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Mathematics LibreTexts

10: Polar Coordinates & Parametric Equations

  • Page ID
    511
  • [ "article:topic-guide", "authorname:guichard" ]

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    Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems.

    • 10.1: Polar Coordinates
      Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems.
    • 10.2: Slopes in Polar Coordinates
      When we describe a curve using polar coordinates, it is still a curve in the x−y plane. We would like to be able to compute slopes and areas for these curves using polar coordinates.
    • 10.3: Areas in Polar Coordinates
      We can use the equation of a curve in polar coordinates to compute some areas bounded by such curves. The basic approach is the same as with any application of integration: find an approximation that approaches the true value. For areas in rectangular coordinates, we approximated the region using rectangles; in polar coordinates, we use sectors of circles.
    • 10.4: Parametric Equations
      Suppose \(f(t)\) and \(g(t)\) are functions. Then the equations \(x=f(t)\) and \(y=g(t)\) describe a curve in the plane. But \(t\) in general is simply an arbitrary variable, often called in this case a parameter, and this method of specifying a curve is known as parametric equations. One important interpretation of \(t\) is time. In this interpretation, the equations \(x=f(t)\) and \(y=g(t)\) give the position of an object at time \(t\).
    • 10.5: Calculus with Parametric Equations
      We have already seen how to compute slopes of curves given by parametric equations---it is how we computed slopes in polar coordinates. Areas can be a bit trickier with parametric equations, depending on the curve and the area desired
    • 10.E: Polar Coordinates, Parametric Equations (Exercises)
      These are homework exercises to accompany David Guichard's "General Calculus" Textmap.