Polar coordinates

Definition

Polar coordinates are a two-dimensional coordinate system that specifies each point on a plane by a distance from a reference point and an angle from a reference direction. This system uses the radial distance from the origin and the angle measured from the positive x-axis, allowing for a more intuitive representation of circular and rotational motion. Polar coordinates are particularly useful in multidimensional integration, especially when dealing with functions that exhibit radial symmetry or when integrating over circular regions.

5 Must Know Facts For Your Next Test

1. In polar coordinates, each point is represented by an ordered pair (r, θ), where r is the radial distance from the origin and θ is the angle from the positive x-axis.
2. The conversion between polar and Cartesian coordinates can be done using the equations: $$x = r \cos(\theta)$$ and $$y = r \sin(\theta)$$.
3. Polar coordinates simplify the process of integrating functions over circular regions, as it aligns better with the symmetry of these shapes.
4. When using polar coordinates, the area element in integration changes from $$dx \, dy$$ to $$r \, dr \, d\theta$$, which accounts for the geometry of the polar system.
5. Graphs in polar coordinates can represent many curves more easily than Cartesian coordinates, such as circles, spirals, and other periodic functions.

Review Questions

• How do you convert between polar coordinates and Cartesian coordinates, and why is this conversion important in multidimensional integration?
  • To convert between polar coordinates (r, θ) and Cartesian coordinates (x, y), you use the formulas $$x = r \cos(\theta)$$ and $$y = r \sin(\theta)$$ for conversion to Cartesian. Conversely, to convert back to polar from Cartesian, use $$r = \sqrt{x^2 + y^2}$$ and $$\theta = \tan^{-1}(\frac{y}{x})$$. This conversion is crucial in multidimensional integration because certain problems are easier to solve in one coordinate system over another, especially when dealing with circular or rotational symmetries.
• Explain how the Jacobian is used when changing variables from Cartesian to polar coordinates in double integrals.
  • When changing variables from Cartesian to polar coordinates in double integrals, the Jacobian determinant comes into play. The area element changes from $$dx \, dy$$ to $$r \, dr \, d\theta$$ due to this transformation. The Jacobian helps adjust for the scaling that occurs when moving between coordinate systems, ensuring that the area calculations remain accurate. This adjustment is vital for correctly evaluating integrals over circular domains.
• Analyze how integrating functions with radial symmetry benefits from using polar coordinates instead of Cartesian coordinates.
  • Integrating functions with radial symmetry greatly benefits from using polar coordinates because it simplifies the setup of the integral. In Cartesian coordinates, you may need complex boundaries and multiple integrations across different regions. However, in polar coordinates, you can express circular boundaries naturally with a constant radius and an angle range. This not only reduces complexity but also often leads to more straightforward integrals that are easier to evaluate analytically or numerically.

Related terms

Cartesian coordinates: A coordinate system that specifies each point by an ordered pair of numbers, representing its horizontal and vertical distances from two perpendicular axes.

Jacobian: A matrix of first-order partial derivatives used in transforming coordinate systems, crucial for changing variables in multiple integrals.

radians: A unit of angular measure used in mathematics where one radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.