     Calculus Animations,Graphics and Lecture Notes Polar Coordinate System    Home About Kelly Liakos Sponsors Calculus 1- Limits and Derivatives Calculus 1 - The Second Derivative Calculus 1 and 3 Formula Sheets Chain Rule Computer Lab Assignments Conic Sections Differential Equations Directional Derivatives/Gradient Double Integrals Equillibrium Solutions 1st order DEs 1st order Diff Eqs -Motion Flux Integrals and Surface Integrals Infinite Sequences Infinie Series Level Curves and Level Surfaces Line Integrals Optimization and Related Rates Optimization for Functions of 2 Variables Parametric Equations 2-space Parametric Equations 3-space Partial Derivatives Polar Coordinate System Polar Coordinates- Derivatives and Integrals PreCalculus Riemann Sums and the Fundamental Theorem of Calculus 2d order Diff EQS-Motion 2d Partial Derivatives Supplemental Exercises and Solutions Tangent Planes/ Differential for f(x,y) Trigonometry Trigonometry Applications Triple Integrals Unit Tangent Vectors/Unit Normal Vectors Vectors in General Vector Valued Functions Visualizing Limits of Functions of 2 Variables Work Links Special Topics Scratch Paper 2 Poems  Lecture Polar Coordinates Basic Graph Animation CirclesThe most basic circle is r = constant  i.e. the set of all points equidistant from the origin.There are s special circles r = acos(Θ) and r = asin(Θ)Below we see the animations but why are these circles? This will be one of the few times we'll compare polar and rectangulat coordinates. x = rcos(Θ) and y = r sin(Θ)x2 + y2 = r2 and tan(Θ) = y/x .Suppose r = cos(Θ) . Multiply each side by r :  r2 = rcos(Θ). We obtain:x2 + y2 =  x  rearranging we obtain  x2 - x  + y2 =  0.  ompleting the square we obtain(x-1/2)2 + y2 = 1/4  which we recognize as the equation of a circle of radius 1/2centered at (1/2,0). Show r = sin(Θ) is a circle centered at (0,1/2) of radius 1/2. r = cos(a) r = sin(a) SpiralsFunctions of the form r = f(Θ) where f(Θ) is not a trig function generate spirals.To plot spirals simply plot values using the usual suspects : 0,π/2,π,3π/2,2π etc.In the following animations r =Θ is a a  spiral  where the bands are equally spaced .With  the exponential spiral r = e .1Θ the spacings increase  and with the logarithmic spiral r =ln(Θ) the spacings decrease.Then we consider two spirals which spiral inward an exponential f =e -.1Θ and a hyperbolic spiral r = 1/(1+Θ) spiral exponential spiral logarithmic spiral inward exponential file inward hyperbolic spiral Lecture Notes Cardioids Animation Cardioid Sines Animation Cardioid Cosines The lecture notes on limacons also conrtain a breif discussion of derivatives in Polar coordinates. For a complete discussion see the page Polar Coordinates - Derivatives and Integrals. Lecture Notes Limacons Animation Limacon1 Lecture Notes sin(nt) cos(nt) Animation cos(3t) Animation sin(2t)   The Smart Bunny-A very short story by Kurt Vonnegut Jr.  