Glossary
Acceleration (vector-valued function)
The first derivative of the velocity vector-valued function, a(t) = v'(t) = r''(t) = <x''(t), y''(t)>, representing the instantaneous rate of change of velocity.
Example:
If a car's velocity is v(t) = <2t, 5>, its acceleration vector is a(t) = <2, 0>, meaning it's only accelerating in the x-direction.
Angle (theta θ)
In polar coordinates, 'θ' represents the angle measured counterclockwise from the positive x-axis to the line connecting the point to the origin.
Example:
For the polar point (2, π/2), the angle (theta θ) is π/2 radians, placing the point on the positive y-axis.
Arc length (of parametric curve)
The total distance along a curve defined by parametric equations over a given interval of the parameter 't'. It is found by integrating the square root of (dx/dt)² + (dy/dt)² with respect to 't'.
Example:
To find the arc length of a spiral defined by x(t) = t cos(t) and y(t) = t sin(t) from t=0 to t=2π, you would use the integral formula involving the derivatives of x(t) and y(t).
Area (of a polar region/single polar curve)
The area enclosed by a single polar curve r = f(θ) from θ=a to θ=b, calculated by the integral (1/2) ∫[a,b] r² dθ.
Example:
To find the area of a circle with radius 2 defined by r = 2, you would integrate (1/2) ∫[0,2π] (2)² dθ.
Cartesian plane
A two-dimensional coordinate system (ℝ^2 or xy-plane) that uses two perpendicular number lines, the x-axis and y-axis, to locate points.
Example:
Plotting the point (3, 4) involves moving 3 units right along the x-axis and 4 units up along the y-axis on the Cartesian plane.
Chain rule (in polar context)
Used to find the derivative dy/dx for polar functions by treating x and y as parametric equations in terms of θ, applying dy/dx = (dy/dθ)/(dx/dθ).
Example:
When differentiating r = sin(2θ) to find dy/dx, the chain rule is essential for correctly calculating d(r sin θ)/dθ and d(r cos θ)/dθ.
Closed curves (in polar context)
Polar curves that start and end at the same point, forming a complete loop, which is a common condition for applying the standard area formula.
Example:
A cardioid, like r = 1 + cos(θ), is a closed curve that completes one loop as θ goes from 0 to 2π.
Components (of a vector-valued function)
The individual scalar functions (e.g., f(t) and g(t)) that make up a vector-valued function, defining its behavior along each axis.
Example:
In the vector-valued function r(t) = <e^t, cos(t)>, e^t and cos(t) are the components that define the x and y positions, respectively.
Convert (between polar and Cartesian)
The process of transforming coordinates or equations from polar form (r, θ) to Cartesian form (x, y) or vice versa, using relations like x = r cos θ, y = r sin θ, and r² = x² + y².
Example:
To convert the polar point (√2, π/4) to Cartesian, you'd use x = √2 cos(π/4) = 1 and y = √2 sin(π/4) = 1, resulting in (1, 1).
Dependent variables
Variables whose values are determined by the value of one or more other variables. In parametric equations, x and y are dependent variables.
Example:
In the parametric equations x = t² and y = 2t, both x and y are dependent variables because their values depend on the value of 't'.
Derivative (of parametric function)
The slope of the tangent line to a parametric curve, found by dividing dy/dt by dx/dt (dy/dx = (dy/dt)/(dx/dt)).
Example:
To find the instantaneous velocity of a particle moving along x(t) = t³ and y(t) = t² at t=1, you'd calculate the derivative dy/dx = (2t)/(3t²) = 2/(3t), which at t=1 is 2/3.
Derivative (of polar function)
The slope of the tangent line to a polar curve (dy/dx), derived using the chain rule after converting x and y to functions of θ (x = r cos θ, y = r sin θ).
Example:
To find the horizontal tangents of a cardioid r = 1 - cos(θ), you would set the numerator of the derivative dy/dx to zero.
Distance from the origin (r)
In polar coordinates, 'r' represents the radial distance of a point from the pole (origin).
Example:
For the polar point (5, π/4), the distance from the origin (r) is 5 units.
Independent variable
A variable whose value does not depend on that of another. In parametric equations, the parameter 't' (time) is the independent variable.
Example:
When modeling the motion of a car with parametric equations, 't' (time) is the independent variable that dictates the car's position (x, y).
Inner curve
In the context of finding the area between two polar curves, the *inner curve* is the curve closer to the origin within the region of interest.
Example:
When finding the area between r = 1 and r = 2 + cos(θ), r = 1 would be the inner curve in the region where the cardioid extends beyond the circle.
Integration (of vector-valued functions)
The process of finding an antiderivative of a vector-valued function by integrating each of its components individually.
Example:
To find the displacement from a velocity vector v(t) = <2t, 3t²>, you would perform integration on each component to get s(t) = <t² + C₁, t³ + C₂>.
Outer curve
In the context of finding the area between two polar curves, the *outer curve* is the curve farther from the origin within the region of interest.
Example:
When finding the area between r = 1 and r = 2 + cos(θ), r = 2 + cos(θ) would be the outer curve in the region where the cardioid extends beyond the circle.
Parametric functions
A set of equations that define the x and y coordinates of a point as functions of a third independent variable, typically 't' (time). They are useful for describing motion or curves that fail the vertical line test.
Example:
The flight path of a projectile can be described by parametric functions like x(t) = (v₀ cos θ)t and y(t) = (v₀ sin θ)t - (1/2)gt², where 't' is time.
Polar coordinates (r, θ)
An ordered pair (r, θ) that specifies the position of a point in the polar plane using its distance from the origin (r) and its angle from the positive x-axis (θ).
Example:
The point (3, π/6) represents a location 3 units from the origin at an angle of π/6 radians in polar coordinates.
Polar function
A function of the form r = f(θ), where the radial distance 'r' is expressed as a function of the angle 'θ'.
Example:
The equation r = 2 cos(3θ) describes a rose curve, which is a type of polar function.
Polar plane
A two-dimensional coordinate system where points are located by their distance from the origin (r) and the angle (θ) they make with the positive x-axis.
Example:
Graphing a cardioid like r = 1 + cos(θ) is done on a polar plane, where 'r' represents distance from the center and 'θ' represents the angle.
Position (vector-valued function)
A vector-valued function, r(t) = <x(t), y(t)>, that describes the location of an object in space at any given time 't'.
Example:
If a particle's position is given by r(t) = <t² + 1, sin(t)>, then at t=0, its position is <1, 0>.
Second derivative (of parametric function)
Measures the concavity of a parametric curve, calculated as the derivative of (dy/dx) with respect to 't', divided by dx/dt.
Example:
If dy/dx = sin(t)/cos(t), finding the second derivative d²y/dx² involves differentiating sin(t)/cos(t) with respect to 't' and then dividing by dx/dt.
Two polar curves (area between)
The area of the region bounded by two polar curves, R (outer curve) and r (inner curve), calculated by the integral (1/2) ∫[a,b] (R² - r²) dθ.
Example:
To find the area between two polar curves, such as a circle and a cardioid, you subtract the square of the inner curve's radius from the square of the outer curve's radius before integrating.
Vector-valued function
A function that maps a real number (often 't') to a vector, typically represented as r(t) = <f(t), g(t)> or f(t)i + g(t)j.
Example:
The position of a drone in flight can be modeled by a vector-valued function r(t) = <3t, 5t - 0.5t²>, where 't' is time.
Velocity (vector-valued function)
The first derivative of the position vector-valued function, v(t) = r'(t) = <x'(t), y'(t)>, representing the instantaneous rate of change of position.
Example:
If a rocket's position is r(t) = <t³, 4t>, its velocity vector is v(t) = <3t², 4>, indicating its speed and direction at any time 't'.