Problem 2 - Active Calculus Multivariable

Question

Find a parametrization of the circle of radius 6 in the xy-plane, centered at the origin, oriented clockwise. The point (6,0) should correspond to \(t=0 .\) Use \(t\) as the parameter for all of your answers. \(x(t)=\)_________ \(y(t)=\)_________

Step-by-Step Solution

Answer

The parametrization of the circle of radius 6 in the xy-plane, centered at the origin, oriented clockwise, and with the point (6,0) corresponding to \(t=0\) is: \(x(t) = 6\cos(-t)\) \(y(t) = 6\sin(-t)\)

Step 1: Write the equation of circle centered at origin

The equation of a circle with radius 6 and centered at (0,0) in xy-plane is: \(x^2 + y^2 = 36\)

Step 2: Parametrize x(t) and y(t) using trigonometric functions and clockwise orientation

To consider clockwise orientation and the starting point (6,0) at t = 0, we have: \(x(t) = 6\cos(-t)\) \(y(t) = 6\sin(-t)\)

Step 3: Provide the final parametrization

The final parametrization for the circle of radius 6 in the xy-plane, centered at the origin and oriented clockwise is: \(x(t) = 6\cos(-t)\) \(y(t) = 6\sin(-t)\) This parametrization satisfies the properties of the circle described in the exercise (radius of 6, centered at the origin, clockwise orientation starting at the point (6,0) for t = 0).

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are essential mathematical tools that relate the angles of a triangle to the lengths of its sides. In the context of parametrizing a circle, sine and cosine functions play a pivotal role. The fundamental trigonometric functions are sine (\text{sin}), cosine (\text{cos}), and tangent (\text{tan}), each of which evaluates to a specific ratio of a right-angled triangle's sides.

For a circle of radius r, the parametric equations using trigonometric functions for x and y coordinates can be expressed as:
\[ x(t) = r\cos(t) \]\[ y(t) = r\sin(t) \]
where t is the parameter, typically representing the angle in radians from the positive x-axis to the radius vector of the point on the circle. To adjust for the clockwise direction, we use negative t in the functions, resulting in the parametrization: \( x(t) = r\cos(-t) \) and \( y(t) = r\sin(-t) \). This change in sign dictates the direction of motion along the circle, countering the default counterclockwise movement.

Polar Coordinates and Their Relation to Circles

Polar coordinates offer a different way to represent points in the plane, as opposed to the traditional Cartesian coordinates. A point in polar coordinates is given by a pair (r, θ), where r is the radial distance from the origin, and θ is the angle formed with the positive x-axis.

In the case of a circle, polar coordinates simplify the representation. Every point on the circle has the same radial distance r, equal to the circle's radius. Thus, a circle's equation in polar form is simply r = constant. For our example, where the radius is 6, any point on the circle has polar coordinates (6, θ).

To convert polar coordinates to Cartesian coordinates, the equations \( x = r\cos(θ) \) and \( y = r\sin(θ) \) are used. When parametrizing a circle in the xy-plane, like in our exercise, we essentially use the concepts of polar coordinates and convert them into parametric equations with a trigonometric twist to include the orientation and starting point.

Multivariable Calculus' Role in Parametrization

Multivariable calculus extends the concepts of calculus to functions with more than one variable. In terms of parametrization, this field of study is vital because it allows us to describe curves and surfaces not just in two dimensions, but in three dimensions and beyond.

Parametric equations, such as those for a circle, are one of the fundamental topics in multivariable calculus. These equations express the coordinates of the points of a curve as functions of one or more parameters. Instead of representing a circle as an equation in x and y, we describe x and y separately in terms of a third variable, t, which is the parameter.

The importance of multivariable calculus in this context lies in its ability to analyze and understand the properties of these curves. For example, we can use calculus tools to find the tangent line at a point on the circle or to determine the exact length of the curve defined by these parametric equations. In any field that requires precise motion along paths, such as physics or engineering, understanding the principles of multivariable calculus and parametrization is critical.