Under the square root: (x+y+1 \ge 0) ⇒ (y \ge -x - 1).
Domain is the region above the line (y = -x - 1), excluding the line (x=1).
Example 2: g(x,y) = x \ln(y^2 - x).
Domain constraints:
The argument of the natural log must be positive: (y^2 - x > 0) ⇒ (x < y^2).
Domain is the region to the left of the curve (x = y^2) (a sideways parabola).
Graph of a two-variable function:
For a function (f: \mathbb{R}^2 \to \mathbb{R}), the graph is the set of triples
\(x, y, z)) with (z = f(x,y)) and ((x,y)) in the domain.
This graph is a surface in (\mathbb{R}^3) whenever the domain has nonempty interior (i.e., contains a nonempty open set).
If the domain collapses to a curve or a point, the graph is not a surface.
Notation and generalization:
In higher dimensions, a function can be written as
f: \mathbb{R}^n \to \mathbb{R},with domain contained in (\mathbb{R}^n) and range a subset of (\mathbb{R}).
The same ideas (domain, range, graph) extend naturally to (n) variables.
Takeaways:
The domain is defined by where the expression makes sense (no division by zero, no invalid roots, log arguments positive, etc.).
The graph in higher dimensions is a natural generalization of a curve to a surface (or a higher-dimensional hypersurface).
Connecting to the Exam and Course Structure
The midterm is next week (Monday); no class or workshops next week; the quiz is pushed to Thursday morning.
The mock exam question coverage: topics that were quantitatively covered in lecture today are fair game; topics not covered by end of Week 4 may not be tested.
The course direction: after polar coordinates, the course moves to functions of several variables, marking a turning point to higher-dimensional thinking and new visualization techniques.
Quick Reference: Key Formulas (for easy review)
Polar to Cartesian conversion:
x = r(\theta) \cos\theta, \quad y = r(\theta) \sin\theta.
Parametric arc length (general):
L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt.
Polar arc length (special case):
L = \int_a^b \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r^2}\, d\theta.
Cardioid example length (summary):
For (r(\theta) = 1 + \sin\theta), (L = \int_0^{2\pi} \sqrt{2 + 2\sin\theta}\, d\theta = 8.)
Area in polar coordinates:
A = \int_a^b \frac{1}{2} r(\theta)^2 \, d\theta.
Area between polar curves example:
Intersections from solving r1(\theta) = r2(\theta); for (r1 = 3\sin\theta) and (r2 = 1 + \sin\theta), intersections at \theta = \frac{\pi}{6}, \frac{5\pi}{6}.
Area: A = \int_{\pi/6}^{5\pi/6} \frac{1}{2} \left[ (3\sin\theta)^2 - (1 + \sin\theta)^2 \right] d\theta = \pi.
Two-variable function domain examples (summary):
(f(x,y) = \frac{\sqrt{x+y+1}}{x-1}) with domain (x \neq 1) and (y \ge -x - 1).
(g(x,y) = x \ln(y^2 - x)) with domain (y^2 - x > 0) (i.e., (x < y^2)).
Graph of a two-variable function: surface in (\mathbb{R}^3) when the domain contains an open set; otherwise, the graph may fail to be a surface.
Generalization: for (n) variables, (f: \mathbb{R}^n \to \mathbb{R}) with domain in (\mathbb{R}^n) and graph in (\mathbb{R}^{n+1}).