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):

  • 
    $x(\theta) = r(\theta)\cos\theta,<br>\quad y(\theta) = r(\theta)\sin\theta,$
    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 $r<em>1(\theta) = r</em>2(\theta)$; 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}).